Puzzle Museum



1. BIOGRAPHICAL MATERIAL -- in chronological order

ALCUIN (c735-804)

Phillip Drennon Thomas. Alcuin of York. DSB I, 104-105.

Robert Adamson. Alcuin, or Albinus. DNB, (I, 239-240), 20.

Andrew Fleming West. Alcuin and the Rise of the Christian Schools. (The Great Educators -- III.) Heinemann, 1893. The only book on Alcuin that I found which deals with the Propositiones.

Stephen Allott. Alcuin of York c. A.D. 732 to 804 -- his life and letters. William Sessions, York, 1974.

FIBONACCI [LEONARDO PISANO] (c1170->1240)

See also the entries for Fibonacci in Common References.

Fibonacci. (1202 -- first paragraph); 1228 -- second paragraph, on p. 1. In this paragraph he narrates almost everything we know about him. [In the second ed., he inserted a dedication as the first paragraph.]

The paragraph ends with the notable sentence which I have used as a motto for this work. "Si quid forte minus aut plus iusto vel necessario intermisi, mihi deprecor indulgeatur, cum nemo sit qui vitio careat et in omnibus undique sit circumspectus." (If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. [Grimm's translation.])

Richard E. Grimm. The autobiography of Leonardo Pisano. Fibonacci Quarterly 11:1 (Feb 1973) 99-104. He has collated six MSS of the autobiographical paragraph and presents his critical version of it, with English translation and notes. Sigler, below, gives another translation. I give Grimm's translation, omitting his notes.

After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business, I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.

F. Bonaini. Memoria unica sincrona di Leonardo Fibonacci novamente scoperta. Giornale Storico degli Archivi Toscani 1:4 (Oct-Dec 1857) 239-246. This reports the discovery of a 1241 memorial of the Comune of Pisa, which I reproduce as it is not well known. This grants Leonardo an annual honorarium of 20 pounds. In 1867, a plaque bearing this inscription and an appropriate heading was placed in the atrium of the Archivio di Stato in Pisa.

"Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferunter; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus."

A translation follows, but it can probably be improved. My thanks to Steph Maury Gannon for many improvements over my initial version.

Considering the honour and progress of our city and its citizens that is brought to them through both the knowledge and the diligent application of the discreet and wise Maestro Leonardo Bigallo in the art of calculation for valuations and accounts for the city and its officials and others, as often as necessary; we declare by this present decree that there shall be given to the same Leonardo, from the Comune and on behalf of the Comune, by reason of affection and gratitude, and for his excellence in science, in recompense for the labour which he has done in auditing and consolidating the above mentioned valuations and accounts for the Comune and the public bodies, as his wages or salary, 20 pounds in money each year and his usual fees (the same Pisano shall continue to render his usual services to the Comune and its officials in the art of calculation etc.).

Bonaini also quotes a 1506 reference to Lionardo Fibonacci.

Mario Lazzarini. Leonardo Fibonacci Le sue Opere e la sua Famiglia. Bolletino di Bibliografia e Storia delle Scienze Matematiche 6 (1903) 98-102 & 7 (1904) 1-7. Traces the family to late 11C, saying Leonardo's father was Guglielmo and his grandfather was probably Bonaccio. He estimates the birth date as c1170. He describes a contract of 28 Aug 1226 in which Leonardo Bigollo, his father, Guglielmo, and his brother, Bonaccingo, buy a piece of land from a relative. This land included a tower and other buildings, outside the city, near S. Pietro in Vincoli. [G. Milanesi; Documento inedito intorno a Leonardo Fibonacci; Rome, 1867 -- ??NYS]. Says nothing is known of the 1202 ed of Liber Abbaci. Quotes the above memorial.

R. B. McClenon. Leonardo of Pisa and his liber quadratorum. AMM 26:1 (Jan 1919) 1-8.

Gino Loria. Leonardo Fibonacci. Gli Scienziati Italiana dall'inizio del medio evo ai nostri giorni. Ed. by Aldo Mieli. (Dott. Attilio Nardecchia Editore, Rome, 1921;) Casa Editrice Leonardo da Vinci, Rome, 1923. Vol. 1, pp. 4-12. This reproduces much of the material in Lazzarini and the opening biographical paragraph of Liber Abaci.

Ettore Bortolotti. Article on Fibonacci in: Enciclopedia Italiana. G. Treccani, Rome, 1949 (reprint of 1932 ed.).

Charles King. Leonardo Fibonacci. Fibonacci Quarterly 1:4 (Dec 1963) 15-19.

Gino Arrighi, ed. Leonardo Fibonacci: La Practica di Geometria -- Volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano. Dal Codice 2186 della Biblioteca Riccardiana di Firenze. Domus Galilaeana, Pisa, 1966. The Frontispiece is the mythical portrait of Fibonacci, taken from I Benefattori dell'Umanità, vol. VI; Ducci, Florence, 1850. (Smith, History II 214 says it is a "Modern engraving. The portrait is not based on authentic sources".) P. 15 shows the plaque erected in the Archivio di Stato di Pisa in 1855 which reproduces the above memorial with an appropriate heading, but Arrighi has no discussion of it. P. 19 is a photo of the statue in Pisa and p. 16 describes its commissioning in 1859.

Joseph and Francis Gies. Leonard of Pisa and the New Mathematics of the Middle Ages. Crowell, NY, 1969. This is a book for school students and contains a number of dubious statements and several false statements.

Kurt Vogel. Fibonacci, Leonardo, or Leonardo of Pisa. DSB IV, 604-613.

A. F. Horadam. Eight hundred years young. Australian Mathematics Teacher 31 (1975) 123-134. Good survey of Fibonacci's life & work. Gives English of a few problems. This is available on Kimberling's website - see below.

Ettore Picutti. Leonardo Pisano. Le Scienze 164 (Apr 1982) ??NYS. = Le Scienze, Quaderni; 1984, pp. 30-39. (Le Scienze is a magazine; the Quaderni are collections of articles into books.) Mostly concerned with the Liber Quadratorum, but surveys Fibonacci's life and work. Says he was born around 1170. Includes photo of the plaque in the Archivo di Stato di Pisa.

Leonardo Pisano Fibonacci. Liber quadratorum, 1225. Translated and edited by L. E. Sigler as: The Book of Squares; Academic Press, NY, 1987. Introduction: A brief biography of Leonardo Pisano (Fibonacci) [1170 - post 1240], pp. xv-xx. This is the best recent biography, summarizing Picutti's article. Says he was born in 1170 and his father's name was Guilielmo -- cf Loria above. Gives another translation of the biographical paragraph of the Liber Abbaci.

A. F. Horadam & J. Lahr. Letter to the Editor. Fibonacci Quarterly 28:1 (Feb 1990) 90. The authors volunteer to act as coordinators for work on the life and work of Fibonacci. Addresses: A. F. Horadam, Mathematics etc., Univ. of New England, Armidale, New South Wales, 2351, Australia; J. Lahr, 14 rue des Sept Arpents, L-1139 Luxembourg, Luxembourg.

Thomas Koshy. Fibonacci and Lucas Numbers with Applications. Wiley-Interscience, Wiley, 2001. Claims to be 'the first attempt to compile a definitive history and authoritative analysis' of the Fibonacci numbers, but the history is generally second-hand and marred with a substantial number of errors, The mathematical work is extensive, covering many topics not organised before, and is better done, but there are more errors than one would like.

Laurence E. Sigler. Translation of Liber Abaci as: Fibonacci's Liber Abaci A Translation into Modern English of Leonardo Pisano's Book of Calculation. Springer, 2002.

Clark Kimberling's site web includes biographical material on Fibonacci and other similar number theorists. .

Ron Knott has a huge website on Fibonacci numbers and their applications, with material on many related topics, e.g. continued fractions, π, etc. with some history. ee.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html .

Luca PACIOLI (c1445-1517)

S. A. Jayawardene. Luca Pacioli. BDM 4, 1897-1900.

Bernardino Baldi (Catagallina) (1553-1617). Vita di Pacioli. (1589, first published in his Cronica de Mathematici of 1707.) Reprinted in: Bollettino di bibliografia e di storia delle scienze matematiche e fisiche 12 (1879) 421-427. ??NYS -- cited by Taylor, p. 338.

Enrico Narducci. Intorno a due edizioni della "Summa de arithmetica" di Fra Luca Pacioli. Rome, 1863. ??NYS -- cited by Riccardi [Biblioteca Matematica Italiana, 1952]

D. Ivano Ricci. Luca Pacioli, l'uomo e lo scienziato. San Sepolcro, 1940. ??NYS -- cited in BDM.

R. Emmett Taylor. No Royal Road Luca Pacioli and His Times. Univ. of North Carolina Press, Chapel Hill, 1942. BDM describes this as lively but unreliable.

Ettore Bortolotti. La Storia della Matematica nella Università di Bologna. Nicola Zanichelli Editore, Bologna, 1947. Chap. I, § V, pp. 27-33: Luca Pacioli.

Margaret Daly Davis. Piero della Francesca's Mathematical Treatises The "Trattato d'abaco" and "Libellus de quinque corporibus regularibus". Longo Editore, Ravenna, 1977. This discusses Piero's reuse of his own material and Pacioli's reuse of Piero's material.

Fenella K. C. Rankin. The Arithmetic and Algebra of Luca Pacioli. PhD thesis, Univ. of London, 1992 (copy at the Warburg Institute), ??NYR.

Enrico Giusti, ed. Descriptive booklet accompanying the 1994 facsimile of the Summa -- qv in Common References.

Edward A. Fennell. Figures in Proportion: Art, Science and the Business Renaissance. The contribution of Luca Pacioli to culture and commerce in the High Renaissance. Catalogue for the exhibition, The Institute of Chartered Accountants in England and Wales, London, 1994.

Claude-Gaspar BACHET de Méziriac (1581-1638)

C.-G. Collet & J. Itard. Un mathématicien humaniste -- Claude-Gaspar Bachet de Méziriac (1581-1638). Revue d'Histoire des Sciences et leurs Applications 1 (1947) 26-50.

J. Itard. Avant-propos. IN: Bachet; Problemes; 1959 reprint, pp. v-viii. Based on the previous article.

There is a Frontispiece portrait in the reprint.

Underwood Dudley. The first recreational mathematics book. JRM 3 (1970) 164-169. On Bachet's Problemes.

William Schaaf. Bachet de Méziriac, Claude-Gaspar. DSB I, 367-368.

Jean LEURECHON (c1591-1670) and Henrik VAN ETTEN

A. Deblaye. Étude sur la récréation mathématique du P. Jean Leurechon, Jésuite. Mémoires de la Société Philotechnique de Pont-à-Mousson 1 (1874) 171-183. [MUS #314. Schaaf. Hall, OCB, pp. 86, 88 & 114, says the only known copy of this journal is at Harvard, which has kindly supplied me with a photocopy of this article. Hall indicates the article is in vol. II and says it is 12 pages, but only cites pp. 171 & 174.] This simply assumes Leurechon is the author and gives a summary of his life. The essential content is described by Hall.

G. Eneström. Girard Desargues und D.A.L.G. Biblioteca Mathematica (3) 14 (1914) 253-258. D.A.L.G. was an annotator of van Etten's book in c1630. Although D.A.L.G. was used by Mydorge on one of his other books, it had been conjectured that this stood for Des Argues Lyonnais Girard (or Géomètre). Eneström can find no real evidence for this and feels that Mydorge is the most likely person.

Trevor H. Hall. Mathematicall Recreations. An Exercise in Seventeenth Century Bibliography. Leeds Studies in Bibliography and Textual Criticism, No. 1. The Bibliography Room, School of English, University of Leeds, 1969, 38pp. Pp. 18-38 discuss the question of authorship and Hall feels that van Etten probably was the author and that there is very little evidence for Leurechon being the author. Much of the mathematical content is in Bachet's Problemes and may have been copied from it or some common source. [This booklet is reproduced as pp. 83-119 of Hall, OCB, with the title page of the 1633 first English edition reproduced as plate 5, opp. p. 112. Some changes have been made in the form of references since OCB is a big book, but the only other substantial change is that he spells the name of the dedicatee of the book as Verreyken rather than Verreycken.]

William Schaaf. Leurechon, Jean. DSB VIII, 271-272.

Jacques Voignier. Who was the author of "Recreation Mathematique" (1624)? The Perennial Mystics #9 (1991) 5-48 (& 1-2 which are the cover and its reverse). [This journal is edited and published by James Hagy, 2373 Arbeleda Lane, Northbrook, Illinois, 60062, USA.] Presents some indirect evidence for Leurechon's authorship.

Jacques OZANAM (1640-1717)

On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait labelled Ozanam -- the only one I know of. This copy is at the Institut für Geschichte der Naturwissenschaft in Munich. Hofmann published the picture -- see below.

Charles Hutton. A Mathematical and Philosophical Dictionary. 1795-1796. Vol. II, pp. 184-185. ??NYS [Hall, OCB, p. 166.]

Charles Hutton. On the life and writings of Ozanam, the first author of these Mathematical Recreations. Ozanam-Hutton. Vol. I. 1803: xiii-xv; 1814: ix-xi.

William L. Schaaf. Jacques Ozanam on mathematics .... MTr 50 (1957) 385-389. Mostly based on Hutton. Includes a sketchy bibliography of Ozanam's works, generally ignoring the Recreations.

Joseph Ehrenfried Hofmann. Leibniz und Ozanams Problem, drei Zahlen so zu bestimmen, dass ihre Summe eine Quadratzahl und ihre Quadratsumme eine Biquadratzahl ergibt. Studia Leibnitiana 1:2 (1969) 103-126. Outlines Ozanam's life, gives a bibliography of his works and reproduces the above-mentioned drawing as a plate opp. p. 124. (My thanks to Menso Folkerts for this information and a copy of Hofmann's article.)

William L. Schaaf. Ozanam, Jacques. DSB X, 263-265.

Jean Étienne MONTUCLA (1725-1799)

Charles Hutton. Some account of the life and writings of Montucla. Ozanam-Hutton. Vol. I. 1803: viii-xii; 1814: iv-viii.

Charles Hutton. A Philosophical and Mathematical Dictionary. 2nd ed. of the Dictionary cited under Ozanam, 1815, Vol. II, pp. 63-64. ??NYS. According to Hall, OCB, p. 167, this is not in the 1795-1796 ed. and is a reworking of the previous item.

Lewis CARROLL (1832-1898)

Pseudonym of Charles Lutwidge Dodgson. There is so much written on Carroll that I will only give references to his specifically recreational work and some basic references.

The Diaries of Lewis Carroll. Edited by Roger Lancelyn Green. (OUP, 1954); 2 vols, Greenwood Publishers, Westport, Connecticut, 1971, HB.

Lewis Carroll's Diaries The private journals of Charles Lutwidge Dodgson (Lewis Carroll) The first complete version of the nine surviving volumes with notes and annotations by Edward Wakeling. Introduction by Roger Lancelyn-Green. The Lewis Carroll Society, Publications Unit, Luton, Bedfordshire. [There were 13 journals, but 4 are lost.]

Vol. 1. Journal 2, Jan-Sep 1855. 1993, 158pp.

Vol. 2. Journal 4, Jan-Dec 1856. 1994, 158pp.

Vol. 3. Journal 5, Jan 1857 - Apr 1858. 1995, 199pp.

Vol. 4. Journal 8, May 1862 - Sep 1864 and a reconstruction of the four missing

years, 1858-1862. 1997, 399pp.

Vol. 5. Journal 9, Sep 1864 - Jan 1868, including the Russian Journal.

1999, 416pp.

Vol. 6. Journal 10, Apr 1868 - Dec 1876. 2001, 552pp.

Vol. 7. Journal 11, Jan 1877 - Jun 1883. 2003, 606pp.

The Letters of Lewis Carroll. Edited by Morton N. Cohen with the assistance of Roger Lancelyn Green. Volume One ca.1837 - 1885; Volume Two 1886 - 1898. Macmillan London, 1979.

Stuart Dodgson Collingwood. The Life and Letters of Lewis Carroll. T. Fisher Unwin, London, 1898.

Stuart Dodgson Collingwood, ed. The Lewis Carroll Picture Book. T. Fisher Unwin, London, 1899. = Diversions and Digressions of Lewis Carroll, Dover, 1961. = The Unknown Lewis Carroll, Dover, 1961(?). Reprint, in reduced format, Collins, c1910. The pagination of the main text is the same in the 1899 and in both Dover reprints, but is quite different than the Collins. Cited as: Carroll-Collingwood, qv in Common References.

R. B. Braithwaite. Lewis Carroll as logician. MG 16 (No. 219) (Jul 1932) 174-178. He notes that Carroll assumed that a universal statement implied the existence of an object satisfying the antecedent, e.g. 'all unicorns are blue' would imply the existence of unicorns, contrary to modern convention.

Derek Hudson. Lewis Carroll -- An Illustrated Biography. Constable, 1954; new illustrated ed., 1976.

Warren Weaver. Lewis Carroll: Mathematician. SA 194:4 (Apr 1956) 116-128. + Letters and response. SA 194:6 (Jun 1956) 19-22.

Martin Gardner. The Annotated Alice. C. N. Potter, NY, 1960. Penguin, 1965; 2nd ed., 1971. Revised as: More Annotated Alice, 1990, qv.

Martin Gardner. The Annotated Snark. Bramhall House, 1962. Penguin, 1967; revised, 1973 & 1974.

John Fisher. The Magic of Lewis Carroll. Nelson, 1973. Penguin, 1975.

Morton N. Cohen, ed. The Selected Letters of Lewis Carroll. Papermac (Macmillan), 1982.

Martin Gardner. More Annotated Alice. [Extension of The Annotated Alice.] Random House, 1990.

Edward Wakeling. Lewis Carroll's Games and Puzzles. Dover and the Lewis Carroll Birthplace Trust, 1992. Cited as Carroll-Wakeling, qv in Common References.

Francine F. Abeles, ed. The Pamphlets of Lewis Carroll -- Vol. 2: The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces. Lewis Carroll Society of North America, distributed by University Press of Virginia, Charlottesville, 1994.

Edward Wakeling. Rediscovered Lewis Carroll Puzzles. Dover, 1995. Cited as Carroll-Wakeling II, qv in Common References.

Martin Gardner. The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations, Games, Puzzles and Word Plays. Copernicus (Springer, NY), 1996. Cited as Carroll-Gardner, qv in Common References.

Martin Gardner. The Annotated Alice: The Definitive Edition. 1999. [A combined version of The Annotated Alice and More Annotated Alice.]

Professor Louis HOFFMANN (1839-1919)

Pseudonym of Angelo John Lewis.

Joseph Foster. Men-at-the-Bar: A biographical Hand-List of the Members of the Various Inns of Court, including Her Majesty's Judges, etc. 2nd ed, the author, 1885. P. 277 is the entry for Lewis. Born in London, eldest son of John Lewis. Graduated from Wadham College, Oxford. Entered Lincoln's Inn as a student in 1858, called to the bar there in 1861. Married Mary Ann Avery in 1864. Author of Manual of Indian Penal Code and Manual of Indian Civil Procedure. Address: 12 Crescent Place, Mornington Crescent, London, NW. (My thanks to the Library of Lincoln's Inn for this information.)

Anonymous. Professor Hoffmann. Mahatma 4:1 (Jul 1900) 377-378. A brief note, with photograph, stating that he is Mr. Angelo Lewis, M.A. and Barrister-at-Law.

Will Goldston. Will Goldston's Who's Who in Magic. My version is included in a compendium called: Tricks that Mystify; Will Goldston, London, nd [1934-NUC]. Pp. 106-107. Says he was a barrister, retired to Hastings about 1903 and died in 1917.

Who Was Who, 1916-1928, p. 627. This says he attended North London Collegiate School and that he only practised law until 1876. He was on the staff of the Saturday Review and a contributor to many journals. Won the £100 prize offered by Youth's Companion (Boston) for best short story for boys. Lists 36 books by him and 9 card games he invented. Address: Manningford, Upper Bolebrooke Road, Bexhill-on-Sea. (My thanks to the Library of Lincoln's Inn for this information.)

J. B. Findlay & Thomas A. Sawyer. Professor Hoffmann: A Study. Published by Thomas A. Sawyer, Tustin, California, 1977. A short book, 12 + 67 pp, with two portraits (one from Mahatma) and 27pp of bibliography. He was born at 3 Crescent Place, Mornington Crescent, London. He was a barrister and wrote two books on Indian law.

Charles Reynolds. Introduction -- to the reprint of Hoffmann's Modern Magic, Dover, 1978, pp. v-xiv. This says Lewis was a barrister, which is mentioned in another reprint of a Hoffmann book and in S. H. Sharpe's translation of Ponsin on Conjuring.

Edward Hordern. Foreword to this edition. In: Hoffmann's Puzzles Old and New (see under Common References), 1988 reprint, pp. v-vi. This says he was the Reverend Lewis, but this is corrected in Hoffmann-Hordern to saying he was a barrister.

Hoffmann-Hordern, p. viii, is a version of the photograph in Mahatma.

Hall, OCB, p. 189, gives Hoffmann's address as Ireton Lodge, Cromwell Ave., N. -- presumably the Cromwell Ave. in Highgate.

Toole Stott 386 gives a little information about Hoffmann and Modern Magic, including an address in Mornington Crescent in 1877.

No DNB or DSB entry -- I have suggested a DNB entry.

Sam LOYD (1841-1911) and Sam LOYD JR. (1873-1934)

[W. R. Henry.] Samuel Loyd. [Biography.] Dubuque Chess Journal, No. 66 (Aug-Sep 1875) 361-365. ??NX -- o/o (11 Jul 91).

Loyd. US Design 4793 -- Design for Puzzle-Blocks. 11 April 1871. These are solid pieces, but unfortunately the drawing did not come with this, so I am not clear what they are. ??Need drawing -- o/o (11 Jul 91).

Anonymous & Sam Loyd. Loyd's puzzles (Introductory column). Brooklyn Daily Eagle (22 Mar 1896) 23. Says he lives at 153 Halsey St., Brooklyn.

L. D. Broughton Jr. Samuel Loyd. [A Biography.] Lasker's Chess Magazine 1:2 (Dec 1904) 83-85. About his chess problems with a mention of some of his puzzles.

G. G. Bain. The prince of puzzle-makers. An interview with Sam Loyd. Strand Magazine 34 (No. 204) (Dec 1907) 771-777. Solutions of Sam Loyd's puzzles. Ibid. 35 (No. 205) (Jan 1908) 110.

Walter Prichard Eaton. My fifty years in puzzleland -- Sam Loyd and his ten thousand brain-teasers. The Delineator (New York) (April 1911) 274 & 328. Drawn portrait of Loyd, age 69.

Anon. Puzzle inventor dead. New-York Daily Tribune (12 Apr 1911) 7. Says he died at his house, 153 Halsey St. "He declared no one had ever succeeded in solving [the "Disappearing Chinaman"]." Says he is survived by a son and two daughters (!! -- has anyone ever tracked the daughters and their descendents??).

Anon. Sam Loyd, puzzle man, dies. New York Times (12 Apr 1911) 13. Says he was for some time editor of The Sanitary Engineer and a shrewd operator on Wall Street.

Anon. Sam Loyd. SA (22 Apr 1911) 40-41?? Says he was for some years chess editor of SA and was puzzle editor of Woman's Home Companion when he died.

W. P. Eaton. Sam Loyd. The American Magazine 72 (May 1911) 50, 51, 53. Abridged version of Eaton's earlier article. Photo of Loyd on p. 50.

P. J. Doyle. Letter to the Chess column. The Sunday Call [Newark, NJ] (21 May 1911), section III, p. 10.

A. C. White. Sam Loyd and His Chess Problems. Whitehead and Miller, Leeds, UK, 1913; corrected, Dover, 1962.

Alain C. White. Supplement to Sam Loyd and His Chess Problems. Good Companion Chess Problem Club, Philadelphia, vol. I, nos. 11-12 (Aug 1914), 12pp. This is mostly corrections of the chess problems, but adds a few family details with a picture of the Loyd Homestead and Grist Mill in Moylan, Pennsylvania.

Alain C. White. Reminiscences of Sam Loyd's family. The Problem [Pittsburgh] (28 Mar 1914) 2, 3, 6, 7.

Louis C. Karpinski. Loyd, Samuel. Dictionary of American Biography, Scribner's, NY, vol. XI, 1933, pp. 479-480.

Loyd Jr. SLAHP. 1928. Preface gives some details of his life, making little mention of his father, "who was a famous mathematician and chess player". He claims to have created over 10,000 puzzles. There are some vague biographical details on pp. 1-22, e.g. 'Father conducted a printing establishment.' 'My "Missing Chinaman Puzzle"'. (It may have been some such assertion that led me to estimate his birthdate as 1865, but I now see it is well known to be 1873.)

Anonymous. Sam Loyd dead; puzzle creator. New York Times (25 Feb 1934). Obituary of Sam Loyd Jr. Says he resided at 153 Halsey St., Brooklyn -- the same address as his father -- see the Brooklyn Daily Eagle article of 1896, above. He worked from a studio at 246 Fulton St., Brooklyn. It says Jr. invented 'How Old is Ann?'.

Clark Kinnaird. Encyclopedia of Puzzles and Pastimes. Grosset & Dunlap, NY, 1946. Pp. 263-267: Sam Loyd. Asserts that Loyd Jr. invented 'How Old is Ann?'

Gardner. Sam Loyd: America's greatest puzzlist. SA (Aug 1957) c= First Book, Chap. 9.

Gardner. Advertising premiums. SA (Nov 1971) c= Wheels, chap. 12.

Will Shortz is working on a biography.

No DSB entry.

François Anatole Édouard LUCAS (1842-1891)

Jeux Scientifiques de Ed. Lucas. Advertisement by Chambon & Baye (14 rue Etienne-Marcel, Paris) for the 1re Serie of six games. Cosmos. Revue des Sciences et Leurs Applications 39 (NS No. 254) (7 Dec 1889) no page number on my photocopy.

B. Bailly [name not given, but supplied by Hinz]. Article on Lucas's puzzles. Cosmos. Revue des Sciences et Leurs Applications. NS, 39 (No. 259) (11 Jan 1890) 156-159. NEED 156-157.

Nécrologie: Édouard Lucas. La Nature 19 (1891) II, 302.

Obituary notice: "La Nature announces the death of Prof. Edouard Lucas ...." Nature 44 (15 Oct 1891) 574-575.

Duncan Harkin. On the mathematical work of François-Édouard-Anatole Lucas. L'Enseignement Math. (2) 3 (1957) 276-288. Pp. 282-288 is a bibliography of 184 items. I have found many Lucas publication not listed here and have started a new Bibliography -- see below.

P. J. Campbell. Lucas' solution to the non-attacking rooks problem. JRM 9 (1976/77) 195-200. Gives life of Lucas.

A photo of Lucas is available from Bibliothèque Nationale, Service Photographique, 58 rue Richelieu, F-75084 Paris Cedex 02, France. Quote Cote du Document Ln27 . 43345 and Cote du Cliche 83 A 51772. (??*) I have obtained a copy, about 55 x 85 mm, with the photo in an oval surround. It looks like a carte-de-visite, but has Édouard LUCAS (1842-1891). -- Phot. Zagel. underneath. (Thanks to H. W. Lenstra for the information.)

Norman T. Gridgeman. Lucas, François-Édouard-Anatole. DSB VIII, 531-532.

Susanna S. Epp. Discrete Mathematics with Applications. Wadsworth, Belmont, Calif., 1990, p. 477 gives a small photo of Lucas which looks nothing like the photo from the BN. I have since received a note from Epp via Paul Campbell that a wrong photo was used in the first edition, but this was corrected in later editions.

Alain Zalmanski. Edouard Lucas Quand l'arithmétique devient amusante. Jouer Jeux Mathématiques 3 (Jul/Sep 1991) 5. Brief notice of his life and work.

Andreas M. Hinz. Pascal's triangle and the Tower of Hanoi. AMM 99 (1992) 538-544. Sketches Lucas' life and work, giving details that are not in the above items.

David Singmaster. The publications of Édouard Lucas. Draft version, 14pp, 1998. I discovered many items in Dickson's History of the Theory of Numbers and elsewhere which are not given by Harkin (cf above). This has 248 items, though many of these are multiple items so the actual count is perhaps 275. However, Dickson does not give article titles, and may not give the pages of the entire article, so the same article may be cited more than once, at different pages. I hope to fill in the missing information at some time.

Hermann Cäsar Hannibal SCHUBERT (1848-1911)

Acta Mathematica 1882-1912. Table Générale des Tomes 1-35. 1913. P. 169. Portrait of Schubert.

Werner Burau. Schubert, Hermann Cäsar Hannibal. DSB XII, 227-229.

Walter William Rouse BALL (1850-1925)

Anon. Obituary: Mr. Rouse Ball. The Times (6 Apr 1925) 16.

Anon. Funeral notice: Mr. W. W. R. Ball. The Times (9 Apr 1925) 13.

(Lord) Phillimore. Letter: Mr. Rouse Ball. The Times (9 Apr 1925) 15.

"An old pupil". The late Mr. Rouse Ball. The Times (13 Apr 1925) 12.

J. J. Thomson. W. W. Rouse Ball. The Cambridge Review (24 Apr 1925) 341-342.

Anon. Obituary of W. W. Rouse Ball. Nature 115 (23 May 1925) 808-809.

Anon. The late Mr. W. W. Rouse Ball. The Trinity Magazine (Jun 1925) 53-54.

Anon. Entry in Who's Who, 1925, p. 127.

Anon. Wills and bequests: Mr. Walter William Rouse Ball. The Times (7 Sep 1925) 15.

E. T. Whittaker. Obituary. W. W. Rouse Ball. Math. Gaz. 12 (No. 178) (Oct 1925) 449-454, with photo opp. p. 449.

F. Cajori. Walter William Rouse Ball. Isis 8 (1926) 321-324. Photo on plate 15, opp. p. 321. Copy of Ball's 1924 Xmas card on p. 324.

J. A. Venn. Alumni Cantabrigienses. Part II: From 1752 to 1900. Vol. I, p. 136. CUP, 1940.

David Singmaster. Walter William Rouse Ball (1850-1925). 6pp handout for 1st UK Meeting on the History of Recreational Mathematics, 24 Oct 1992. Plus extended biographical (6pp) and bibliographical (8pp) notes which repeat some of the material in the handout.

No DNB or DSB entry -- however I have offered to write a DNB entry. I have since seen the proposed list of names for the next edition and Ball is already on it.

Henry Ernest DUDENEY (1857-1930)

Anon. & Dudeney. A chat with the puzzle king. The Captain 2 (Dec? 1899) 314-320, with photo. Partly an interview. Includes photos of Littlewick Meadow.

Anon. Solutions to "Sphinx's puzzles". The Captain 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89.

Anon. Master of the breakfast table problem. Daily Mail (1 Feb 1905) 7. An interview with Dudeney in which he gives the better version of his spider and fly problem.

Fenn Sherie. The Puzzle King: An Interview with Henry E. Dudeney. Strand Magazine 71 (Apr 1926) 398-4O4.

Alice Dudeney. Preface to PCP, dated Dec 1931, pp. vii-x. The date of his death is erroneously given as 1931.

Gardner. Henry Ernest Dudeney: England's greatest puzzlist. SA (Jun 1958) c= Second Book, chap. 3.

Angela Newing. The Life and Work of H. E. Dudeney. MS 21 (1988/89) 37-44.

Angela Newing is working on a biography.

No DNB or DSB entry. I have suggested a DNB entry.

Wilhelm Ernst Martin Georg AHRENS (1872-1927)

Wilhelm Lorey. Wilhelm Ahrens zum Gedächtnis. Archiv für Geschichte der Mathematik, der Naturwissenschaften und der Technik 10 (1927/28) 328-333. Photo on p. 328.

O. Staude. Dem Andenken an Dr. Wilhelm Ahrens. Jahresbericht DMV 37 (1928) 286-287.

No DSB entry.

Yakov Isidorovich PERELMAN [Я. И. Перелман] (1882-1942)

Perelman. FMP. 1984. P. 2 (opp. TP) is a sketch of his life and the history of the book. There is a small drawing of Perelman at the top of the page.

Patricio Barros. Website -- Yakov I. Perelman [in Spanish]: yakov_perelman/index.html. This includes a four page biography, in collaboration with Antonio Bravo, and two photos.

Hubert PHILLIPS (1891-1964)

Hubert Phillips. Journey to Nowhere. A Discursive Autobiography. Macgibbon & Kee, London, 1960. ??NYR

No DNB entry -- I have suggested one.

2. GENERAL PUZZLE COLLECTIONS AND SURVEYS

H. E. Dudeney. Great puzzle crazes. London Magazine 13?? (Nov 1904) 478-482. Fifteen Puzzle. Pigs in Clover, Answers, Pick-me-up (spiral ramp) and other dexterity puzzles. Get Off the Earth. Conjurer's Medal (ring maze). Chinese Rings. Chinese Cross (six piece burr). Puzzle rings. Solitaire. The Mathematician's Puzzle (square, circle, triangle). Imperial Scale. Heart and Balls.

H. E. Dudeney. Puzzles from games. Strand Magazine 35 (No. 207) (Mar 1908) 339-344. Solutions. Ibid. 35 (No. 208) (Apr 1908) 455-458.

H. E. Dudeney. Some much-discussed puzzles. Strand Magazine 35 (No. 209) (May 1908) 580-584. Solutions. Ibid. 35 (No. 210) (Jun 1908) 696.

H. E. Dudeney. The world's best puzzles. Strand Magazine 36 (No. 216) (Dec 1908) 779-787. Solutions. Ibid. 37 (No. 217) (Jan 1909) 113-116.

H. E. Dudeney. The psychology of puzzle crazes. The Nineteenth Century 100:6 (Dec 1926) 868-879. Repeats much of his 1904 article.

Sam Loyd Jr. Are you good at solving puzzles? The American Magazine (Sep 1931) 61-63, 133-137.

Orville A. Sullivan. Problems involving unusual situations. SM 9 (1943) 114-118 & 13 (1947) 102-104.

3. GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL

I have tried to divide this material into historical and bibliographical parts, but the two overlap considerably.

3.A. GENERAL HISTORICAL MATERIAL

Raffaella Franci. Giochi matematici in trattati d'abaco del medioevo e del rinascimento. Atti del Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981. Tipografia Senese for GIOCREA (Società Italiana Giochi Creativi), 1981. Pp. 18-43. Describes and quotes many typical problems. 17 references, several previously unknown to me.

Heinrich Hermelink. Arabische Unterhaltungsmathematik als Spiegel Jahrtausendealter Kulturbeziehungen zwischen Ost und West. Janus 65 (1978) 105-117, with English summary. An English translation appeared as: Arabic recreational mathematics as a mirror of age-old cultural relations between Eastern and Western civilizations; in: Ahmad Y. Al-Hassan, Ghada Karmi & Nizar Namnum, eds.; Proceedings of the First International Symposium for the History of Arabic Science, April 1976 -- Vol. Two: Papers in European Languages; Institute for the History of Arabic Science, Aleppo, 1978, pp. 44-52. (There are a few translation and typographical errors, which make it clear that the English version is a translation of the German.)

D. E. Smith. On the origin of certain typical problems. AMM 24 (1917) 64-71. (This is mostly contained in his History, vol. II, pp. 536-548.)

3.B. BIBLIOGRAPHICAL MATERIAL

Many of the items cited in the Common References have extensive bibliographies. In particular: BLC; BMC; BNC; DNB; DSB; Halwas; NUC; Schaaf; Smith & De Morgan: Rara; Suter are basic bibliographical sources. Datta & Singh; Dickson; Heath: HGM; Murray; Sanford: H&S & Short History; Smith: History & Source Book; Struik; Tropfke are histories with extensive bibliographical references. AR; BR are editions of early texts with substantial bibliographical material. Ahrens: MUS; Ball: MRE; Berlekamp, Conway & Guy: Winning Ways; Gardner; Lucas: RM are recreational books with some useful bibliographical material. Of these, the material in Ahrens is by far the most useful. The magic bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Price (see HPL), Toole Stott and Volkmann & Tummers have considerable overlap with the present material, particularly for older books, though Hall, Heyl and Toole Stott restrict themselves to English material, while Volkmann & Tummers only considers German. Santi is also very useful. Below I give some additional bibliographical material which may be useful, arranged in author order.

Anonymous. Mathematical bibliography. SSM 48 (1948) 757-760. Covers recreations.

Wilhelm Ahrens. Mathematische Spiele. Section I G 1 of Encyklopadie der Math. Wiss., Vol. I, part 2, Teubner, Leipzig, 1900-1904, pp. 1080-1093.

Raymond Clare Archibald. Notes on some minor English mathematical serials. MG 14 (1928-29) 379-400.

Elliott M. Avedon & Brian Sutton-Smith. The Study of Games. (Wiley, NY, 1971); Krieger, Huntington, NY, 1979.

Anthony S. M. Dickins. A Catalogue of Fairy Chess Books and Opuscules Donated to Cambridge University Library, 1972-1973, by Anthony Dickins M.A. Third ed., Q Press, Kew Gardens, UK, 1983.

Underwood Dudley. An annotated list of recreational mathematics books. JRM 2:1 (Jan 1969) 13-20. 61 titles, in English and in print at the time.

Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related Material. There have been several versions with slightly varying titles. The most recent printed version is: 400 items, 28 pp., including 4 pp of text, Sep 1990. Technical Report CS90-23, Weizmann Institute of Science, Rehovot, Israel. = Proc. Symp. Appl. Math. 43 (1991) 191-226. Fraenkel has since produced Update 1 to this which lists 430 items on 31pp, Aug 1992; and Update 2, 480 items on 33pp, with 5 pp of text, accidentally dated Aug 1992 at the top but produced in Feb 1994. On 22 Nov 1994, it became a dynamic survey on the Electronic J. Combinatorics and can be accessed from:

.

It can also be accessed via anonymous ftp from ftp.wisdom.weizmann.ac.il. After logging in, do cd pub/fraenkel and then get one of the following three compressed files: games.tex.z; games.dvi.z; games.ps.z.

Martin P. Gaffney & Lynn Arthur Steen. Annotated Bibliography of Expository Writing in the Mathematical Sciences. MAA, 1976.

JoAnne S. Growney. Mathematics and the arts -- A bibliography. Humanistic Mathematics Network Journal 8 (1993) 22-36. General references. Aesthetic standards for mathematics and other arts. Biographies/autobiographies of mathematicians. Mathematics and display of information (including mapmaking). Mathematics and humor. Mathematics and literature (fiction and fantasy). Mathematics and music. Mathematics and poetry. Mathematics and the visual arts.

JoAnne S. Growney. Mathematics in Literature and Poetry. Humanistic Mathematics Network Journal 10 (Aug 1994) 25-30. Short survey. 3 pages of annotated references to 29 authors, some of several books.

R. C. Gupta. A bibliography of selected book [sic] on history of mathematics. The Mathematics Education 23 (1989) 21-29.

Trevor H. Hall. Mathematicall Recreations. Op. cit. in 1. This is primarily concerned with the history of the book by van Etten. [This booklet is revised as pp. 83-119 of Hall, OCB -- see Section 1.]

Catherine Perry Hargrave. A History of Playing Cards and a Bibliography of Cards and Gaming. (Houghton Mifflin, Boston, 1930); Dover, 1966.

Susan Hill. Catalogue of the Turner Collection of the History of Mathematics Held in the Library of the University of Keele. University Library, Keele, 1982. (Sadly this collection was secretly sold by Keele University in 1998 and has now been dispersed.)

Honeyman Collection -- see: Sotheby's.

Horblit Collection -- see: Sotheby's and H. P. Kraus.

Else Høyrup. Books about Mathematics. Roskilde Univ. Center, PO Box 260, DK-4000, Roskilde, Denmark, 1979.

D. O. Koehler. Mathematics and literature. MM 55 (1982) 81-95. 64 references. See Utz for some further material.

H. P. Kraus (16 East 46th Street, New York, 10017). The History of Science including Navigation.

Catalogue 168. A First Selection of Books from the Library of Harrison D. Horblit. Nd [c1976].

Catalogue 169. A Further Selection of Books, 1641-1700 (Wing Period) from the Library of Harrison D. Horblit. Nd [c1976].

Catalogue 171. Another Selection of Books from the Library of Harrison D. Horblit. Nd [c1976].

These are the continuations of the catalogues issued by Sotheby's, qv.

John S. Lew. Mathematical references in literature. Humanistic Mathematics Network Journal 7 (1992) 26-47.

Antonius van der Linde. Das erst Jartausend [sic] der Schachlitteratur -- (850-1880). (1880); Facsimile reprint by Caissa Limited Editions, Yorklyn, Delaware, 1979, HB.

Andy Liu. Appendix III: A selected bibliography on popular mathematics. Delta-k 27:3 (Apr 1989) -- Special issue: Mathematics for Gifted Students, 55-83.

Édouard Lucas. Récréations mathématiques, vol 1 (i.e. RM1), pp. 237-248 is an Index Bibliographique.

Felix Müller. Führer durch die mathematische Literature mit besonderer Berücksichtigung der historisch wichtigen Schriften. Abhandlungen zur Geschichte der Mathematik 27 (1903).

Charles W. Newhall. "Recreations" in secondary mathematics. SSM 15 (1915) 277-293.

Mathematical Association. 259 London Road, Leicester, LE2 3BE.

Catalogue of Books and Pamphlets in the Library. No details, [c1912], 19pp, bound in at end of Mathematical Gazette, vol. 6 (1911-1912).

A First List of Books & Pamphlets in the Library of the Mathematical Association -- Books and Pamphlets acquired before 1924. Bell, London, 1926.

A Second List of Books & Pamphlets in the Library of the Mathematical Association -- Books and Pamphlets acquired during 1924 and 1925. Bell, London, 1929.

A Third List of Books & Pamphlets in the Library of the Mathematical Association -- Books and Pamphlets added from 1926 to 1929. Bell, London, 1930.

A Fourth List of Books & Pamphlets in the Library of the Mathematical Association -- Books and Pamphlets added from 1930 to 1935. Bell, London, 1936.

Lists 1-4 edited by E. H. Neville.

Books and Periodicals in the Library of the Mathematical Association. Ed. by R. L. Goodstein. MA, 1962. Includes the four previous lists and additions through 1961.

SEE ALSO: Riley; Rollett; F. R. Watson.

Stanley Rabinowitz. Index to Mathematical Problems 1980-1984. MathPro Press, Westford, Massachusetts, 1992.

Cecil B. Read & James K. Bidwell.

Selected articles dealing with the history of elementary Mathematics. SSM 76 (1976) 477-483.

Periodical articles dealing with the history of advanced mathematics -- Parts I & II. SSM 76 (1976) 581-598 & 687-703.

Rudolf H. Rheinhardt. Bibliography on Whist and Playing Cards. From: Whist Scores and Card-table Talk, Chicago, 1887. Reprinted by L. & P. Parris, Llandrindod Wells, nd [1980s].

Pietro Riccardi. Biblioteca Matematica Italiana dalla Origine della Stampa ai Primi Anni del Secolo XIX. G. G. Görlich, Milan, 1952, 2 vols. This work appeared in several parts and supplements in the late 19C and early 20C, mostly published by the Società Tipografica Modense, Modena, 1878-1893. Because it appeared in parts, the contents of early copies are variable and even the reprints may vary. The contents of this set are as follows.

I. 20pp prelims + Col. 1 - 656 (Abaco - Kirchoffer). [= original Vol. I.]

Col. 1 - 676 (La Cometa - Zuzzeri) + 2pp correzioni. [= original Vol. II.]

II. 4pp titles and reverses. Correzioni ed Aggiunte. [= original Appendice.]

Serie I.a Col. 1 - 78 + 1½pp Continuazione delle Correzioni (note that these

have Pag. when they mean Col.).

Serie II.a. Col. 81 - 156.

Serie III.a. Col. 157 - 192 + Aggiunte al Catalogo delle Opere di sovente citate,

col. 193-194 + 1p Continuazione delle Correzioni (note that these have

Pag. when they mean Col.).

Serie IV.a. Col. 197 - 208 + Seconda Aggiunta al Catalogo delle Opere più di

sovente citate, col. 209 - 212 + Continuazione delle Correzioni in

col. 211-212.

Serie V.a. Col. 1 - 180.

Serie VI.a. Col. 179 - 200.

Serie V & VI must have been published as one volume as Serie V ends

halfway down a page and then Serie VI begins on the same page.

Serie VII.a. 2pp introductory note by Ettore Bortolotti in 1928 saying that this

material was left as a manuscript by Riccardi and never previously

published + Col. 1 - 106.

Indice Alfabetico, of authors, covering the original material and all seven Series

of Correzioni ed Aggiunte, in 34 unnumbered columns.

Parte Seconda. Classificazione per materie delle opere nella Parte I. 18pp

(including a chronological table) + subject index, pp. 1 - 294.

Catalogo Delle opere più di sovente citate, col. 1 - 54.

[I have seen an early version which had the following parts: Vol. I, 1893, col. 1-656; Vol. II, 1873, col. 1-676; Appendice, 1878-1880-1893, col. 1-228. Appendice, nd, col. 1-212. Serie V, col. 1-228. Parte 2, Vol. 1, 1880, pp. 1-294. Renner Katalog 87 describes it as 5 in 2 vols.]

A. W. Riley. School Library Mathematics List -- Supplement No. 1. MA, 1973.

SEE ALSO: Rollett.

Tom Rodgers. Catalog of his collection of books on recreational mathematics, etc. The author, Atlanta, May 1991, 40pp.

Leo F. Rogers. Finding Out in the History of Mathematics. Produced by the author, London, c1985, 52pp.

A. P. Rollett. School Library Mathematics List. Bell, London, for MA, 1966.

SEE ALSO: Riley.

Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3-5.

José A. Sánchez Pérez. Las Matematicas en la Biblioteca del Escorial. Imprenta de Estanislao Maestre, Madrid, 1929.

William L. Schaaf.

List of works on recreational mathematics. SM 10 (1944) 193-200.

PLUS: A. Gloden; Additions to Schaaf's "List of works on mathematical recreations"; SM 13 (1947) 127.

A Bibliography of Recreational Mathematics. Op. cit. in Common References, 4 vols., 1955-1978. In these volumes he gives several lists of relevant books.

Books for the periods 1900-1925 and 1925-c1956 are given as Sections 1.1 (pp. 2-3) and 1.2 (pp. 4-12) in Vol. 1.

Chapter 9, pp. 144-148, of Vol. 1, is a Supplement, generally covering c1954-c1962, but with some older items.

In Vol. 2, 1970, the Appendix, pp. 181-191, extends to c1969, including some older items and repeating a few from the Supplement of Vol. 1.

Appendix A of Vol. 3, 1973, pp. 111-113, adds some more items up through 1972.

Appendix A, pp. 134-137, of Vol. 4, 1978, extends up through 1977.

The following VESTPOCKET BIBLIOGRAPHIES are extensions of the material

in his Bibliographies.

No. 1: Pythagoras and rational triangles; Geoboards and lattices. JRM 16:2 (1983-84) 81-88.

No. 2: Combinatorics; Gambling and sports. JRM 16:3 (1983-84) 170-181.

No. 3: Tessellations and polyominoes; Art and music. JRM 16:4 (1983-84) 268-280.

No. 4: Recreational miscellany. JRM 17:1 (1984-85) 22-31.

No. 5: Polyhedra; Topology; Map coloring. JRM 17:2 (1984-85) 95-105.

No. 6: Sundry algebraic notes. JRM 17:3 (1984-85) 195-203.

No. 7: Sundry geometric notes. JRM 18:1 (1985-86) 36-44.

No. 8: Probability; Gambling. JRM 18:2 (1985-86) 101-109.

No. 9: Games and puzzles. JRM 18:3 (1985-86) 161-167.

No. 10: Recreational mathematics; Logical puzzles; Expository mathematics. JRM 18:4 (1985-86) 241-246.

No. 11: Logic, Artificial intelligence, and Mathematical foundations. JRM 19:1 (1987) 3-9.

No. 12: Magic squares and cubes; Latin squares; Mystic arrays and Number patterns. JRM 19:2 (1987) 81-86.

The High School Mathematics Library. NCTM, (1960, 1963, 1967, 1970, 1973); 6th ed., 1976; 7th ed., 1982; 8th ed., 1987.

SEE ALSO: Wheeler; Wheeler & Hardgrove.

Early Books on Magic Squares. JRM 16:1 (1983-84) 1-6.

William L. Schaaf & David Singmaster. Books on Recreational Mathematics. A Supplement to the Lists in William L. Schaaf's A Bibliography of Recreational Mathematics. Collected by William L. Schaaf; typed and annotated by David Singmaster. School of Computing, Information Systems and Mathematics, South Bank University, London, SE1 0AA. 18pp, Dec 1992 and revised several times afterwards.

Peter Schreiber.

Mathematik und belletristik [1.] & 2. Teil. Mitteilungen der Mathematischen Gesellschaft der Deutschen Demokratischer Republik. (1986), no. 4, 57-71 & (1988), no. 1-2, 55-61. Good on German works relating mathematics and arts.

Mathematiker als Memoirenschreiber. Alpha (Berlin) (1991), no. 4, no page numbers on copy received from author. Extends previous work.

S. N. Sen. Scientific works in Sanskrit, translated into foreign languages and vice-versa in the 18th and 19th century A.D. Indian J. History of Science 7 (1972) 44-70.

Will Shortz. Puzzleana [catalogue of his puzzle books]. Produced by the author. 14 editions have appeared. The latest is: May 1992, 88pp with 1175 entries in 26 categories, with indexes of authors and anonymous titles. Some entries cover multiple items. In Jan 1995, he produced a 19pp Supplement extending to a total of 1451 entries.

David Singmaster.

The Bibliography of Some Recreational Mathematics Books. School of Computing, Information Systems

and Mathematics, South Bank Univ.

13 Nov 1994, 39pp. Technical Report SBU-CISM-94-09.

2nd ed., Aug 1995, 41pp. Technical Report SBU-CISM-95-08.

3rd ed., Jun 1996, 42pp. Technical Report SBU-CISM-96-12.

4th ed., Jun 1998, 44pp. Technical Report SBU-CISM-98-02.

(Current version is 61pp.)

Books on Recreational Mathematics. School of Computing, Information Systems and

Mathematics, South Bank Univ., until 1996.

21 Jan 1991. Approx. 2951 items on 120pp, ringbound.

30 Jan 1992. Approx. 3314 items on 138pp, ringbound.

10 Jan 1993. Approx. 3606 items on 95pp, ringbound.

10 Dec 1994. Approx. 4303 items plus 67 Old Books on 110pp. Technical

Report SBU-CISM-94-11.

10 Oct 1996. Approx. 4842 items plus 84 Old Books on 127pp. Technical

Report SBU-CISM-96-17.

24 May 1999. Approx. 6015 items plus 133 Old Books on 166pp. Technical

Report SBU-CISM-99-14.

26 Feb 2002. Approx. 7185 items plus 192 Old Books plus Supplement of

Calculating Devices, on 220pp. thermal bound.

22 Nov 2003. Approx. 7811 items plus 202 Old Books plus Supplement of

Calculating Devices, on 244pp. thermal bound.

Index to Martin Gardner's Columns and Cross Reference to His Books. (Oct 1993.) Slightly revised as: Technical Report SBU-CISM-95-09; School of Computing, Information Systems, and Mathematics; South Bank University, London, Aug 1995, 22pp. (Current version is 23pp and Don Knuth has sent 9pp of additional material and I will combine these at some time.)

Harold Adrian Smith. Dick and Fitzgerald Publishers. Books at Brown 34 (1987) 108-114.

Sotheby's [Sotheby Parke Bernet].

Catalogue of the J. B. Findlay Collection Books and Periodicals on Conjuring and the Allied Arts. Part I: A-O 5-6 Jul 1979. Part II: P-Z plus: Mimeographed Books and Instructions; Flick Books Catalogues of Apparatus and Tricks Autograph Letters, Manuscripts, and Typescripts 4-5 Oct 1979. Part III: Posters and Playbills 3-4 Jul 1980. Each with estimates and results lists.

The Celebrated Library of Harrison D. Horblit Esq. Early Science Navigation & Travel Including Americana with a few medical books. Part I A - C 10/11 Jun 1974. Part II D - G 11 Nov 1974. HB. The sale was then cancelled and the library was sold to E. P. Kraus, qv, who issued three further catalogues, c1976.

The Honeyman Collection of Scientific Books and Manuscripts. Seven volumes, each

with estimates and results booklets.

Part I: Printed Books A-B, 30-31 Oct 1978.

Part II: Printed Books C-E, 30 Apr - 1 May 1979.

Part III: Manuscripts and Autograph Letters of the 12th to the 20th Centuries.

Part IV: Printed Books F-J, 5-6 Nov 1979.

Part V: Printed Books K-M, 12-13 May 1980.

Part VI: Printed Books N-Sa, 10-11 Nov 1980.

Part VII: Printed Books Sc-Z and Addenda, 19-20 May 1981.

Lynn A. Steen, ed.

Library Recommendations for Undergraduate Mathematics. MAA Reports No. 4, 1992.

Two-Year College Mathematics Library Recommendations. MAA Reports No. 5, 1992.

Strens/Guy Collection. Author/Title Listing. Univ. of Calgary. Preliminary Catalogue, 319 pp., July 1986. [The original has a lot of blank space. I have a computer version which is reduced to 67pp.]

Eva Germaine Rimington Taylor. The Mathematical Practitioners of Tudor & Stuart England 1485-1714. CUP for the Institute of Navigation, 1970.

Eva Germaine Rimington Taylor. The Mathematical Practitioners of Hanoverian England 1714-1840. CUP for the Institute of Navigation, 1966.

PLUS: Kate Bostock, Susan Hurt & Michael Hart; An Index to the Mathematical Practitioners of Hanoverian England 1714-1840; Harriet Wynter Ltd., London, 1980.

W. R. Utz. Letter: Mathematics in literature. MM 55 (1982) 249-250. Utz has sent his 3pp original more detailed version along with 4pp of further citations. This extends Koehler's article.

George Walker. The Art of Chess-Play: A New Treatise on the Game of Chess. 4th ed., Sherwood, Gilbert & Piper, London, 1846. Appendix: Bibliographical Catalogue of the chief printed books, writers, and miscellaneous articles on chess, up to the present time, pp. 339-375.

Frank R. [Joe] Watson, ed. Booklists. MA.

Puzzles, Problems, Games and Mathematical Recreations. 16pp, 1980.

Selections from the Recommended Books. 18pp, 1980.

Full List of Recommended Books. 105pp, 1984.

Margariete Montague Wheeler. Mathematics Library -- Elementary and Junior High School. 5th ed., NCTM, 1986.

SEE ALSO: Schaaf; Wheeler & Hardgrove.

Margariete Montague Wheeler & Clarence Ethel Hardgrove. Mathematics Library -- Elementary and Junior High School. NCTM, (1960; 1968; 1973); 4th ed., 1978.

SEE ALSO: Schaaf; Wheeler.

Ernst Wölffing. Mathematischer Bücherschatz. Systematisches Verzeichnis der wichtigsten deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts auf dem Gebiete der mathematischen Wissenschaften. I: Reine Mathematik; (II: Angewandte Mathematik never appeared). AGM 16, part I (1903).

4. MATHEMATICAL GAMES

Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related Material. Op. cit. in 3.B.

4.A. GENERAL THEORY AND NIM-LIKE GAMES

Conway's extension of this theory is well described in Winning Ways and later work is listed in Fraenkel's Bibliography -- see section 3.B & 4 -- so I will not cover such material here.

4.A.1. ONE PILE GAME

See MUS I 145-147.

(a, b) denotes the game where one can take 1, 2, ..., or a away from one pile, starting with b in the pile, with the last player winning. The version (10, 100) is sometimes called Piquet des Cavaliers or Piquet à Cheval, a name which initially perplexed me. Piquet is one of the older card games, being well known to Rabelais (1534) and was known in the 16C as Cent (or Saunt or Saint) because of its goal of 100 points. See: David Parlett; (Oxford Guide to Card Games, 1990 =) A History of Card Games; OUP, 1991, pp. 24 & 175-181. The connection with horses undoubtedly indicates that (10, 100) was viewed as a game which could be played without cards, while riding -- see Les Amusemens, Decremps.

INDEX

( 3, 13) Dudeney, Stong

( 3, 15) Mittenzwey, Hoffmann, Mr. X, Dudeney, Blyth,

( 3, 17) Fourrey,

( 3, 21) Blyth, Hummerston,

( 4, 15) Mittenzwey,

( 6, 30) Pacioli, Leske, Mittenzwey, Ducret,

( 6, 31) Baker,

( 6, 50) Ball-FitzPatrick,

( 6, 52) Rational Recreations

( 6, 57) Hummerston,

( 7, 40) Mittenzwey,

( 7, 41) Sprague,

( 7, 45) Mittenzwey,

( 7, 50) Decremps,

( 7, 60) Fourrey,

( 8, 100) Bachet, Carroll,

( 9, 100) Bachet, Ozanam, Alberti

(10, 100) Bachet, Henrion, Ozanam, Alberti, Les Amusemens, Hooper, Decremps,

Badcock, Jackson, Rational Recreations, Manuel des Sorciers,

Boy's Own Book, Nuts to Crack, Young Man's Book, Carroll,

Magician's Own Book, Book of 500 Puzzles, Secret Out,

Boy's Own Conjuring Book, Vinot, Riecke, Fourrey, Ducret, Devant,

(10, 120) Bachet,

(12, 134) Decremps,

General case: Bachet, Ozanam, Alberti, Decremps, Boy's Own Book, Young Man's Book, Vinot, Mittenzwey, (others ?? check)

Versions with limited numbers of each value or using a die -- see 4.A.1.a.

Version where an odd number in total has to be taken: Dudeney, Grossman & Kramer, Sprague.

Versions with last player losing: Mittenzwey,

Pacioli. De Viribus. c1500. Ff. 73v - 76v. XXXIIII effecto afinire qualunch' numero na'ze al compagno anon prendere piu de un termi(n)ato .n. (34th effect to finish whatever number is before the company, not taking more than a limiting number) = Peirani 109-112. Phrases it as an addition problem. Considers (6, 30) and the general problem.

David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games. Penguin, 1991, pp. 174-175. "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)."

Cardan. Practica Arithmetice. 1539. Chap. 61, section 18, ff. T.iiii.v - T.v.r (p. 113). "Ludi mentales". One has 1, 3, 6 and the other has 2, 4, 5; or one has 1, 3, 5, 8, 9 and the other has 2, 4, 6, 7, 10; one one wants to make 100. "Sunt magnæ inventionis, & ego inveni æquitando & sine aliquo auxilio cum socio potes ludere & memorium exercere ...."

Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670: pp. 353-354. ??NX. (6, 31).

Bachet. Problemes. 1612. Prob. XIX: 1612, 99-103. Prob. XXII, 1624: 170-173; 1884: 115-117. Phrases it as an addition problem. First considers (10, 100), then (10, 120), (8, 100), (9, 100), and the general case. Labosne omits the demonstration.

Dennis Henrion. Nottes to van Etten. 1630. Pp. 19-20. (10, 100) as an addition problem, citing Bachet.

Ozanam. 1694. Prob. 21, 1696: 71-72; 1708: 63-64. Prob. 25, 1725: 182-184. Prob. 14, 1778: 162-164; 1803: 163-164; 1814: 143-145. Prob. 13, 1840: 73-74. Phrases it as an addition problem. Considers (10, 100) and (9, 100) and remarks on the general case.

Alberti. 1747. Due persone essendo convenuto ..., pp. 105-108 (66-67). This is a slight recasting of Ozanam.

Les Amusemens. 1749. Prob. 10, p. 130: Le Piquet des Cavaliers. (10, 100) in additive form. "Deux amis voyagent à cheval, l'un propose à l'autre un cent de Piquet sans carte."

William Hooper. Rational Recreations, In which the Principles of Numbers and Natural Philosophy Are clearly and copiously elucidated, by a series of Easy, Entertaining, Interesting Experiments. Among which are All those commonly performed with the cards. [Taken from my 2nd ed.] 4 vols., L. Davis et al., London, 1774; 2nd ed., corrected, L. Davis et al., London, 1783-1782 (vol. 1 says 1783, the others say 1782; BMC gives 1783-82); 3rd ed., corrected, 1787; 4th ed., corrected, B. Law et al., London, 1794. [Hall, BCB 180-184 & Toole Stott 389-392. Hall says the first four eds. have identical pagination. I have not seen any difference in the first four editions, except as noted in Section 6.P.2. Hall, OCB, p. 155. Heyl 177 notes the different datings of the 2nd ed, Hall, BCB 184 and Toole Stott 393 is a 2 vol. 4th ed., corrected, London, 1802. Toole Stott 394 is a 2 vol. ed. from Perth, 1801. I have a note that there was an 1816 ed, but I have no details. Since all relevant material seems the same in all volumes, I will cite this as 1774.] Vol. 1, recreation VIII: The magical century. (10, 100) in additive form. Mentions other versions and the general rule.

I don't see any connection between this and Rational Recreations, 1824.

Henri Decremps. Codicile de Jérôme Sharp, Professeur de Physique amusante; Où l'on trouve parmi plusieurs Tours dont il n'est point parlé dans son Testament, diverses récréations relatives aux Sciences & Beaux-Arts; Pour servir de troisième suite À La Magie Blanche Dévoilée. Lesclapart, Paris, 1788. Chap. XXVII, pp. 177-184: Principes mathématiques sur le piquet à cheval, ou l'art de gagner son diner en se promenant. Does (10, 100) in additive form, then discusses the general method, illustrating with (7, 50) and (12, 134).

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33-34, no. 48: A curious recreation with a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50 counters. Discusses general case, but doesn't notice that the limitation to 50 counters each considerably changes the game!

Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 47, pp. 11 & 64. Additive form of (10, 100).

Rational Recreations. 1824. Exercise 12(?), pp. 57-58. As in Badcock. Then says it can be generalised and gives (6, 52).

Manuel des Sorciers. 1825. Pp. 57-58, art. 30: Le piquet sans cartes. ??NX (10, 100) done subtractively.

The Boy's Own Book.

The certain game. 1828: 177; 1828-2: 236; 1829 (US): 104; 1855: 386-387; 1868: 427.

The magical century. 1828: 180; 1828-2: 236-237; 1829 (US): 104-105; 1855: 391-392.

Both are additive phrasings of (10, 100). The latter mentions using other numbers and how to win then.

Nuts to Crack V (1836), no. 70. An arithmetical problem. (10, 100).

Young Man's Book. 1839. Pp. 294-295. A curious Recreation with a Hundred Numbers, usually called the Magical Century. Almost identical to Boy's Own Book.

Lewis Carroll.

Diary entry for 5 Feb 1856. In Carroll-Gardner, pp. 42-43. (10, 100). Wakeling's note in the Diaries indicates he is not familiar with this game.

Diary entry for 24 Oct 1872. Says he has written out the rules for Arithmetical Croquet, a game he recently invented. Roger Lancelyn Green's abridged version of the Diaries, 1954, prints a MS version dated 22 Apr 1889. Carroll-Wakeling, prob. 38, pp. 52-53 and Carroll-Gardner, pp. 39 & 42 reprint this, but Gardner has a misprinted date of 1899. Basically (8, 100), but passing the values 10, 20, ..., requires special moves and one may have to go backward. Also, when a move is made, some moves are then barred for the next player. Overall, the rules are typically Carrollian-baroque.

Magician's Own Book. 1857.

The certain game, p. 243. As in Boy's Own Book.

The magical century, pp. 244-245. As in Boy's Own Book.

Book of 500 Puzzles. 1859.

The certain game, p. 57. As in Boy's Own Book.

The magical century, pp. 58-59. As in Boy's Own Book.

The Secret Out. 1859. Piquet on horseback, pp. 397-398 (UK: 130-131) -- additive (10, 100) unclearly explained.

Boy's Own Conjuring Book. 1860.

The certain game, pp. 213-214. As in Boy's Own Book.

Magical century, pp. 215. As in Boy's Own Book.

Vinot. 1860. Art. XI: Un cent de piquet sans cartes, pp. 19-20. (10. 100). Says the idea can be generalised, giving (7, 52) as an example.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 563-III, pp. 247: Wer von 30 Rechenpfennigen den letzen wegnimmt, hat gewonnen. (6, 30).

F. J. P. Riecke. Mathematische Unterhaltungen. 3 vols., Karl Aue, Stuttgart, 1867, 1868 & 1873; reprint in one vol., Sändig, Wiesbaden, 1973. Vol. 3, art 22.2, p. 44. Additive form of (10, 100).

Mittenzwey. 1880. Probs. 286-287, pp. 52 & 101-102; 1895?: 315-317, pp. 56 & 103-104; 1917: 315-317, pp. 51 & 98.

(6, 30), last player wins.

(4, 15), last player loses, the solution discusses other cases: (7, 40), (7, 45) and indicates the general solution.

(added in 1895?) (3, 15), last player loses.

Hoffmann. 1893. Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300-301 = Hoffmann-Hordern, p. 197. (3, 15). c= Benson, 1904, The fifteen match puzzle, pp. 241-242.

Ball-FitzPatrick. 1st ed., 1898. Deuxième exemple, pp. 29-30. (6, 50).

E. Fourrey. Récréations Arithmétiques. (Nony, Paris, 1899; 2nd ed., 1901); 3rd ed., Vuibert & Nony, Paris, 1904; (4th ed., 1907); 8th ed., Librairie Vuibert, Paris, 1947. [The 3rd and 8th eds are identical except for the title page, so presumably are identical to the 1st ed.] Sections 65-66: Le jeu du piquet à cheval, pp. 48-49. Additive forms of (10, 100) and (7, 60). Then gives subtractive form for a pile of matches for (3, 17).

Étienne Ducret. Récréations Mathématiques. Garnier Frères, Paris, nd [not in BN, but a similar book, nouv. ed., is 1892]. Pp. 102-104: Le piquet à cheval. Additive version of (10, 100) with some explanation of the use of the term piquet. Discusses (6, 30).

Mr. X [possibly J. K. Benson -- see entry for Benson in Abbreviations]. His Pages. The Royal Magazine 9:3 (Jan 1903) 298-299. A good game for two. (3, 15) as a subtraction game.

David Devant. Tricks for Everyone. Clever Conjuring with Everyday Objects. C. Arthur Pearson, London, 1910. A counting race, pp. 52-53. (10, 100).

Dudeney. AM. 1917. Prob. 392: The pebble game, pp. 117 & 240. (3, 15) & (3, 13) with the object being to take an odd number in total. For 15, first player wins; for 13, second player wins. (Barnard (50 Telegraph ..., 1985) gives the case (3, 13).)

Blyth. Match-Stick Magic. 1921.

Fifteen matchstick game, pp. 87-88. (3, 15).

Majority matchstick game, p. 88. (3, 21).

Hummerston. Fun, Mirth & Mystery. 1924.

Two second-sight tricks (no. 2), p. 84. (6, 57), last player losing.

A match mystery, p. 99. (3, 21), last player losing.

H. D. Grossman & David Kramer. A new match-game. AMM 52 (1945) 441-443. Cites Dudeney and says Games Digest (April 1938) also gave a version, but without solution. Gives a general solution whether one wants to take an odd total or an even total.

C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. How to design a "Pircuit" or Puzzle circuit, pp. 388-394. On pp. 388-391, Harry Rudloe describes a relay circuit for playing the subtractive form of (3, 13), which he calls the "battle of numbers" game.

Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 24: "Ungerade" gewinnt, pp. 16 & 44-45. (= 'Odd' is the winner, pp. 18 & 53-55.) (7, 41) with the winner being the one who takes an odd number in total. Solves (7, b) and states the structure for (a, b).

I also have some other recent references to this problem. Lewis (1983) gives a general solution which seems to be wrong.

4.A.1.a. THE 31 GAME

Numerical variations: Badcock, Gibson, McKay.

Die versions: Secret Out (UK), Loyd, Mott-Smith, Murphy.

Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670: pp. 353-354. ??NX. (6, 31). ??CHECK if this has the limited use of numbers.

John Fisher. Never Give a Sucker an Even Break. (1976); Sphere Books, London, 1978. Thirty-one, pp. 102-104. (6, 31) additively, but played with just 4 of each value, the 24 cards of ranks 1 -- 6, and the first to exceed 31 loses. He says it is played extensively in Australia and often referred to as "The Australian Gambling Game of 31". Cites the 19C gambling expert Jonathan Harrington Green who says it was invented by Charles James Fox (1749-1806). Gives some analysis.

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33-34, no. 48: A curious recreation with a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50 counters. Discusses general case, but doesn't notice that the limitation to 50 counters each considerably changes the game!

Nuts to Crack V (1836), no. 71. (6, 31) additively, with four of each value. "Set down on a slate, four rows of figures, thus:-- ... You agree to rub out one figure alternately, to see who shall first make the number thirty-one."

Magician's Own Book. 1857. Art. 31: The trick of thirty-one, pp. 70-71. (6, 31) additively, but played with just 4 of each value -- e.g. the 24 cards of ranks 1 -- 6. The author advises you not to play it for money with "sporting men" and says it it due to Mr. Fox. Cf Fisher. = Boy's Own Conjuring Book; 1860; Art. 29: The trick of thirty-one, pp. 78-79. = The Secret Out; 1859, pp. 65-66, which adds a footnote that the trick is taken from the book One Hundred Gambler Tricks with Cards by J. H. Green, reformed gambler, published by Dick & Fitzgerald.

The Secret Out (UK), c1860. To throw thirty-one with a die before your antagonist, p. 7. This is incomprehensible, but is probably the version discussed by Mott-Smith.

Edward S. Sackett. US Patent 275,526 -- Game. Filed: 9 Dec 1882; patented: 10 Apr 1883. 1p + 1p diagrams. Frame of six rows holding four blocks which can be slid from one side to the other to play the 31 game, though other numbers of rows, blocks and goal may be used. Gives an example of a play, but doesn't go into the strategy at all.

Larry Freeman. Yesterday's Games. Taken from "an 1880 text" of games. (American edition by H. Chadwick.) Century House, Watkins Glen, NY, 1970. P. 107: Thirty-one. (6, 31) with 4 of each value -- as in Magician's Own Book.

Algernon Bray. Letter: "31" game. Knowledge 3 (4 May 1883) 268, item 806. "... has lately made its appearance in New York, ...." Seems to have no idea as how to win.

Loyd. Problem 38: The twenty-five up puzzle. Tit-Bits 32 (12 Jun & 3 Jul 1897) 193 & 258. = Cyclopedia. 1914. The dice game, pp. 243 & 372. = SLAHP: How games originate, pp. 73 & 114. The first play is arbitrary. The second play is by throwing a die. Further values are obtained by rolling the die by a quarter turn.

Ball-FitzPatrick. 1st ed., 1898. Généralization récente de cette question, pp. 30-31. (6, 50) with each number usable at most 3 times. Some analysis.

Ball. MRE, 4th ed., 1905, p. 20. Some analysis of (6, 50) where each player can play a value at most 3 times -- as in Ball-FitzPatrick, but with the additional sentence: "I have never seen this extension described in print ...." He also mentions playing with values limited to two times. In the 5th ed., 1911, pp. 19-21, he elaborates his analysis.

Dudeney. CP. 1907. Prob. 79: The thirty-one game, pp. 125-127 & 224. Says it used to be popular with card-sharpers at racecourses, etc. States the first player can win if he starts with 1, 2 or 5, but the analysis of cases 1 and 2 is complicated. This occurs as No. 459: The thirty-one puzzle, Weekly Dispatch (17 Aug 1902) 13 & (31 Aug 1902) 13, but he leaves the case of opening move 2 to the reader, but I don't see the answer given in the next few columns.

Devant. Tricks for Everyone. Op. cit. in 4.A.1. 1910. The thirty-one trick, pp. 53-54. Says to get to 3, 10, 17, 24.

Hummerston. Fun, Mirth & Mystery. 1924. Thirty-one -- a game of skill, pp. 95-96. This uses a layout of four copies of the numbers 1, 2, 3, 4, 5, 6 with one copy of 20 in a 5 x 5 square with the 20 in the centre. Says to get to 3, 10, 17, 24, but that this will lose to an experienced player.

Loyd Jr. SLAHP. 1928. The "31 Puzzle Game", pp. 3 & 87. Loyd Jr says that as a boy, he often had to play it against all comers with a $50 prize to anyone who could beat 'Loyd's boy'. This is the game that Loyd Sr called 'Blind Luck', but I haven't found it in the Cyclopedia. States the first player wins with 1, 2 or 5, but only sketches the case for opening with 5. I have seen an example of Blind Luck -- it has four each of the numbers 1 - 6 arranged around a frame containing a horseshoe with 13 in it.

McKay. Party Night. 1940. The 21 race, pp. 166. Using the numbers 1, 2, 3, 4, at most four times, achieve 21. Says to get 1, 6, 11, 16. He doesn't realise that the sucker can be mislead into playing first with a 1 and losing! Says that with 1, ..., 5 at most four times, one wants to achieve 26 and that with 1, ..., 6 at most four times, one wants to achieve 31. Gives just the key numbers each time.

Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and Enthusiasts. (Blakiston, 1946); revised 2nd ed., Dover, 1954.

Prob. 179: The thirty-one game, pp. 117-119 & 231-232. As in Dudeney.

Prob. 180: Thirty-one with dice, p. 119 & 232-233. Throw a die, then make quarter turns to produce a total of 31. Analysis based on digital roots (i.e. remainders (mod 9)). First player wins if the die comes up 4, otherwise the second player can win. He doesn't treat any other totals.

"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon, 1947. "Trente et un", pp. 56-57. Says he doesn't know any name for this. Get 31 using 4 each of the cards A, 2, ..., 6. Says first player loses easily if he starts with 4, 5, 6 (not true according to Dudeney) and that gamblers dupe the sucker by starting with 3 and winning enough that the sucker thinks he can win by starting with 3. But if he starts with a 1 or 2, then the second player must play low and hope for a break.

Walter B. Gibson. Fell's Guide to Papercraft Tricks, Games and Puzzles. Frederick Fell, NY, 1963. Pp. 54-55: First to fifty. First describes (50, 6), but then adds a version with slips of paper: eight marked 1 and seven marked with 2, 3, 4, 5, 6 and you secretly extract a 6 slip when the other player starts.

Harold Newman. The 31 Game. JRM 23:3 (1991) 205-209. Extended analysis. Confirms Dudeney. Only cites Dudeney & Mott-Smith.

Bernard Murphy. The rotating die game. Plus 27 (Summer 1994) 14-16. Analyses the die version as described by Mott-Smith and finds the set, S(n), of winning moves for achieving a count of n by the first player, is periodic with period 9 from n = 8, i.e. S(n+9) = S(n) for n ( 8. There is no first player winning move if and only if n is a multiple of 9. [I have confirmed this independently.]

Ken de Courcy. The Australian Gambling Game of 31. Supreme Magic Publication, Bideford, Devon, nd [1980s?]. Brief description of the game and some indications of how to win. He then plays the game with face-down cards! However, he insures that the cards by him are one of of each rank and he knows where they are.

4.A.2. SYMMETRY ARGUMENTS

Loyd?? Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349. (= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57, pp. 40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles with 13 objects. Solution uses a symmetry argument -- but the Tit-Bits solution was written by Dudeney.

Dudeney. Problem 500: The cigar puzzle. Weekly Dispatch (7 Jun, 21 Jun, 5 Jul, 1903) all p. 16. (= AM, prob. 398, pp. 119, 242.) Symmetry in placement game, using cigars on a table.

Loyd. Cyclopedia. 1914. The great Columbus problem, pp. 169 & 361. (= MPSL1, prob. 65, pp. 62 & 144. = SLAHP: When men laid eggs, pp. 75 & 115.) Placing eggs on a table.

Maurice Kraitchik. La Mathématique des Jeux. Stevens, Bruxelles, 1930. Section XII, prob. 1, p. 296. (= Mathematical Recreations; Allen & Unwin, London, 1943; Problem 1, pp. 13-14.) Child plays black and white against two chess players and guarantees to win one game. [MJ cites L'Echiquier (1925) 84, 151.]

CAUTION. The 2nd edition of Math. des Jeux, 1953, is a translation of Mathematical Recreations and hence omits much of the earlier edition.

Leopold. At Ease! 1943. Chess wizardry in two minutes, pp. 105-106. Same as Kraitchik.

4.A.3. KAYLES

This has objects in a line or a circle and one can remove one object or two adjacent objects (or more adjacent objects in a generalized version of the game). This derives from earlier games with an array of pins at which one throws a ball or stick.

Murray 442 cites Act 17 of Edward IV, c.3 (1477): "Diversez novelx ymagines jeuez appellez Cloishe Kayles ..." This outlawed such games. A 14C picture is given in [J. A. R. Pimlott; Recreations; Studio Vista, 1968, plate 9, from BM Royal MS 10 E IV f.99] showing a 3 x 3 array of pins. A version is shown in Pieter Bruegel's painting "Children's Games" of 1560 with balls being thrown at a row of pins by a wall, in the back right of the scene. Versions of the game are given in the works of Strutt and Gomme cited in 4.B.1. Gomme II 115-116 discusses it under Roly-poly, citing Strutt and some other sources. Strutt 270-271 (= Strutt-Cox 219-220) calls it "Kayles, written also cayles and keiles, derived from the French word quilles". He has redrawings of two 14C engravings (neither that in Pimlott) showing lines of pins at which one throws a stick (= plate opp. 220 in Strutt-Cox). He also says Closh or Cloish seems to be the same game and cites prohibitions of it in c1478 et seq. Loggats was analogous and was prohibited under Henry VIII and is mentioned in Hamlet.

14C MS in the British Museum, Royal Library, No. 2, B. vii. Reproduced in Strutt, p. 271. Shows a monk(?) standing by a line of eight conical pins and another monk(?) throwing a stick at the pins.

Anonymous. Games of the 16th Century. The Rockliff New Project Series. Devised by Arthur B. Allen. The Spacious Days of Queen Elizabeth. Background Book No. 5. Rockliff Publishing, London, ©1950, 4th ptg. The Background Books seem to be consecutively paginated as this booklet is paginated 129-152. Pp. 133-134 describes loggats, quoting Hamlet and an unknown poet of 1611. P. 137 is a photograph of the above 14C illustration. The caption is "Skittles, or "Kayals", and Throwing a Whirling Stick".

van Etten. 1624. Prob. 72 (misnumbered 58) (65), pp 68-69 (97-98): Du jeu des quilles (Of the play at Keyles or Nine-Pins). Describes the game as a kind of ninepins.

Loyd. Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349. (= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57, pp. 40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles with 13 objects. See also 4.A.2.

Dudeney. Sharpshooters puzzle. Problem 430. Weekly Dispatch (26 Jan, 9 Feb, 1902) both p. 13. Simple version of Kayles.

Ball. MRE, 4th ed., 1905, pp. 19-20. Cites Loyd in Tit-Bits. Gives the general version: place p counters in a circle and one can take not more than m adjacent ones.

Dudeney. CP. 1907. Prob. 73: The game of Kayles, pp. 118-119 & 220. Kayles with 13 objects.

Loyd. Cyclopedia. 1914. Rip van Winkle puzzle, pp. 232 & 369-370. (c= MPSL2, prob. 6, pp. 5 & 122.) Linear version with 13 pins and the second knocked down. Gardner asserts that Dudeney invented Kayles, but it seems to be an abstraction from the old form of the game.

Rohrbough. Puzzle Craft, later version, 1940s?. Daisy Game, p. 22. Kayles with 13 petals of a daisy.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 45, pp. 48 & 95. Circular kayles with five objects.

Doubleday - 2. 1971. Take your pick, pp. 63-65. This is Kayles with a row of 10, but he says the first player can only take one.

4.A.4. NIM

Nim is the game with a number of piles and a player can take any number from one of the piles. Normally the last one to play wins.

David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games. Penguin, 1991. Pp. 174-175. "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)."

Charles L. Bouton. Nim: a game with a complete mathematical theory. Annals of Math. (2) 3 (1901/02) 35-39. He says Nim is played at American colleges and "has been called Fan-Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it." He says Paul E. More showed him the misère (= last player loses) version in 1899, so it seems that Bouton did not actually invent the game himself.

Ahrens. "Nim", ein amerikanisches Spiel mit mathematischer Theorie. Naturwissenschaftliche Wochenschrift 17:22 (2 Mar 1902) 258-260. He says that Bouton has admitted that he had confused Nim and Fan-Tan. Fan-Tan is a Chinese game where you bet on the number of counters (mod 4) in someone's hand. Parker, Ancient Ceylon, op. cit. in 4.B.1, pp. 570-571, describes a similar game, based on odd and even, as popular in Ceylon and "certainly one of the earliest of all games".

For more about Fan-Tan, see the following.

Stewart Culin. Chess and playing cards. Catalogue of games and implements for divination exhibited by the United States National Museum in connection with the Department of Archæology and Paleontology of the University of Pennsylvania at the Cotton States and International Exposition, Atlanta, Georgia, 1895. IN: Report of the U. S. National Museum, year ending June 30, 1896. Government Printing Office, Washington, 1898, HB, pp. 665-942. [There is a reprint by Ayer Co., Salem, Mass., c1990.] Fan-Tan (= Fán t‘án = repeatedly spreading out) is described on pp. 891 & 896, with discussion of related games on pp. 889-902.

Alan S. C. Ross. Note 2334: The name of the game of Nim. MG 37 (No. 320) (May 1953) 119-120. Conjectures Bouton formed the word 'nim' from the German 'nimm'. Gives some discussion of Fan-Tan and quotes MUS I 72.

J. L. Walsh. Letter: The name of the game of Nim. MG 37 (No. 322) (Dec 1953) 290. Relates that Bouton said that he had chosen the word from the German 'nimm' and dropped one 'm'.

W. A. Wythoff. A modification of the game of Nim. Nieuw Archief voor Wiskunde (Groningen) (2) 7 (1907) 199-202. He considers a Nim game with two piles allows the extra move of taking the same amount from both piles. [Is there a version with more piles where one can take any number from one pile or equal amounts from two piles?? See Barnard, below for a three pile version.]

Ahrens. MUS I. 1910. III.3.VII: Nim, pp. 72-88. Notes that Nim is not the same as Fan-Tan, has been known in Germany for decades and is played in China. Gives a thorough discussion of the theory of Nim and of an equivalent game and of Wythoff's game.

E. H. Moore. A generalization of the game called Nim. Annals of Math. (2) 11 (1910) 93-94. He considers a Nim game with n piles and one is allowed to take any number from at most k piles.

Ball. MRE, 5th ed., 1911, p. 21. Sketches the game of Nim and its theory.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 13: The last match, pp. 10-11. Thirty matches divided at random into three heaps. Last player loses. Explanation of how to win is rather cryptic: "you must try and take away ... sufficient ... to leave the matches in the two or three heaps remaining, paired in ones, twos, fours, etc., in respect of each other."

Loyd Jr. SLAHP. 1928. A tricky game, pp. 47 & 102. Nim (3, 4, 8).

Emanuel Lasker. Brettspiele der Völker. 1931. See comments in 4.A.5. Jörg Bewersdorff [email of 6 Jun 1999] says that Lasker considered a three person Nim and found an equilibrium for it -- see: Jörg Bewersdorff; Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.3 Ein Spiel zu dritt, pp. 110-115.

Lynn Rohrbough, ed. Fun in Small Spaces. Handy Series, Kit Q, Cooperative Recreation Service, Delaware, Ohio, nd [c1935]. Take Last, p. 10. Last player loses Nim (3, 5, 7).

Rohrbough. Puzzle Craft. 1932.

Japanese Corn Game, p. 6 (= p. 6 of 1940s?). Last player loses Nim (1, 2, 3, 4, 5).

Japanese Corn Game, p. 23. Last player loses Nim (3, 5, 7).

René de Possel. Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion. Actualités Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and also the misère version.

Depew. Cokesbury Game Book. 1939. Make him take it, pp. 187-188. Nim (3, 4, 5), last player loses.

Edward U. Condon, Gereld L. Tawney & Willard A. Derr. US Patent 2,215,544 -- Machine to Play Game of Nim. Filed: 26 Apr 1940; patented: 24 Sep 1940. 10pp + 11pp diagrams.

E. U. Condon. The Nimatron. AMM 49 (1942) 330-332. Has photo of the machine.

Benedict Nixon & Len Johnson. Letters to the Notes & Queries Column. The Guardian (4 Dec 1989) 27. Reprinted in: Notes & Queries, Vol. 1; Fourth Estate, London, 1990, pp. 14-15. These describe the Ferranti Nimrod machine for playing Nim at the Festival of Britain, 1951. Johnson says it played Nim (3, 5, 6) with a maximum move of 3. The Catalogue of the Exhibition of Science shows this as taking place in the Science Museum.

H. S. M. Coxeter. The golden section, phyllotaxis, and Wythoff's game. SM 19 (1953) 135-143. Sketches history and interconnections.

H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Chap. 11: The golden section and phyllotaxis, pp. 160-172. Extends his 1953 material.

A. P. Domoryad. Mathematical Games and Pastimes. (Moscow, 1961). Translated by Halina Moss. Pergamon, Oxford, 1963. Chap. 10: Games with piles of objects, pp. 61-70. On p. 62, he asserts that Wythoff's game is 'the Chinese national game tsyanshidzi ("picking stones")'. However M.-K. Siu cannot recognise such a Chinese game, unless it refers to a form of jacks, which has no obvious connection with Wythoff's game or other Nim games. He says there is a Chinese character, 'nian', which is pronounced 'nim' in Cantonese and means to pick up or take things.

N. L. Haddock. Note 2973: A note on the game of Nim. MG 45 (No. 353) (Oct 1961) 245-246. Wonders if the game of Nim is related to Mancala games.

T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Section on misère version of Wythoff's game, p. 133. Richard Guy (letter of 27 Feb 1985) says this is one of O'Beirne's few mistakes -- cf next entry.

Winning Ways. 1982. P. 407 says Wythoff's game is also called Chinese Nim or Tsyan-shizi. No reference given. See comment under Domoryad above. This says many authors have done this incorrectly.

D. St. P. Barnard. 50 Daily Telegraph Brain-Twisters. Javelin Books, Poole, Dorset, 1985. Prob. 30: All buttoned up, pp. 49-50, 91 & 115. He suggests three pile game where one can take any number from one pile or an equal number from any two or all three piles. [See my note to Wythoff, above.]

Matthias Mala. Schnelle Spiele. Hugendubel, Munich, 1988. San Shan, p. 66. This describes a nim-like game named San Shan and says it was played in ancient China.

Jagannath V. Badami. Musings on Arithmetical Numbers Plus Delightful Magic Squares. Published by the author, Bangalore, India, nd [Preface dated 9 Sep 1999]. Section 4.16: The game of Nim, pp. 124-125. This is a rather confused description of one pile games (21, 5) and (41, 5), but he refers to solving them by (mentally) dividing the pile into piles. This makes me think of combining the two games, i.e. playing Nim with several piles but with a limit on the number one can take in a move.

4.A.5. GENERAL THEORY

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. Ff. 134-144 are: Essay 10 Part 5. See 4.B.1 for more details. At the top of f. 134.r, he has added a note: "This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". He then describes Tit Tat To and makes some simple analysis, but he never uses a name for it.

Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205. ??NX. See 4.B.1 for more details. On f. 304, he starts on analysis of games. Ff. 310-383 are almost entirely devoted to Tit-Tat-To, with some general discussions. F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general. F. 324-333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of positions in Tit Tat To as 9! + 8! + ... + 1! = 409,113. F. 333 has an idea of the tree structure of a game.

John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 & 125-130. See 4.B.1 for more details. He discusses the above Babbage material. On p. 127, Dubbey has: "The basic problem is one that appears not to have been previously considered in the history of mathematics." Dubbey, on p. 129, says: "This analysis ... must count as the first recorded stochastic process in the history of mathematics." However, it is really a deterministic two-person game.

E. Zermelo. Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc. 5th ICM (1912), CUP, 1913, vol. II, 501-504. Gives general idea of first and second person games.

Ahrens. A&N. 1918. P. 154, note. Says that each particular Dots and Boxes board, with rational play, has a definite outcome.

W. Rivier. Archives des Sciences Physiques et Naturelles (Nov/Dec 1921). ??NYS -- cited by Rivier (1935) who says that the later article is a new and simpler version of this one.

H. Steinhaus. Difinicje potrzebne do teorji gry i pościgu (Definitions for a theory of games and pursuit). Myśl Akademicka (Lwów) 1:1 (Dec 1925) 13-14 (in Polish). Translated, with an introduction by Kuhn and a letter from Steinhaus in: Naval Research Logistics Quarterly 7 (1960) 105-108.

Dénès König. Über eine Schlussweise aus dem Endlichen ins Unendliche. Mitteilungen der Universitä Szeged 3 (1927) 121-130. ??NYS -- cited by Rivier (1935). Kalmár cites it to the same Acta as his article.

László Kalmár. Zur Theorie der abstracten Spiele. Acta Litt. Sci. Regia Univ. Hungaricae Francisco-Josephine (Szeged) 4 (1927) 62-85. Says there is a gap in Zermelo which has been mended by König. Lengthy approach, but clearly gets the idea of first and second person games.

Max Euwe. Proc. Koninklijke Akadamie van Wetenschappen te Amsterdam 32:5 (1929). ??NYS -- cited by Rivier (1935).

Emanuel Lasker. Brettspiele der Völker. Rätsel- und mathematische Spiele. A. Scherl, Berlin, 1931, pp. 170-203. Studies the one pile game (100, 5) and the sum of two one-pile games: (100, 5) + (50, 3). Discusses Nimm, "an old Chinese game according to Ahrens" and says the solver is unknown. Gives Lasker's Nim -- one can take any amount from a pile or split it in two -- and several other variants. Notes that 2nd person + 2nd person is 2nd person while 2nd person + 1st person is 1st person. Gives the idea of equivalent positions. Studies three (and more) person games, assuming the pay-offs are all different. Studies some probabilistic games. Jörg Bewersdorff [email of 6 Jun 1999] observes that Lasker's analysis of his Nim got very close to the idea of the Sprague-Grundy number. See: Jörg Bewersdorff; Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.5 Lasker-Nim: Gewinn auf verborgenem Weg, pp. 118-124.

W. Rivier. Une theorie mathématique des jeux de combinaisions. Comptes-Rendus du Premier Congrès International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 106-113. A revised and simplified version of his 1921 article. He cites and briefly discusses Zermelo, König and Euwe. He seems to be classifying games as first player or second player.

René de Possel. Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion. Actualités Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and also the misère version. Shows that any combinatorial game is a win, loss or draw and describes the nature of first and second person positions. He then goes on to consider games with chance and/or bluffing, based on von Neumann's 1927 paper.

R. Sprague. Über mathematische Kampfspiele. Tôhoku Math. J. 41 (1935/36) 438-444.

P. M. Grundy. Mathematics and games. Eureka 2 (1939) 6-8. Reprinted, ibid. 27 (1964) 9-11. These two papers develop the Sprague-Grundy Number of a game.

D. W. Davies. A theory of chess and noughts and crosses. Penguin Science News 16 (Jun 1950) 40-64. Sketches general ideas of tree structure, Sprague-Grundy number, rational play, etc.

H. Steinhaus. Games, an informal talk. AMM 72 (1965) 457-468. Discusses Zermelo and says he wasn't aware of Zermelo in 1925. Gives Mycielski's formulation and proof via de Morgan's laws. Goes into pursuit and infinite games and their relation to the Axiom of Choice.

H. Steinhaus. (Proof that a game without ties has a strategy.) In: M. Kac; Hugo Steinhaus -- a reminiscence and a tribute; AMM 81 (1974) 572-581. Repeats idea of his 1965 talk.

4.B. PARTICULAR GAMES

See 5.M for Sim and 5.R.5 for Fox and Geese, etc.

Most of the board games described here are classic and have been extensively described and illustrated in the various standard books on board games, particularly the works of Robert C. Bell, especially his Board and Table Games from Many Civilizations; OUP, vol. I, 1960, vol. II, 1969; combined and revised ed., Dover, 1979 and the older work of Edward G. Falkener; Games Ancient and Oriental and How to Play Them; Longmans, Green, 1892; Dover, 1961. The works by Culin (see 4.A.4, 4.B.5 and 4.B.9) are often useful. Several general works on games are cited in 4.B.1 and 4.B.5 -- I have read Murray's History of Board Games Other than Chess, but not yet entered the material. Note that many of these works are more concerned with the game than with its history and have a tendency to exaggerate the ages of games by assuming, e.g. that a 3 x 3 board must have been used for Tic-Tac-Toe. I will not try to duplicate the descriptions by Bell, Falkener and others, but will try to outline the earliest history, especially when it is at variance with common belief. The most detailed mathematical analyses are generally in Winning Ways.

4.B.1. TIC-TAC-TOE = NOUGHTS AND CROSSES

Popular belief is that the game is ancient and universal -- e.g. see Brandreth, 1976. However the game appears to have evolved from earlier three-in-a-row games, e.g. Nine Holes or Three Men's Morris, in the early 19C. See also the historical material in 4.B.5. The game is not mentioned in Strutt nor most other 19C books on games, not even in Kate Greenaway's Book of Games (1889), nor in Halliwell's section on slate games (op. cit. in 7.L.1, 1849, pp. 103-104), but there may be an 1875 description in Strutt-Cox of 1903. Babbage refers to it in his unpublished MSS of c1820 as a children's game, but without giving it a name. In 1842, he calls it Tit Tat To and he uses slight variations on this name in his extended studies of the game -- see below. The OED's earliest references are: 1849 for Tip-tap-toe; 1855 for Tit-tat-toe; 1861 for Oughts and Crosses. However, the first two entries may be referring to some other game -- e.g. the entries for Tick-tack-toe for 1884 & 1899 are clearly to the game that Gomme calls Tit-tat-toe. Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name. Using the works of Strutt, Gomme, Strutt-Cox, Fiske, Murray, the OED and some personal communications, I have compiled a separate index of 121 variant names which refer to 5 basic games, with a few variants and a few unknown games. The Murray and Parker material is given first, as it deals generally with the ancient history. Then I list several standard sources and then summarize their content. Other material follows that. Fiske says that van der Linde and von der Lasa (see 5.F.1) mention early appearances of Morris games, but rather briefly and I don't always have that material.

The usual # shape board will be so indicated. If one is setting down pieces, then the board is often drawn as a 'crossed square', i.e. a square with its horizontal and vertical midlines drawn, and one plays on the intersections. Fiske 127 says this form is common in Germany, but unknown in England and the US. In addition, the diagonals are often drawn, producing a 'doubly crossed square'. The squares are sometime drawn as circles giving a 'crossed circle' and a 'doubly crossed circle', though it is hard to identify the corners in a crossed circle. The 3 x 3 array of dots sometimes occurs. The standard # pattern is sometimes surrounded by a square producing a '3 x 3 chessboard'.

Fiske 129 says the English play with O and +, while the Swedes play with O and 1. My experience is that English and Americans play with O and X. One English friend said that where she grew up, it was called 'Exeter's Nose' as a deliberate corruption of 'Xs and Os'.

The first clear references to the standard game of Noughts and Crosses are Babbage (1820) and the items discussed under Tic-tac-toe below. Further clear references are: Cassell's, Berg, A wrangler ..., Dudeney, White and everything entered below after White.

Misère version: Gardner (1957); Scotts (1975);

Murray mentions Morris, which he generally calls Merels, many times. Besides the many specific references mentioned below and in 4.B.5, he shows, on p. 614, under Nine Holes and Three Men's Morris, a number of 3 x 3 diagrams.

Kurna, Egypt, (-14C) -- a double crossed square and a double crossed circle -- see Parker below.

Ptolemaic Egypt (in the BM, no. 14315) -- a square with # drawn inside. See below where I describe this, from a recent exhibition, as just a # board.

Ceylon -- a doubly crossed square -- see Parker below.

Rome and Pompeii -- doubly crossed circles.

Under Nine Holes, he says a piece can be moved to any vacant point; under Three Men's Morris, he says a man can only be moved along a marked line to an adjacent point, i.e. horizontally, vertically or along a main diagonal.

Under Nine Holes, he shows the # board for English Noughts and Crosses. He specifically notes that the pieces do not move. His only other mention of this board is for a Swedish game called Tripp, Trapp, Trull, but he does not state that the pieces do not move. He gives no other examples of the # board nor of non-moving pieces.

He also mentions Five (or Six) Men's Morris, of which little is known. On p. 133, he mentions a 3 x 3 "board of nine points used for a game essentially identical with the 'three men's merels', which has existed in China from at least the time of the Liang dynasty (A.D. 502-557). The 'Swei shu' (first half of the 7th c.) gives the names of twenty books on this game."

H. Parker. Ancient Ceylon. ??, London, 1909; Asian Educational Services, New Delhi, 1981. Nerenchi keliya, pp. 577-580 & 644. There is a crossed square with small holes at the intersections at the Temple of Kurna, Upper Egypt, -14C. [Rohrbough, loc. cit. in 4.B.5, says this temple was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.] On p. 644, he shows 34 mason's diagrams from Kurna, which include #, # in a circle, crossed square with small holes at the intersections, doubly crossed square, doubly crossed circle. He cites Bell, Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for for a doubly crossed square in Ceylon, c1C, but Noughts and Crosses is not found in the interior of Ceylon. The doubly crossed square was used in 18C Ireland. On pp. 643-665, he discusses appearances of the crossed square and doubly crossed circle as designs or characters and claims they have mystic significance. On p. 662, he lists many early appearances of the # pattern.

Murray 440, note 63, includes a reference to Soutendam; Keurboek van Delft; Delft, c1425, f. 78 (or p. 78?); who says games of subtlety are allowed, e.g. ... ticktacken. There is no indication if this may be our game and the OED indicates that such names were used for backgammon back to 1558. The OED doesn't cite: W. Shakespeare; Measure for Measure, c1604. Act I, scene ii, line 180 (or 196): "foolishly lost at a game of ticktack". Later it was more common as Tric-trac.

Murray 746 notes a Welsh game Gwyddbwyll mentioned in the Mabinogion (14C). The name is cognate with the Irish Fidchell and may be a Three Men's Morris, but the game was already forgotten by the 15C.

STANDARD SOURCES ON GAMES

Joseph Strutt. The Sports and Pastimes of the People of England. (With title starting: Glig-Gamena Angel-Ðeod., or the Sports ...; J. White, London, 1791, 1801, 1810). A new edition, with a copious index, by William Hone. Tegg, London, 1830, 1831, 1833, 1834, 1838, 1841, 1850, 1855, 1875, 1876, 1891. [The 1830 ed. has a preface, omitted in 1833, stating that the 1810 ed. is the same as the 1801 ed. and that Hone has only changed it by adding the Index and incorporating some footnotes into the text.] [Hall, BCB 263-266 are: 1801, 1810, 1830, 1831. Toole Stott 647-656 are: 1791; 1801; 1810; 1828-1830 in 10 monthly parts with Index by Hone; 1830; 1830; 1833; 1838; 1841; 1876, an expanded ed, ed by Hone. Heyl 300-302 gives 1830; 1838; 1850. Toole Stott 653 says the sheets were remaindered to Hone, who omitted the first 8pp and issued it in 1833, 1834, 1838, 1841. I have seen an 1855 ed. C&B list 1801, 1810, 1830, 1903. BMC has 1801, 1810, 1830, 1833, 1834, 1838, 1841, 1875, 1876, 1898.]

Strutt-Cox. The Sports and Pastimes of the People of England. By Joseph Strutt. 1801. A new edition, much enlarged and corrected by J. Charles Cox. Methuen, 1903. The Preface sketches Strutt's life and says this is based on the 'original' 1801 in quarto, with separate plates which were often hand coloured, but not consistently, while the 1810 reissue had them all done in a terra-cotta shade. Hone reissued it in octavo in 1830 with the plates replaced by woodcuts in the text and this was reissued in 1837, 1841 and 1875. (From above we see that there were other reissues.) "Mr. Strutt has been left for the most part to speak in his own characteristic fashion .... A few obvious mistakes and rash conclusions have been corrected, ... certain unimportant omissions have been made. ... Nearly a third of the book is new." Reprinted in 1969 and in the 1960s?

J. T. Micklethwaite. On the indoor games of school boys in the middle ages. Archaeological Journal 49 (Dec 1892) 319-328. Describes various 3 x 3 boards and games on them, including Nine Holes and "tick, tack, toe; or oughts and crosses, which I suppose still survives wherever slate and pencil are used as implements of education", Three Men's Morris and also Nine Men's Morris, Fox and Geese, etc.

Alice B. Gomme. The Traditional Games of England, Scotland, and Ireland. 2 vols., David Nutt, London, 1894 & 1898. Reprinted in one vol., Thames & Hudson, London, 1984.

Willard Fiske. Chess in Iceland and in Icelandic Literature with Historical Notes on Other Table-Games. The Florentine Typographical Society, Florence, 1905. Esp. pp. 97-156 of the Stray Notes. P. 122 lists a number of works on ancient games.

These and the OED have several entries on Noughts and Crosses and Tic-tac-toe and many on related games, which are summarised below. Gomme often cites or quotes Strutt. The OED often gives the same quotes as Gomme. Gomme's references are highly abbreviated but full details of the sources can usually be found in the OED.

(Nine Men's) Morris, where Morris is spelled about 30 different ways, e.g. Marl, Merelles, Mill, Miracles, Morals, and Nine Men's may be given as, e.g. Nine-peg, Nine Penny, Nine Pin. Also known as Peg Morris and Shepherd's Mill. Gomme I 80 & 414-419 and Strutt 317-318 (c= Strutt-Cox 256-258 & plate opp. 246, which adds reference to Micklethwaite) are the main entries. See 4.B.5 for material more specifically on this game.

Nine Holes, also known as Bubble-justice, Bumble-puppy, Crates, and possibly Troll-madam, Troule-in-Madame. Gomme I 413-414 and Strutt 274-275 & 384 (c= Strutt-Cox 222-223 & 304) are the main entries. Twelve Holes is similar [Gomme II 321 gives a quote from 1611]. There seem to be cases where Nine Men's Morris was used in referring to Nine Holes [Gomme I 414-419]. There are two forms of the game: one form has holes in an upright board that one must roll a ball or marble through; the other form has holes in the ground, usually in a 3  x  3 array, that one must roll balls into. Unfortunately, none of the references implies that one has to get three in a row -- see Every Little Boys Book for a version where this is certainly not the case. There are references going back to 1572 for Crates (but mentioning eleven holes) [Gomme I 81 & II 309] and 1573 [OED] for Nine Holes. Botermans et al.; The World of Games; op. cit. in 4.B.5; 1989; p. 213, shows a 17C engraving by Ménian showing Le Jeu de Troumadame as having a board with holes in it, held vertically on a table and one must roll marbles through the holes. They say it is nowadays known as 'bridge'.

Three Men's Morris. This is less common, but occurs in several variant spellings corresponding to the variants of Nine Men's Morris, including, e.g. Three-penny Morris, Tremerel. The game is played on a 3  x 3 board and each player has three men. After making three plays each, consisting of setting men on the cells, further play consists of picking up one of your own men and placing it on a vacant cell, with the object of getting three in a row. There are several versions of this game, depending on which cells one may play to, but the descriptions given rarely make this clear. [Gomme I 414-419] quotes from F. Douce; Illustrations of Shakespeare and of Ancient Manners; 1807, i.184. "In the French merelles each party had three counters only, which were to be placed in a line to win the game. It appears to have been the tremerel mentioned in an old fabliau. See Le Grand, Fabliaux et Contes, ii.208. Dr. Hyde thinks the morris, or merrils, was known during the time that the Normans continued in possession of England, and that the name was afterwards corrupted into three men's morals, or nine men's morals." [Hyde. Hist. Nederluddi [sic], p. 202.] In practice, the board is often or usually drawn as a crossed square. If one can move along all winning lines, then it would be natural to draw a doubly crossed square. See under Alfonso MS (1283) in 4.B.5 for versions called marro, tres en raya and riga di tre. Again, much of the material on this game is in 4.B.5.

Five-penny Morris. None of the references make it clear, but this seems to be (a form of) Three Men's Morris. Gomme I 122 and the OED [under Morrell] quote: W. Hawkins; Apollo Shroving (a play of 1627), act III, scene iv, pp. 48-49.

"..., Ovid hath honour'd my exercises. He describes in verse our boyes play.

Twise three stones, set in a crossed square where he wins the game

That can set his three along in a row,

And that is fippeny morrell I trow."

Most of the references (and myself) are perplexed by the reference to five, though the fact that one has at most five moves in Tic-tac-toe might have something to do with it?? Since Three Men's Morris is less well known, some writers have assumed Five-penny Morris was Nine Men's Morris and others have called all such games by the same name. A few lines later, Hawkins has: "I challenge him at all games from blowpoint upward to football, and so on to mumchance, and ticketacke. ... rather than sit out, I will give Apollo three of the nine at Ticketacke, ..."

Corsicrown [Gomme I 80] seems to be a version of Three Men's Morris, but using seven of the nine cells, omitting two opposite side cells. Gomme quotes from J. Mactaggart; The Scottish Gallovidian Encyclopedia; (1871 or possibly 1824?): "each has three men .... there are seven points for these men to move about on, six on the edges of the square and one at the centre."

Tic-tac-toe. The earliest clearly described versions are given in Babbage (with no name given), c1820, and Gomme I 311, under Kit-cat-cannio, where she quotes from: Edward Moor; Suffolk Words and Phrases; 1823 (This word does not occur in the OED). Gomme also gives entries for Noughts and Crosses [I 420-421] and Tip-tap-toe [II 295-296] with variants Tick-tack-toe and Tit-tat-toe. In 1842-1865, Babbage uses Tit Tat To and slight variants. Under Tip-tap-toe, Gomme says the players make squares and crosses and that a tie game is a score for Old Nick or Old Tom. (When I was young, we called it Cat's Game, and this is an old Scottish term [James T. R. Ritchie; The Singing Street Scottish Children's Games, Rhymes and Sayings; (O&B, 1964); Mercat Press, Edinburgh, 2000, p. 61].) She quotes regional glossaries for Tip-tap-toe (1877), Tit-tat-toe (1866 & 1888), Tick-tack-toe (1892). The OED entry for Oughts and Crosses seems to be this game and gives an 1861 quote. Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name.

Tit-tat-toe [Gomme II 296-298]. This is a game using a slate marked with a circle and numbered sectors. The player closes his eyes and taps three times with a pencil and tries to land on a good sector. Gomme gives the verse:

Tit, tat, toe, my first go,

Three jolly butcher boys all in a row

Stick one up, stick one down,

Stick one in the old man's ground.

But cf Games and Sports for Young Boys, 1859, below.

The OED entries under Tick-tack, Tip-tap and Tit give a number of variant spellings and several quotations, which are often clearly to this game, but are sometimes unclear. Also some forms seem to refer to backgammon.

In her 'Memoir on the study of children's games' [Gomme II 472-473], Gomme gives a somewhat Victorian explanation of the origin of Old Nick as the winner of a tie game as stemming from "the primitive custom of assigning a certain proportion of the crops or pieces of land to the devil, or other earth spirit."

Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. P. 81 says examples of boards were discovered in the lowest level of Troy and in the Bronze Age tombs in Co. Wicklow, Ireland. Their description is a bit vague but indicates that the Italian version of Tic-tac-toe is actually Three Men's Morris.

Anonymous. Play the game. Guardian Education section (21 Sep 1993) 18-19. Shows a stone board with the # incised on it 'from Bet Shamesh, Israel, 2000 BC'. This might be the same as the first board below??

A small exhibition of board games organized by Irving Finkel at the British Museum, 1991, displayed the following.

Stone slab with the usual # Tick-Tac-Toe board incised on it, but really a 4 x 3 board. With nine stone men. From Giza, >-850. BM items EA 14315 & 14309, donated by W. M. Flinders Petrie. Now on display in Room 63, Case C.

Stone Nine Holes board from the Temple of Artemis, Ephesus, 2C-4C. Item BM GR 1873.5.5.150. This is a 3 x 3 array of depressions. Now on display in Room 69, Case 9.

Robbie Bell & Michael Cornelius. Board Games Round the World. CUP, 1988. P. 6 states that the crossed square board has been found at Kurna (c-1400) and at the Ptolemaic temple at Komombo (c-300). They state that Three Men's Morris is the game mentioned by Ovid in Ars Amatoria. They say that it was known to the Chinese at the time of Confucius (c-500) under the name of Yih, but is now known as Luk tsut k'i. They also say the game is also known as Nine Holes -- which seems wrong to me.

The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototypic Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913. (There was also an edition by Arnald Steiger, Geneva, 1941.) See 4.B.5 for more details of this work. Vol. 2, f. 93v, p. CLXXXVI, shows a doubly crossed square board. ??NX -- need to study text.

Pieter Bruegel (the Elder). Children's Games. Painting dated 1560 at the Kunsthistorisches Museum, Vienna. In the right background, children are playing a game involving throwing balls into holes in the ground, but the holes appear to be in a straight line.

Anonymous. Games of the 16th Century. 1950. Op. cit. in 4.A.3. P. 134 describes nine-holes, quoting an unknown poet of 1611: "To play at loggats, Nine-holes, or Ten-pinnes". The author doesn't specify what positions the balls are to be rolled into. P. 152 describes Troll-my-dames or Troule-in-madame: "they may have in the end of a bench eleven holes made, into which to troll pummets, or bowls of lead, ...."

William Wordsworth. The Prelude, Book 1. Completed 1805, published 1850. Lines 509-513.

At evening, when with pencil, and smooth slate

In square divisions parcelled out and all

With crosses and with cyphers scribbled o'er,

We schemed and puzzled, head opposed to head

In strife too humble to be named in verse.

It is not clear if this is referring to Noughts and Crosses.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. F. 4r is part of the Table of Contents. It shows Noughts and Crosses games played on the # board and on a 4 x 4 board adjacent to entry 4: The Mill. Ff. 124-146 are: Essay 10 -- Of questions requiring the invention of new modes of analysis. On f. 128.r, he refers to a game in which "the relative positions of three of the marks is the object of inquiry." Though the reference is incomplete, a Noughts and Crosses game is drawn on the facing page, f. 127.v. Ff. 134-144 are: Essay 10 Part 5. At the top of f. 134.r, he has added a note: "This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". The Essay begins: "Amongst the simplest of those games requiring any degree of skill which amuse our early years is one which is played at in the following manner." He then describes the game in detail and makes some simple analysis, but he never uses a name for it.

Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205. ??NX. On f. 304, he starts on analysis of games. Ff. 310-383 are almost entirely devoted to Tit-Tat-To, with some general discussions. Most of this material comprises a few sheets of working, carefully dated, sometimes amended and with the date of the amendment. A number of sheets describe parts of the automaton that he was planning to build which would play the game, but no such machine was built until 1949. The sheets are not always in strict chronological order.

F. 310.r is the first discussion of the game, called Tit Tat To, dated 17 Sep 1842. On F. 312.r, 20 Sep 1843, he says he has "Reduced the 3024 cases D to 199 which include many Duplicates by Symmetry." F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general. He refers to Tit-tat-too. F. 322.r continues and he says: "I have found no game of skill more simple that that which children often play and which they call Tit-tat-to." F. 324-333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of positions as 9! + 8! + ... + 1!  = 409,113. F. 333 has an idea of the tree structure of a game. On ff. 337-338, 8 Sep 1848, he has Tit-tat too. On ff. 347.r-347.v, he suggests Nine Men's Morris boards in triangular and pentagonal shapes and does various counting on the different shapes. On ff. 348-349, 26 Oct 1859, he uses Tit-Tat-To.

John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 & 125-130. He discusses the above Babbage material. On p. 127, Dubbey has: "After a surprisingly lengthy explanation of the rules, he attempts a mathematical formulation. The basic problem is one that appears not to have been previously considered in the history of mathematics." Babbage represents the game using roots of unity. Dubbey, on p. 129, says: "This analysis ... must count as the first recorded stochastic process in the history of mathematics." However, it is really a deterministic two-person game.

Games and Sports for Young Boys. Routledge, nd [1859 - BLC]. P. 70, under Rhymes and Calls: "In the game of Tit-tat-toe, which is played by very young boys with slate and pencil, this jingle is used:--

Tit, tat, toe, my first go:

Three jolly butcher boys all in a row;

Stick one up, stick one down.

Stick one on the old man's crown."

Baron Tassilo von Heydebrand und von der Lasa. Ueber die griechischen und römischen Spiele, welche einige ähnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234, 257-264. ??NYS -- described on Fiske 121-122 & 137, who says van der Linde I 40-47 copies much of it. Von der Lasa asserts that the Parva Tabella of Ovid is Kleine Mühle (Three Men's Morris). He says the game is called Tripp, Trapp, Trull in the Swedish book Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57) -- ??NYS. Van der Linde says that the Dutch name is Tik, Tak, Tol. Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses.

C. Babbage. Passages from the Life of a Philosopher. 1864. Chapter XXXIV -- section on Games of Skill, pp. 465-471. (= pp. 152-156 in: Charles Babbage and His Calculating Engines, Dover, 1961.) Partial analysis. He calls it tit-tat-to.

The Play Room: or, In-door Games for Boys and Girls. Dick & Fitzgerald(?), 1866. [Reprinted as: How to Amuse an Evening Party. Dick & Fitzgerald, NY, 1869.] ??NX -- the 1869 was seen at Shortz's. P. 22: Tit-tat-to. Uses O and +. "This is a game that small boys enjoy, and some big ones who won't own it."

Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty illustrations. Routledge, London, nd. HPL gives c1850, but the text is clearly derived from Every Boy's Book, whose first edition was 1856. But the main part of the text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), but is in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously revised edition, with Edmund Routledge as editor. So this may be c1868. This is the first published use of the term Noughts and Crosses found so far -- the OED's 1861 quote is to Oughts and Crosses..

Pp. 46-47: Slate games: Noughts and crosses. "This is a capital game, and one which every school-boy truly enjoys." Though the example shown is a draw, there is no mention of the fact that the game should always be a tie.

Pp. 85-86: Nine-holes. This has nine holes in a row and each player has a hole. The ball is rolled to them and the person in whose hole it lands must run and pick up the ball and try to hit one of the others who are running away. So this has nothing to do with our games or other forms of Nine Holes.

P. 106: Nine-holes or Bridge-board. This has nine holes in an upright board and the object is get one's marbles through the holes. (This material is in the 1856 ed. of Every Boy's Book.)

Correspondent to Notes and Queries (1875) ??NYS -- quoted by Strutt-Cox 257. Describes a game called Three Mans' Marriage [sic] in Derbyshire which seems to be Noughts and Crosses played on a crossed square board. Pieces are not described as moving, but in the next description of a Nine Men's Morris, they are specifically described as moving. However, the use of a crossed square board may indicate that diagonals were not considered.

Cassell's. 1881. Slate Games: Noughts and Crosses, or Tit-Tat-To, p. 84, with cross reference under Tit-Tat-To, p. 87. = Manson, 1911, pp. 202-203 & 208.

Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883. Vol. I, p. 162++. ??NYS -- quoted and described in Fiske 134-136. Description of Tripp, Trapp, Trull, with winning cry: "Tripp, trapp, trull, min qvarn är full." (Qvarn = mill.)

Lucas. RM2, 1883. Pp. 73-99. Analysis of Three Men's Morris, on a board with the main diagonals drawn, with moves of only one square along a winning line. He shows this is a first person game. If the first player is not permitted to play in the centre, then it is a tie game. No mention of Tic-Tac-Toe.

Albert Ellery Berg, ed. The Universal Self-Instructor. Thomas Kelly, NY, 1883. Tit-tat-to, p. 379. Brief description.

Mark Twain. The Adventures of Huckleberry Finn. 1884. Chap. XXXIV, about half-way through. "It's as simple as tit-tat-toe, three-in-a-row, ..., Huck Finn."

"A wrangler and late master at Harrow school." The science of naughts and crosses. Boy's Own Paper 10: (No. 498) (28 Jul 1888) 702-703; (No. 499) (4 Aug 1888) 717; (No. 500) (11 Aug 1888) 735; (No. 501) (18 Aug 1888) 743. Exhaustive analysis, including odds of second player making a correct response to each opening. For first move in: middle, side, corner, the odds of a correct response are: 1/2, 1/2, 1/8. He implies that the analysis is not widely known.

"Tom Wilson". Illustred Spelbok (in Swedish). Nd [late 1880s??]. ??NYS -- described by Fiske 136-137. This gives Tripp, Trapp, Trull as a Three Men's Morris game on the crossed square, with moves "according to one way of playing, to whatever points they please, but according to another, only to the nearest point along the lines on which the pieces stand. This last method is always employed when the board has, in addition to the right lines, or lines joining the middles of the exterior lines, also diagonals connecting the angles". He then describes a drawn version using the # board and 0 and + (or 1 and 2 in the North) which seems to be genuinely Noughts and Crosses. Fiske says the book seems to be based on an early edition of the Encyclopédie des Jeux or a similar book, so it is uncertain how much the above represents the current Swedish game. Fiske was unable to determine the author's real name, though he was still living in Stockholm at the time.

Il Libro del Giuochi. Florence, 1894. ??NYS -- described in Fiske, pp. 109-110. Gives doubly crossed square board and mentions a Three Men's Morris game.

T. de Moulidars. Grande Encyclopédie des Jeux. Montgredien or Librairie Illustree, Paris, nd. ??NYS -- Fiske 115 (in 1905) says it appeared 'not very long ago' and that Gelli seems to be based on it. Fiske quotes the clear description of Three Men's Morris as Marelle Simple, using a doubly crossed square, saying that pieces move to adjoining cells, following a line, and that the first player should win if he plays in the centre. Fiske notes that Noughts and Crosses is not mentioned.

J. Gelli. Come Posso Divertirmi? Milan, 1900. ??NYS -- described in Fiske 107. Fiske quotes the description of Three Men's Morris as Mulinello Semplice, essentially a translation from Moulidars.

Dudeney. CP. 1907. Prob. 109: Noughts and crosses, pp. 156 & 248. (c= MP, prob. 202: Noughts and crosses, pp. 89 & 175-176. = 536, prob. 471: Tic tac toe, pp. 185 & 390-392. Asserts the game is a tie, but gives only a sketchy analysis. MP gives a reasonably exhaustive analysis. Looks at Ovid's game.

A. C. White. Tit-tat-toe. British Chess Magazine (Jul 1919) 217-220. Attempt at a complete analysis, but has a mistake. See Gardner, SA (Mar 1957) = 1st Book, chap. 4.

D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and Playthings, pp. 148-165. On p. 160, he quotes Ovid and says it is Noughts and Crosses, or in Ireland, Tip-top-castle.

The Home Book of Quizzes, Games and Jokes. Blue Ribbon Books, NY, 1941. This is excerpted from several books -- this material is most likely taken from: Clement Wood & Gloria Goddard; Complete Book of Games; same publisher, nd [late 1930s]. P. 150: Tit-tat-toe, noughts and crosses. Brief description of the game on the # board. "To win requires great ingenuity."

G. E. Felton & R. H. Macmillan. Noughts and Crosses. Eureka 11 (Jan 1949) 5-9. Mentions Dudeney's work on the 3 x 3 board and asks for generalizations. Mentions pegotty = go-bang. Then studies the 4 x 4 x 4 game -- see 4.B.1.a. Adds some remarks on pegotty, citing Falkener, Lucas and Tarry.

Stanley Byard. Robots which play games. Penguin Science News 16 (Jun 1950) 65-77. On p. 73, he says D. W. Davies, of the National Physical Laboratory, has built, and exhibited to the Royal Society in May 1949, an electro-mechanical noughts and crosses machine. A photo of the machine is plate 16. He also mentions Babbage's interest in such a machine and an 1874 paper to the US National Academy by a Dr. Rogers -- ??NYS.

P. C. Parks. Building a noughts and crosses machine. Eureka 14 (Oct 1951) 15-17. Cites Babbage, Rogers, Davies, Byard. Parks built a simple machine with wire and tin cans in 1950 at a cost of about £6. Says G. Eastell of Thetford, Norfolk, built a machine using sixty valves for the Festival of Britain.

Gardner. Ticktacktoe. SA (Mar 1957) c= 1st Book, chap. 4. Quotes Wordsworth, discusses Three Men's Morris (citing Ovid) and its variants (including versions on 4 x 4 and 5 x 5 boards), the misère version (the person who makes three in a row loses), three and n dimensional forms (giving L. Moser's result on the number of winning lines on a kn board), go-moku, Babbage's proposed machine, A. C. White's article. Addendum mentions the Opies' assertion that the name comes from the rhyme starting "Tit, tat, toe, My first go,".

C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. A ticktacktoe machine, pp. 384-385. Noel Elliott gives a brief description of a relay logic machine to play the game.

Donald Michie. Trial and error. Penguin Science Survey 2 (1961) 129-145. ??NYS. Describes his matchbox and bead learning machine, MENACE (Matchbox educable noughts and crosses engine), for the game.

Gardner. A matchbox game-learning machine. SA (Mar 1962) c= Unexpected, chap. 8. Describes Michie's MENACE. Says it used 300 matchboxes. Gardner adapts it to Hexapawn, which is much simpler, requiring only 24 matchboxes. Discusses other games playable by 'computers'. Addendum discusses results sent in by readers including other games.

Barnard. 50 Observer Brain-Twisters. 1962. Prob. 34: Noughts and crosses, pp. 39-40, 64 & 93-94. Asserts there are 1884 final winning positions. He doesn't consider equivalence by symmetry and he allows either player to start.

Donald Michie & R. A. Chambers. Boxes: an experiment in adaptive control. Machine Intelligence 2 (1968) 136-152. Discusses MENACE, with photo of the pile of boxes. Says there are 288 boxes, but doesn't explain exactly how he found them. Chambers has implemented MENACE as a general game-learning computer program using adaptive control techniques designed by Michie. Results for various games are given.

S. Sivasankaranarayana Pillai. A pastime common among South Indian school children. In: P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and Reminiscences; 2: Ramanujan -- An Inspiration; Muthialpet High School, Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 2, pp. 81-85. [Taken from Mathematics Student, but no date or details given -- ??] Shows ordinary tic-tac-toe is a draw and considers trying to get t in a row on an n x n board. Shows that n = t ( 3 is a draw and that if t ( n + 1 - ((n/6), then the game is a draw.

L. A. Graham. The Surprise Attack in Mathematical Problems. Dover, 1968. Tic-tac-toe for gamblers, prob. 8, pp. 19-22. Proposed by F. E. Clark, solutions by Robert A. Harrington & Robert E. Corby. Find the probability of the first player winning if the game is played at random. Two detailed analyses shows that the probabilities for first player, second player, tie are (737, 363, 160)/1260.

[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY, 1975. P. 143: Ha-ho-ha. Misère version of noughts and crosses proposed. No discussion.

Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Noughts and Crosses, pp. 11-12. Asserts "It's been played all around the world for hundreds, if not thousands, of years ...." I've included it as a typical example of popular belief about the game. = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, Tic-Tac-Toe, pp. 11-12.

Winning Ways. 1982. Pp. 667-680. Complete and careful analysis, including various uncommon traps. Several equivalent games. Discusses extensions of board size and dimension.

Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Tic-tac-toe squared, pp. 88-89. Get 3 in a row on the 4 x 4 board. Also considers Tic-tac-toe-toe -- get 4 in a row on 5 x 5 board.

George Markovsky. Numerical tic-tac-toe -- I. JRM 22:2 (1990) 114-123. Describes and studies two versions where the moves are numbered 1, 2, .... One is due to Ron Graham, the other to P. H. Nygaard and Markowsky sketches the histories.

Ira Rosenholtz. Solving some variations on a variant of tic-tac-toe using invariant subsets. JRM 25:2 (1993) 128-135. The basic variant is to avoid making three in a row on a 4 x 4 board. By playing symmetrically, the second player avoids losing and 252 of the 256 centrally symmetric positions give a win for the second player. Extends analysis to 2n x 2n, 5 x 5, 4 x 4 x 4, etc.

Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24-25 & 33 (Feb 1994) 32. This describes a sliding piece puzzle on the doubly crossed square board -- see under 5.A.

See: Yuri I. Averbakh; Board games and real events; 1995; in 5.R.5, for a possible connection.

4.B.1.a IN HIGHER DIMENSIONS

C. Planck. Four-fold magics. Part 2 of chap. XIV, pp. 363-375, of W. S. Andrews, et al.; Magic Squares and Cubes; 2nd ed., Open Court, 1917; Dover, 1960. On p. 370, he notes that the number of m-dimensional directions through a cell of the n-dimensional board is the m-th term of the binomial expansion of ½(1+2)n.

Maurice Wilkes says he played 3-D noughts and crosses at Cambridge in the late 1930s, but the game was to get the most lines on a 3 x 3 x 3 board. I recall seeing a commercial version, called Plato?, of this in 1970.

Cedric Smith says he played 3-D and 4-D versions at Cambridge in the early 1940s.

Arthur Stone (letter to me of 9 Aug 1985) says '3 and 4 dimensional forms of tic-tac-toe produced by Brooks, Smith, Tutte and myself', but it's not quite clear if they invented these. Tutte became expert on the 43 board and thought it was a first person game. They only played the 54 game once - it took a long time.

Funkenbusch & Eagle, National Mathematics Mag. (1944) ??NYR.

G. E. Felton & R. H. Macmillan. Noughts and crosses. Eureka 11 (1949) 5-9. They say they first met the 4 x 4 x 4 game at Cambridge in 1940 and they give some analysis of it, with tactics and problems.

William Funkenbusch & Edwin Eagle. Hyper-spacial tit-tat-toe or tit-tat-toe in four dimensions. National Mathematics Magazine 19:3 (Dec 1944) 119-122. ??NYR

A. L. Rubinoff, proposer; L. Moser, solver. Problem E773 -- Noughts and crosses. AMM 54 (1947) 281 & 55 (1948) 99. Number of winning lines on a kn board is {(k+2)n - kn}/2. Putting k = 1 gives Planck's result.

L. Buxton. Four dimensions for the fourth form. MG 26 (1964) 38-39. 3 x 3 x 3 and 3 x 3 x 3 x 3 games are obviously first person, but he proposes playing for most lines and with the centre blocked on the 3 x 3 x 3 x 3 board. Suggests 3n and 4 x 4 x 4 games.

Anon. Puzzle page: Noughts and crosses. MTg 33 (1965) 35. Says practice shows that the 4 x 4 x 4 game is a draw. [I only ever had one drawn game!] Conjectures nn is first player and (n+1)n is a draw.

Roland Silver. The group of automorphisms of the game of 3-dimensional ticktacktoe. AMM 74 (1967) 247-254. Finds the group of permutations of cells that preserve winning lines is generated by the rigid motions of the cube and certain 'eviscerations'. [It is believed that this is true for the kn board, but I don't know of a simple proof.]

Ross Honsberger. Mathematical Morsels. MAA, 1978. Prob. 13: X's and O's, p. 26. Obtains L. Moser's result.

Kathleen Ollerenshaw. Presidential Address: The magic of mathematics. Bull. Inst. Math. Appl. 15:1 (Jan 1979) 2-12. P. 6 discusses my rediscovery of L. Moser's 1948 result.

Paul Taylor. Counting lines and planes in generalised noughts and crosses. MG 63 (No. 424) (Jun 1979) 77-82. Determines the number pr(k) of r-sections of a kn board by means of a recurrence pr(k) = [pr-1(k+2) - pr-1(k)]/2r which generalises L. Moser's 1948 result. He then gets an explicit sum for it. Studies some other relationships. This work was done while he was a sixth form student.

Oren Patashnik. Qubic: 4 x 4 x 4 tic-tac-toe. MM 53 (1980) 202-216. Computer assisted proof that 4 x 4 x 4 game is a first player win.

Winning Ways. 1982. Pp. 673-679, esp. 678-679. Discusses getting k in a row on a n x n board. Discusses 43 game (Tic-Toc-Tac-Toe) and kn game.

Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London, 1991.) Chapter 7: Conceptual conflict in multi-dimensional space, pp. 80-94 (1991: 98-112) & answers on pp. 99, 100, 106 & 131 (1991: 115, 116, 122 & 147). He considers various higher dimensional noughts and crosses on the 33, 34 and 35 boards. He finds that there are 49 winning lines on the 33 and he finds how to determine the number of d-facets on an n-cube as the coefficients in the expansion of (2x + 1)n. He also considers games where one has to complete a 3 x 3 plane to win and gives a problem: OXO three hypercube planes, p. 91 (1991: 109) & Answer 29, p. 106 (1991: 122) which asks for the number of planes in the hypercube 34. The answer says there are 123 of them, but in 1985 I found 154 and the general formula for the number of d-sections of a kn board. When I wrote to Serebriakoff, he responded that he could not follow the mathematics and that "I arrived at the figures ... from a simple formula published in one of Art [sic] Gardner's books which checked out as far as I could take it. Several other mathematicians have looked through it and not disagreed." I wrote for a reference to Gardner but never had a response. I presented my work to the British Mathematical Colloquium at Cambridge on 2 Apr 1985 and discovered that the results were known -- I had found the explicit sum given by Taylor above, but not the recurrence.

4.B.2. HEX

David Fielker sent some pages from a Danish book on games, but the TP is not present in his copies, so we don't have details. This says that Hein introduced the game in a lecture to students at the Institute for Theoretical Physics (now the Niels Bohr Institute) in Copenhagen in 1942. After its appearance in Politiken, specially printed pads for playing the game were sold, and a game board was marketed in the US as Hex in 1952.

Piet Hein. Article or column in Politiken (Copenhagen) (26 Dec 1942). ??NYR, but the diagrams show a board of hexagons.

Gardner (1957) and others have related that the game was independently invented by John Nash at Princeton in 1948-1949. Gardner had considerable correspondence after his article which I have examined. The key point is that one of Niels Bohr's sons, who had known the game in Copenhagen, was a visitor at the Institute for Advanced Study at the time and showed it to friends. I concluded that it was likely that some idea of the game had permeated to Nash who had forgotten this and later recalled and extensively developed the idea, thinking it was new to him. I met Harold Kuhn in 1998, who was a student with Nash at the time and he has no doubt that Nash invented the idea. In particular, Nash started with the triangular lattice, i.e. the dual of Hein's board, for some time before realising the convenience of the hexagonal lattice. Nash came to Princeton as a graduate student in autumn 1948 and had invented the game by the spring of 1949. Kuhn says he observed Nash developing the ideas and recognising the connections with the Jordan Curve Theorem, etc. Kuhn also says that there was not much connection between students at Princeton and at the Institute and relates that von Neumann saw the game at Princeton and asked what it was, indicating that it was not well known at the Institute. In view of this, it seems most likely that Nash's invention was independent, but I know from my own experience that it can be difficult to remember the sources of one's ideas -- a casual remark about a hexagonal game could have re-emerged weeks or months later when Nash was studying games, as the idea of looking at hexagonal boards in some form, from which the game would be re-invented. Sylvester was notorious for publishing ideas which he had actually refereed or edited some years earlier, but had completely forgotten the earlier sources. In situations like Hex, we will never know exactly what happened -- even if we were present at the time, it is difficult to know what is going on in the mind of the protagonist and the protagonist himself may not know what subconscious connections his mind is making. Even if we could discover that Nash had been told something about a hexagonal game, we cannot tell how his mind dealt with this information and we cannot assume this was what inspired his work. In other words, even a time machine will not settle such historical questions -- we need something that displays the conscious and the unconscious workings of a person's mind.

Parker Brothers. Literature on Hex, 1952. ??NYS or NYR.

Claude E. Shannon. Computers and automata. Proc. Institute of Radio Engineers 41 (Oct 1953) 1234-1241. Describes his Hex machine on p. 1237.

M. Gardner. The game of Hex. SA (Jul 1957) = 1st Book, chap. 8. Description of Shannon's 8 by 7 'Hoax' machine, pp. 81-82, and its second person strategy, p. 79.

Anatole Beck, Michael N. Bleicher & Donald W. Crowe. Excursions into Mathematics. Worth Publishers, NY, 1969. Chap. 5: Games (by Beck), Section 3: The game of Hex, pp. 327-339 (with photo of Hein on p. 328). Says it has been attributed to Hein and Nash. At Yale in 1952, they played on a 14 x 14 board. Shows it is a first player win, invoking the Jordan Curve Theorem

David Gale. The game of Hex and the Brouwer fixed-point theorem. AMM 86:10 (Dec 1979) 818-827. Shows that the non-existence of ties (Hex Theorem) is equivalent to the Brouwer Fixed-Point Theorem in two and in n dimensions. Says the use of the Jordan Curve Theorem is unnecessary.

Winning Ways. 1982. Pp. 679-680 sketches the game and the strategy stealing argument which is attributed to Nash.

C. E. Shannon. Photo of his Hoax machine sent to me in 1983.

Cameron Browne. Hex Strategy: Making the Right Connections. A. K. Peters, Natick, Massachusetts, 2000.

4.B.3. DOTS AND BOXES

Lucas. Le jeu de l'École Polytechnique. RM2, 1883, pp. 90-91. He gives a brief description, starting: "Depuis quelques années, les élèves de l'École Polytechnique ont imaginé un nouveaux jeu de combinaison assez original." He clearly describes drawing the edges of the game board and that the completer of a box gets to go again. He concludes: "Ce jeu nous a paru assez curieux pour en donner ici la description; mais, jusqu'a présent, nous ne connaissons pas encore d'observations ni de remarques assez importantes pour en dire davantage."

Lucas. Nouveaux jeux scientifiques de M. Édouard Lucas. La Nature 17 (1889) 301-303. Clearly describes a game version of La Pipopipette on p. 302, picture on p. 301, "... un nouveau jeu ... dédié aux élèves de l'école Polytechnique." This is dots and boxes with the outer edges already drawn in.

Lucas. L'Arithmétique Amusante. 1895. Note III: Les jeux scientifiques de Lucas, pp. 203-209 -- includes his booklet: La Pipopipette, Nouveau jeu de combinaisons, Dédié aux élèves de l'École Polytechnique, Par un Antique de la promotion de 1861, (1889), on pp. 204-208. On p. 207, he says the game was devised by several of his former pupils at the École Polytechnique. On p. 37, he remarks that "Pipo est la désignation abrégée de Polytechnique, par les élèves de l'X, ...."

Robert Marquard & Georg Frieckert. German Patent 108,830 -- Gesellschaftsspiel. Patented: 15 Jun 1899. 1p + 1p diagrams. 8 x 8 array of boxes on a board with slots for inserting edges. No indication that the player who completes a box gets to play again. They have some squares with values but also allow all squares to have equal value.

C. Ganse. The dot game. Ladies' Home Journal (Jun 1903) 41. Describes the game and states that one who makes a box gets to go again.

Loyd. The boxer's puzzle. Cyclopedia, 1914, pp. 104 & 352. = MPSL1, prob. 91, pp. 88-89 & 152-153. c= SLAHP: Oriental tit-tat-toe, pp. 28 & 92-93. Loyd doesn't start with the boundaries drawn. He asserts it is 'from the East'.

Ahrens. A&N. 1918. Chap. XIV: Pipopipette, pp. 147-155, describes it in more detail than Lucas does. He says the game appeared recently.

Blyth. Match-Stick Magic. 1921. Boxes, pp. 84-85. "The above game is familiar to most boys and girls ...." No indication that the completer of a box gets to play again.

Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930. Pp. 84-85: Die Käsekiste. Describes a version for two or more players. The first player must start at a corner and players must always connect to previously drawn lines. A player who completes a box gets to play again.

Meyer. Big Fun Book. 1940. Boxes, p. 661. Brief description, somewhat vaguely stating that a player who completes a box can play again.

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 151: Dots and squares. Clearly says the completer gets to play again. "The game calls for great ingenuity."

"Zodiastar". Fun with Matches and Match Boxes. (Cover says: Match Tricks From the 1880s to the 1940s.) Universal Publications, London, nd [late 1940s?]. The game of boxes, pp. 48-49. Starts by laying out four matches in a square and players put down matches which must touch the previous matches. Completing a box gives another play. No indication that matches must be on lattice lines, but perhaps this is intended.

Readers' Research Department. RMM 2 (Apr 1961) 38-41, 3 (Jun 1961) 51-52, 4 (Aug 1961) 52-55. On pp. 40-41 of No. 2, it says that Martin Gardner suggests seeking the best strategy. Editor notes there are two versions of the rules -- where the one who makes a box gets an extra turn, and where he doesn't -- and that the game can be played on other arrays. On p. 51 of No. 3, there is a symmetry analysis of the no-extra-turn game on a board with an odd number of squares. On pp. 52-54 of No. 4, there is some analysis of the extra-turn case on a board with an odd number of boxes.

Everett V. Jackson. Dots and cubes. JRM 6:4 (Fall 1973) 273-279. Studies 3-dimensional game where a play is a square in the cubical lattice.

Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Worm, pp. 18-19. This is a sort of 'anti-boxes' -- one draws segments on the lattice forming a path without any cycles -- last player wins. = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, pp. 18-19.

Winning Ways. 1982. Chap. 16: Dots-and-Boxes, pp. 507-550

David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 44-45 suggests playing on the triangular lattice.

Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987.

Eternal triangles, pp. 80-81. Gives the game on the triangular lattice.

Snakes, pp. 81-82. Same as Brandreth's Worm. I think 'snake' would be a better title as only one path is drawn.

4.B.4. SPROUTS

M. Gardner. SA (Jul 1967) = Carnival, chap. 1. Describes Michael Stewart Paterson and John Horton Conway's invention of the game on 21 Feb 1967 at tea time in the Department common room at Cambridge. The idea of adding a spot was due to Paterson and they agreed the credit for the game should be 60% Paterson to 40% Conway.

Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Sprouts, p. 13. "... actually born in Cambridge about ten years ago." c= Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, p. 13: "... was invented about ten years ago."

Winning Ways. 1982. Sprouts, pp. 564-570 & 573. Says the game was "introduced by M. S. Paterson and J. H. Conway some time ago". Also describes Brussels Sprouts and Stars-and-Stripes. An answer for Brussels Sprouts and some references are on p. 573.

Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Sprouts, pp. 95-97.

Karl-Heinz Koch. Pencil & Paper Games. (As: Spiele mit Papier und Bleistift, no details); translated by Elisabeth E. Reinersmann. Sterling, NY, 1992. Sprouts, pp. 36-37, says it was invented by J. H. Conway & M. S. Paterson on 21 Feb 1976 [sic -- misprint of 1967] during their five o'clock tea hour.

4.B.5. OVID'S GAME AND NINE MEN'S MORRIS

See also 4.B.1 for historical material.

The classic Nine Men's Morris board consists of three concentric squares with their midpoints joined by four lines. The corners are sometimes also joined by another four diagonal lines, but this seems to be used with twelve men per side and is sometimes called Twelve Men's Morris -- see 1891 below. Fiske 108 says this is common in America but infrequent in Europe, though on 127 he says both forms were known in England before 1600, and both were carried to the US, though the Nine form is probably older.

Murray 615 discusses Nine Men's Morris. He cites Kurna, Egypt (-14C), medieval Spain (Alquerque de Nueve), the Gokstad ship and the steps of the Acropolis of Athens. He says the board sometimes has diagonals added and then is played with 9, 11 or 12 pieces.

Dudeney. AM. 1917. Introduction to Moving Counter Problems, pp. 58-59. This gives a brief survey, mentioning a number of details that I have not seen elsewhere, e.g. its occurrence in Poland and on the Amazon. Says the board was found on a Roman tile at Silchester and on the steps of the Acropolis in Athens among other sites.

J. A. Cuddon. The Macmillan Dictionary of Sports and Games. Macmillan, London, 1980. Pp. 563-564. Discusses the history. Says there is a c-1400 board cut in stone at Kurna, Egypt and similar boards were made in years 9 to 21 at Mihintale, Ceylon. Says Ars Amatoria may be describing Three Men's Morris and Tristia may be describing a kind of Tic-tac-toe. Cites numerous medieval descriptions and variants.

Claudia Zaslavsky. Tic Tac Toe and Other Three-in-a-Row Games from Ancient Egypt to the Modern Computer. Crowell, NY, 1982. This is really a book for children and there are no references for the historical statements. I have found most of them elsewhere, and the author has kindly send me a list of source books, but I have not yet tracked down the following items -- ??.

There is an English court record of 1699 of punishment for playing Nine Holes in church.

There is a Nine Men's Morris board on a stone on the temple of Seti I (presumably this is at Kurna). There is a picture in the 13C Spanish 'Book of Games' (presumably the Alfonso MS -- see below) of children playing Alquerque de Tres (c= Three Men's Morris). A 14C inventory of the Duc de Berry lists tables for Mérelles (=? Nine Men's Morris) (see Fiske 113-115 below) and a book by Petrarch shows two apes playing the game.

H. Parker. Ancient Ceylon. Loc. cit. in 4.B.1. Nine Men's Morris board in the Temple of Kurna, Egypt, -14C. [Rohrbough, below, says this temple was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.] Two diagrams for Nine Men's Morris are cut into the great flight of steps at Mihintale, Ceylon and these are dated c1C. He cites Bell; Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for another diagram of similar age.

Jack Botermans, Tony Burrett, Pieter van Delft & Carla van Spluntern. The World of Games. (In Dutch, 1987); Facts on File, NY, 1989.

P. 35 describes Yih, a form of Three Men's Morris, played on a doubly crossed square with a man moving "one step along any line". A note adds that only the French have a rule forbidding the first player to play in the centre, which makes the game more challenging and is recommended.

Pp. 103-107 is the beginning of a section: Games of alignment and configuration and discusses various games, but rather vaguely and without references. They mention Al-Qurna, Mihintale, Gokstad and some other early sites. They say Yih was described by Confucius, was played c-500 and is "the game, that we now know as tic-tac-toe, or three men's morris." They describe Noughts and Crosses in the usual way. They then distinguish Tic-Tac-Toe, saying "In Britain it is generally known as three men's morris ...." and say it is the same as Yih, "which was known in ancient Egypt". They say "Ovid mentions tic-tac-toe" in Ars Amatoria, that several Roman boards have survived and that it was very popular in 14C England with several boards for this and Nine Men's Morris cut into cloister seats. They then describe Three-in-a-Row, which allows pieces to move one step in any direction, as a game played in Egypt. They then describe Five or Six Men's Morris, Nine Men's Morris, Twelve Men's Morris and Nine Men's Morris with Dice, with nice 13C & 15C illustration of Nine Men's Morris.

Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pp. 6-8. They discuss the crossed square board -- see 4.B.1 -- and describe Three Men's Morris with moves only along the lines to an adjacent vacant point. They then describe Achi, from Ghana, on the doubly crossed square with the same rules. They then describe Six Men's Morris which was apparently popular in medieval Europe but became obsolete by c1600.

Ovid. Ars Amatoria. -1. II, 203-208 & III, 353-366. Translated by J. H. Mozley; Loeb Classical Library, 1929, pp. 80-81 & 142-145. Translated by B. P. Moore, 1935, used in A. D. Melville; Ovid The Love Poems; OUP, 1990, pp. 113, 137, 229 & 241.

II, 203-208 are three couplets apparently referring to three games: two dice games and Ludus Latrunculorum. Mozley's prose translation is:

"If she be gaming, and throwing with her hand the ivory dice, do you throw amiss and move your throws amiss; or if is the large dice you are throwing, let no forfeit follow if she lose; see that the ruinous dogs often fall to you; or if the piece be marching under the semblance of a robbers' band, let your warrior fall before his glassy foe."

'Dogs' is the worst throw in Roman dice games.

Moore's verse translation of 207-208 is:

"And when the raiding chessmen take the field, Your champion to his crystal foe must yield."

Melville's note says the original has 'bandits' and says the game is Ludus Latrunculorum.

III, 357-360 is probably a reference to the same game since 'robbers' occurs again, though translated as brigands by Mozley, and again it immediately follows a reference to throwing dice. Mozley's translation of 353-366 is:

"I am ashamed to advise in little things, that she should know the throws of the dice, and thy powers, O flung counter. Now let her throw three dice, and now reflect which side she may fitly join in her cunning, and which challenge, Let her cautiously and not foolishly play the battle of the brigands, when one piece falls before his double foe and the warrior caught without his mate fights on, and the enemy retraces many a time the path he has begun. And let smooth balls be flung into the open net, nor must any ball be moved save that which you will take out. There is a sort of game confined by subtle method into as many lines as the slippery year has months: a small board has three counters on either side, whereon to join your pieces together is to conquer."

Moore's translation of 357-360 is:

"To guide with wary skill the chessmen's fight, When foemen twain o'erpower the single knight, And caught without his queen the king must face The foe and oft his eager steps retrace".

This is clearly not a morris game -- Mozley's note above and the next entry make it clear it is Ludus Latrunculorum, which had a number of forms. Mozley's note on pp. 142-143 refers to Tristia II, 478 and cites a number of other references for Ludus Latrunculorum.

Moore's translation of 363-366 is:

"A game there is marked out in slender zones As many as the fleeting year has moons; A smaller board with three a side is manned, And victory's his who first aligns his band."

Mozley's notes and Melville's notes say the first two lines refer to the Roman game of Ludus Duodecim Scriptorum -- the Twelve Line Game -- which is the ancestor of Backgammon. Mozley says the game in the latter two lines is mentioned in Tristia, "but we have no information about it." Melville says it is "a 'position' game, something like Nine Men's Morris" and cites R. C. Bell's article on 'Board and tile games' in the Encyclopaedia Britannica, 15th ed., Macropaedia ii.1152-1153, ??NYS.

Ovid. Tristia. c10. II, 471-484. Translated by A. L. Wheeler. Loeb Classical Library, 1945, pp. 88-91. This mentions several games and the text parallels that of Ars Amatoria III.

"Others have written of the arts of playing at dice -- this was no light sin in the eyes of our ancestors -- what is the value of the tali, with what throw one can make the highest point, avoiding the ruinous dogs; how the tessera is counted, and when the opponent is challenged, how it is fitting to throw, how to move according to the throws; how the variegated soldier steals to the attack along the straight path when the piece between two enemies is lost, and how he understands warfare by pursuit and how to recall the man before him and to retreat in safety not without escort; how a small board is provided with three men on a side and victory lies in keeping one's men abreast; and the other games -- I will not describe them all -- which are wont to waste that precious thing, our time."

A note says some see a reference to Ludus Duodecim Scriptorum at the beginning of this. The next note says the next text refers to Ludus Latrunculorum, a game on a squared board with 30 men on a side, with at least two kinds of men. The note for the last game says "This game seems to have resembled a game of draughts played with few men." and refers to Ars Amatoria and the German Mühlespiel, which he describes as 'a sort of draughts', but which is Nine Men's Morris.

R. G. Austin. Roman board games -- I & II. Greece and Rome 4 (No. 10) (Oct 1934) 24-34 & 4 (No. 11) (Feb 1935) 76-82. Claims the Ovid references are to Ludus Latrunculorum (a kind of Draughts?), Ludus Duodecim Scriptorum (later Tabula, an ancestor of Backgammon) and (Ars Amatoria.iii.365-366) a kind of Three Men's Morris. In the last, he shows a doubly crossed 3 x 3 board, but it is not clear which rule he adopts for the later movement of pieces, but he says: "the first player is always able to force a win if he places his first man on the centre point, and this suggests that the dice may have been used to determine priority of play, although there is no evidence of this." He says no Roman name for this game has survived. He discusses various known artifacts for all the game, citing several Roman 8 x 8 boards found in Britain. He gives an informal bibliography with comments as to the value of the works.

D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and Playthings, pp. 148-165. On p. 160, he quotes Ovid, Ars Amatoria.iii.365-366 and says it is Noughts and Crosses, or in Ireland, Tip-top-castle.

The British Museum has a Nine Men's Morris board from the Temple of Artemis, Ephesus, 2C-4C. Item BM GR 1872,8-3,44. This was in a small exhibition of board games in 1990. I didn't see it on display in late 1996.

Murray, p. 189. There was an Arabic game called Qirq, which Murray identifies with Morris. "Fourteen was a game played with small stones on a wooden board which had three rows of holes (al-Qâbûnî)." Abû-Hanîfa [the H should have a dot under it], c750, held that Fourteen was illegal and Qirq was held illegal by writers soon afterward. On p. 194, Murray gives a 10C passage mentioning Qirq being played at Mecca.

Fiske 255 cites the Kitāb al Aghāni, c960, for a reference to qirkat, i.e. morris boards.

Paul B. Du Chaillu. The Viking Age. Two vols., John Murray, London, 1889. Vol. II, p.168, fig. 992 -- Fragments of wood from Gokstad ship. Shows a partial board for Nine Men's Morris found in the Gokstad ship burial. There is no description of this illustration and there is only a vague indication that this is 10C, but other sources say it is c900.

Gutorm Gjessing. The Viking Ship Finds. Revised ed., Universitets Oldsaksamling, Oslo, 1957. P. 8: "... there are two boards which were used for two kinds of games; on one side figures appear for use in a game which is frequently played even now (known as "Mølle")."

Thorlief Sjøvold. The Viking Ships in Oslo. Universitets Oldsaksamling, Oslo, 1979. P. 54: "... a gaming board with one antler gaming piece, ...."

In medieval Europe, the game is called Ludus Marellorum or Merellorum or just Marelli or Merelli or Merels, meaning the game of counters. Murray 399 says the connection with Qirq is unclear. However, medieval Spain played various games called Alquerque, which is obviously derived from Qirq. Alquerque de Nueve seems to be Nine Men's Morris. However, in Italy and in medieval France, Marelle or Merels could mean Alquerque (de Doze), a draughts-like game with 12 men on a side played on a 5 x 5 board (Murray 615). Also Marro, Marella can refer to Draughts which seems to originate in Europe somewhat before 1400.

Stewart Culin. Korean Games, with Notes on the Corresponding Games of China and Japan. University of Pennsylvania, Philadelphia, 1895. Reprinted as: Games of the Orient; Tuttle, Rutland, Vermont, 1958. Reprinted under the original title, Dover and The Brooklyn Museum, 1991. P. 102, section 80: Kon-tjil -- merrells. This is the usual Nine Men's Morris. The Chinese name is Sám-k'i (Three Chess). "I am told by a Chinese merchant that this game was invented by Chao Kw'ang-yin (917-975), founder of the Sung dynasty." This is the only indication of an oriental source that I have seen.

Gerhard Leopold. Skulptierte Werkstücke in der Krypta der Wipertikirche zu Quedlinburg. IN: Friedrich Möbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann Böhlaus Nachfolger, Weimar, 1987, pp. 27-43; esp. pp. 37 & 43. Describes and gives photos of several Nine-Men's-Morris boards carved on a pillar of the crypt of the Wipertikirche, Quedlinburg, Sachsen-Anhalt, probably from the 10/11 C.

Richard de Fournivall. De Vetula. 13C. This describes various games, including Merels. Indeed the French title is: Ci parle du gieu des Merelles .... ??NYS -- cited by Murray, pp. 439, 507, 520, 628. Murray 620 cites several MSS and publications of the text.

"Bonus Socius" [Nicolas de Nicolaï?]. This is a collection of chess problems, compiled c1275, which exists in many manuscript forms and languages. See 5.F.1 for more details of these MSS. See Murray 618-642. On pp. 619-624 & 627, he mentions several MSS which include 23, 24, 25 or 28 Merels problems. On p. 621, he cites "Merelles a Neuf" from 14C. Fiske 104 & 110-111 discusses some MSS of this collection.

The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototropic Plates. 2 vols., Karl W. Horseman, Leipzig, 1913. (See in 4.A.1 for another ed.) This is a collection of chess problems produced for Alfonso X, the Wise, King of Castile (Castilla). Vol. 2, ff. 92v-93r, pp. CLXXXIV-CLXXXV, shows Nine Men's Morris boards. ??NX -- need to study text. See: Murray 568-573; van der Linde I 137 & 279 ??NYS & Quellenstudien 73 & 277-278, ??NYS (both cited by Fiske 98); van der Lasa 116, ??NYS (cited by Fiske 99).

Fiske 98-99 says that the MS also mentions Alquerque, Cercar de Liebre and Alquerque de Neuve (with 12 men against one). Fiske 253-255 gives a more detailed study of the MS based on a transcript. He also quotes a communication citing al Querque or al Kirk in Kazirmirski's Arabic dictionary and in the Kitāb al Aghāni, c960.

José Brunet y Bellet. El Ajedrez. Barcelona, 1890. ??NYS -- described by Fiske 98. This has a chapter on the Alfonso MS and refers to Alquerque de Doce, saying that it is known as Tres en Raya in Castilian and Marro in Catalan (Fiske 102 says this word is no longer used in Spanish). Brunet notes that there are five miniatures pertaining to alquerque. Fiske says that all this information leaves us uncertain as to what the games were. Fiske says Brunet's chapter has an appendix dealing with Carrera's 1617 discussion of 'line games' and describing Riga di Tre as the same as Marro or Tres en Raya as a form of Three Men's Morris

Murray gives many brief references to the game, which I will note here simply by his page number and the date of the item.

438-439 (12C); 446 (14C);

449 (c1400 -- 'un marrelier', i.e. a Merels board);

431 (c1430); 447 (1491); 446 (1538).

Anon. Romance of Alexander. 1338. (Bodleian Library, Mss Bodl. 264). ??NYS. Nice illustration clearly showing Nine Men's Morris board. I. Disraeli (Amenities of Literature, vol. I, p. 86) also cites British Museum, Bib. Reg. 15, E.6 as a prose MS version with illustrations. Prof. D. J. A. Ross tells me there is nothing in the text corresponding to the illustrations and that the Bodleian text was edited by M. R. James, c1920, ??NYS. Illustration reproduced in: A. C. Horth; 101 Games to Make and Play; Batsford, London, (1943; 2nd ed., 1944); 3rd ed., 1946; plate VI facing p. 44, in B&W. Also in: Pia Hsiao et al.; Games You Make and Play; Macdonald and Jane's, London, 1975, p. 7, in colour.

Fiske 113-115 gives a number of quotations from medieval French sources as far back as mid 14C, including an inventory of the Duc de Berry in 1416 listing two boards. Fiske notes that the game has given rise to several French phrases. He quotes a 1412 source calling it Ludus Sanct Mederici or Jeu Saint Marry and also mentions references in city statutes of 1404 and 1414.

MS, Montpellier, Faculty of Medicine, H279 (Fonts de Boulier, E.93). 14C. This is a version of the Bonus Socius collection. Described in Murray 623-624, denoted M, and in van der Linde I 301, denoted K. Lucas, RM2, 1883, pp. 98-99 mentions it and RM4, 1894, Quatrième Récréation: Le jeu des mérelles au XIIIe siècle, pp. 67-85 discusses it extensively. This includes 28 Merels problems which are given and analysed by Lucas. Lucas dates the MS to the 13C.

Household accounts of Edward IV, c1470. ??NYS -- see Murray 617. Record of purchase of "two foxis and 46 hounds" to form two sets of "marelles".

Civis Bononiae [Citizen of Bologna]. This is a collection of chess problems compiled c1475, which exists in several MSS. See Murray 643-703. It has 48 or 53 merels problems. On p. 644, 'merelleorum' is quoted.

A Hundred Sons. Chinese scroll of Ming period (1368-1644). 18C copy in BM. ??NYS -- extensively reproduced and described in: Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977. On p. 12 of Fawdry is a scene, apparently from the scroll, in which some children appear to be playing on a Twelve Men's Morris board.

Elaborate boards from Germany (c1530) and Venice (16C) survive in the National Museum, Munich and in South Kensington (Murray 757-758). Murray shows the first in B&W facing p. 757.

William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98-100: "The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green For lack of tread are indistinguishable." Fiske 126 opines that the latter two lines may indicate that the board was made in the turf, though he admits that they may refer just to dancers' tracks, but to me it clearly refers to turf mazes.

J. C. Bulenger. De Ludis Privatis ac Domesticus Veterum. Lyons, 1627. ??NYS Fiske 115 & 119 quote his description of and philological note on Madrellas (Three Men's Morris).

Paul Fleming (1609-1640). In one of his lyrics, he has Mühlen. ??NYS -- quoted by Fiske 132, who says this is the first German mention of Morris.

Fiske 133 gives the earliest Russian reference to Morris as 1675.

Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. Historia Triodii, pp. 202-214, is on morris games. (Described in Fiske 118-124, who says there is further material in the Elenchus at the end of the volume -- ??NYS) Hyde asserts that the game was well known to the Romans, though he cannot find a Roman name for it! He cites and discusses Bulenger, but disagrees with his philology. Gives lots of names for the game, ranging as far as Russian and Armenian. He gives both the Nine and Twelve Men's Morris boards on p. 210, but he has not found the Twelve board in Eastern works. On p. 211, he gives the doubly crossed square board with a title in Chinese characters, pronounced 'Che-lo', meaning 'six places', and having three white and three black men already placed along two sides. He says the Irish name is Cashlan Gherra (Short Castle) and that the name Copped Crown is common in Cumberland and Westmorland. He then describes playing the Twelve Man and Nine Man games, and then he considers the game on the doubly crossed square board. He seems to say there are different rules as to how one can move. ??need to study the Latin in detail. This is said to throw light on the Ovid passages. Hyde believes the game was well known to the Romans and hence must be much older. Fiske remarks that this is history by guesswork.

Murray 383 describes Russian chess. He says Amelung identifies the Russian game "saki with Hölzchenspiel (?merels)". Saki is mentioned on this page as being played at the Tsar's court, c1675.

Archiv der Spiele. 3 volumes, Berlin, 1819-1821. Vol. 2 (1820) 21-27. ??NYS Described and quoted by Fiske 129-132. This only describes the crossed square and the Nine Men's Morris boards. It says that the Three Men's Morris on the crossed square board is a tie, i.e. continues without end, but it is not clear how the pieces are allowed to move. Fiske says this gives the most complete explanation he knows of the rules for Nine Men's Morris.

Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205. ??NX. For more details, see 4.B.1. On ff. 347.r-347.v, 8 Sep 1848, he suggests Nine Men's Morris boards in triangular and pentagonal shapes and does various counting on the different shapes.

The Family Friend (1856) 57. Puzzle 17. -- Two and a Bushel. Shows the standard # board. "This very simple and amusing games, -- which we do not remember to have seen described in any book of games, -- is played, like draughts, by two persons with counters. Each player must have three, ... and the game is won when one of the players succeeds in placing his three men in a row; ...." There is no specification of how the men move. The word 'bushel' occurs in some old descriptions of Three Men's Morris and Nine Men's Morris as the name of the central area.

The Sociable. 1858. Merelles: or, nine men's morris, pp. 279-280. Brief description, notable for the use of Merelles in an English book.

Von der Lasa. Ueber die griechischen und römischen Spiele, welche einige ähnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234, 257-264. ??NYS -- described on Fiske 121-122 & 137, who says van der Linde I 40-47 copies much of it. He asserts that the Parva Tabella of Ovid is Kleine Mühle (Three Men's Morris). Von der Lasa says the game is called Tripp, Trapp, Trull in the Swedish book Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57??). Van der Linde says that the Dutch name is Tik, Tak, Tol. Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses.

Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883. Vol. I, p. 162++. ??NYS -- quoted and described in Fiske 134-136. Plays Nine Men's Morris on a Twelve Men's Morris board.

Webster's Dictionary. 1891. ??NYS -- Fiske 118 quotes a definition (not clear which) which includes "twelve men's morris". Fiske says: "Here we have almost the only, and certainly the first mention of the game by its most common New England name, "twelve men's morris," and also the only hint we have found in print that the more complicated of the morris boards -- with the diagonal lines ... -- is used with twelve men, instead of nine, on each side." Fiske 127 says the name only appears in American dictionaries.

Dudeney. CP. 1907. Prob. 110: Ovid's game, pp. 156-157 & 248. Says the game "is distinctly mentioned in the works of Ovid." He gives Three Men's Morris, with moves to adjacent cells horizontally or diagonally, and says it is a first player win.

Blyth. Match-Stick Magic. 1921. Black versus white, pp. 79-80. 4 x 4 board with four men each. But the men must be initially placed WBWB in the first row and BWBW in the last row. They can move one square "in any direction" and the object is to get four in a row of your colour.

Games and Tricks -- to make the Party "Go". Supplement to "Pearson's Weekly", Nov. 7th, no year indicated [1930s??]. A matchstick game, p. 11. On a 4 x 4 board, place eight men, WBWB on the top row and BWBW on the bottom row. Players alternately move one of their men by one square in any direction -- the object is to make four in a line.

Lynn Rohrbough, ed. Ancient Games. Handy Series, Kit N, Cooperative Recreation Service, Delaware, Ohio, (1938), 1939.

Morris was Player [sic] 3,300 Years Ago, p. 27. Says the temple of Kurna was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.

Three Men's Morris, p. 27. After placing their three men, players 'then move trying to get three men in a row.' Contributor says he played it in Cardiff more than 50 years ago.

Winning Ways. 1982. Pp. 672-673. Says Ovid's Game is conjectured to be Three Men's Morris. The current version allows moves by one square orthogonally and is a first person win if the first person plays in the centre. If the first player cannot play in the centre, it is a draw. They use Three Men's Morris for the case with one step moves along winning lines, i.e. orthogonally or along main diagonals. An American Indian game, Hopscotch, permits one step moves orthogonally or diagonally (along any diagonal). A French game, Les Pendus, allows any move to a vacant cell. All of these are draws, even allowing the first player to play in the centre. They briefly describe Six and Nine Men's Morris.

Ralph Gasser & J. Nievergelt. Es ist entscheiden: Das Muehle-Spiel ist unentscheiden. Informatik Spektrum 17 (1994) 314-317. ??NYS -- cited by Jörg Bewersdorff [email of 6 Jun 1999].

L. V. Allis. Beating the World Champion -- The state of the art in computer game playing. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 155-175. On p. 163, he states that Ralph Gasser showed that Nine Men's Morris is a draw in Oct 1993, but the only reference is to a letter from Gasser.

Ralph Gasser. Solving Nine Men's Morris. IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1996, pp. 101-113. ??NYS -- cited by Bewersdorff [loc. cit.] and described in William Hartston; What mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2. Demonstrates that Nine Men's Morris is a draw. Gasser's abstract: "We describe the combination of two search methods used to solve Nine Men's Morris. An improved analysis algorithm computes endgame databases comprising about 1010 states. An 18-ply alpha-beta search the used these databases to prove that the value of the initial position is a draw. Nine Men's Morris is the first non-trivial game to be solved that does not seem to benefit from knowledge-based methods." I'm not sure about the last statement -- 4 x 4 x 4 noughts and crosses (see 4.B.1.a) and Connect-4 were solved in 1980 and 1988, though the first was a computer aided proof and the original brute force solution of Connect-4 by James Allen in Sep 1988 was improved to a knowledge-based approach by L. V. Allis by Aug 1989. The five-in-a-row version of Connect-4 was shown to be a first person win in 1993. Bewersdorff [email of 11 Jun 1999] clarifies this by observing that draw here means a game that continues forever -- one cannot come to a stalemate where neither side can move.

4.B.6. PHUTBALL

Winning Ways. 1982. Philosopher's football, pp. 688-691. In 1985, Guy said this was the only published description of the game.

4.B.7. BRIDG-IT

This is best viewed as played on a n x n array of squares. The n(n+1) vertical edges belong to one player, say red, while the n(n+1) horizontal edges belong to black. Players alternate marking a square with a line of their colour between edges of their colour. A square cannot be marked twice. The object is to complete a path across the board. In practice, the edges are replaced by coloured dots which are joined by lines. As with Hex, there can be no ties and there must be a first person strategy.

M. Gardner. SA (Oct 1958) c= 2nd Book, Chap. 7. Introduces David Gales's game, later called Bridg-it. Addendum in the book notes that it is identical to Shannon's 'Bird Cage' game of 1951 and that it was marketed as Bridg-it in 1960.

M. Gardner. SA (Jul 1961) c= New MD, Chap. 18. Describes Oliver Gross's simple strategy for the first player to win. Addendum in the book gives references to other solutions and mentions.

M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Article says Bridg-it was still on the market.

Winning Ways. 1982. Pp. 680-682. Covers Bridg-it and Shannon Switching Game.

In Oct 2000, I bought a second-hand copy of a 5 x 5 version called Connections, attributed to Tom McNamara, but with no date.

4.B.8. CHOMP

Fred Schuh. Spel van delers (Game of divisors). Nieuw Tijdschrift vor Wiskunde 39 (1951-52) 299-304. ??NYS -- cited by Gardner, below.

M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Gives David Gale's description of his game and results on it. Addendum in Knotted points out that it is equivalent to Schuh's game and gives other references.

David Gale. A curious Nim-type game. AMM 81 (1974) 876-879. Describes the game and the basic results. Wonders if the winning move is unique. Considers three dimensional and infinite forms. A note added in proof refers to Gardner's article, says two programmers have consequently found that the 8 x 10 game has two winning first moves and mentions Schuh's game.

Winning Ways. 1982. Pp. 598-600. Brief description with extensive table of good moves. Cites an earlier paper of Gale and Stewart which does not deal with this game.

4.B.9. SNAKES AND LADDERS

I have included this because it has an interesting history and because I found a nice way to express it as a kind of Markov process or random walk, and this gives an expression for the average time the game lasts. I then found that the paper by Daykin et al. gives most of these ideas.

The game has two or three rules for finishing.

A. One finishes by going exactly to the last square, or beyond it.

B. One finishes by going exactly to the last square. If one throws too much, then one stands still.

C. One finishes by going exactly to the last square. If one throws too much, one must count back from the last square. E.g., if there are 100 squares and one is at 98 and one throws 6, then one counts: 99, 100, 99, 98, 97, 96 and winds up on 96. (I learned this from a neighbour's child but have only seen it in one place -- in the first Culin item below.)

Games of this generic form are often called promotion games. If one considers the game with no snakes or ladders, then it is a straightforward race game, and these date back to Egyptian and Babylonian times, if not earlier.

In fact, the same theory applies to random walks of various sorts, e.g. random walks of pieces on a chessboard, where the ending is arrival exactly at the desired square.

In the British Museum, Room 52, Case 24 has a Babylonian ceramic board (WA 1991-7.20,I) for a kind of snakes and ladders from c-1000. The label says this game was popular during the second and first millennia BC.

Sheng-kuan t'u [The game of promotion]. 7C. Chinese game. This is described in: Nagao Tatsuzo; Shina Minzoku-shi [Manners and Customs of the Chinese]; Tokyo, 1940-1942, perhaps vol. 2, p. 707, ??NYS This is cited in: Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977, p. 183, where the game is described as "played on a board or plan representing an official career from the lowest to the highest grade, according to the imperial examination system. It is a kind of Snakes and Ladders, played with four dice; the object of each player being to secure promotion over the others."

Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De ludo promotionis mandarinorum, pp. 70-101 -- ??NX. This is a long description of Shing quon tu, a game on a board of 98 spaces, each of which has a specific description which Hyde gives. There is a folding plate showing the Chinese board, but the copy in the Graves collection is too fragile to photocopy. I did not see any date given for the game.

Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum for 1893, pp. 489-537. Pp. 502-507 describes several versions of the Japanese Sugoroku (Double Sixes) which is a generic name for games using dice to determine moves, including backgammon and simple race games, as well as Snakes and Ladders games. One version has ending in the form C. Then says Shing Kún T’ò (The Game of the Promotion of Officials) is described by Hyde as The Game of the Promotion of the Mandarins and gives an extended description of it. There is a legend that the game was invented when the Emperor Kienlung (1736-1796) heard a candidate playing dice and the candidate was summoned to explain. He made up a story about the game, saying that it was a way for him and his friends to learn the different ranks of the civil service. He was sent off to bring back the game and then made up a board overnight. However Hyde had described the game a century before this date. It seems that this is not really a Snakes and Ladders game as the moves are determined by the throw of the dice and the position -- there are no interconnections between cells. But Culin notes that the game is complicated by being played for money or counters which permit bribery and rewards, etc.

Culin. Chess and Playing Cards. Op. cit. in 4.A.4. 1898.

Pp. 820-822 & plates 24 & 25 between 821 & 822. Says Shing Kún T’ò (The Game of the Promotion of Officials) is described by Hyde as The Game of the Promotion of the Mandarins and refers to the above for an extended description. Describes the Korean version: Tjyong-Kyeng-To (The Game of Dignitaries) and several others from Korea and Tibet, with 108, 144, 169 and 64 squares.

Pp. 840-842 & plate 28, opp. p. 841 describes Chong ün Ch’au (Game of the Chief of the Literati) as 'in many respects analogous' to Shing Kún T’ò and the Japanese game Sugoroku (Double Sixes) -- in several versions. Then mentions modern western versions -- Jeu de L'Oie, Giuoco dell'Oca, Juego de la Oca, Snake Game. Pp. 843-848 is a table listing 122 versions of the game in the University of Pennsylvania Museum of Archaeology and Paleontology. These are in 11 languages, varying from 22 to 409 squares.

Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Snakes and Ladders and the Chinese Promotion Game, pp. 65-67. They describe the Hindu version of Snakes and Ladders, called Moksha-patamu. Then they discuss Shing Kun t'o (Promotion of the Mandarins), which was played in the Ming (1368-1616) with four or more players racing on a board with 98 spaces and throwing 6 dice to see how many equal faces appeared. They describe numerous modern variants.

Deepak Shimkhada. A preliminary study of the game of Karma in India, Nepal, and Tibet. Artibus Asiae 44 (1983) 4. ??NYS - cited in Belloli et al, p. 68.

Andrew Topsfield. The Indian game of snakes and ladders. Artibus Asiae 46:3 (1985) 203-214 + 14 figures. Basically a catalogue of extant Indian boards. He says the game is called Gyān caupad [the d should have an underdot] or Gyān chaupar in Hindi. He states that Moksha-patamu sounds like it is Telugu and that this name appeared in Grunfield's Games of the World (1975) with no reference to a source and that Bell has repeated this. Game boards were drawn or painted on paper or cloth and hence were perishable. The oldest extant version is believed to be an 84 square board of 1735, in the Museum of Indology, Jaipur. There were Hindu, Jain, Muslim and Tibetan versions representing a kind of Pilgrim's Progress, finally arriving at God or Heaven or Nirvana. The number of squares varies from 72 to 360.

He gives many references and further details. An Indian version of the game was described by F. E. Pargiter; An Indian game: Heaven or Hell; J. Royal Asiatic Soc. (1916) 539-542, ??NYS. He cites the version by Ayres (and Love's reproduction of it -- see below) as the first English version. He cites several other late 19C versions.

F. H. Ayres. [Snakes and ladders game.] No. 200682 Regd. Example in the Bethnal Green Museum, Misc. 8 - 1974. Reproduced in: Brian Love; Play The Game; Michael Joseph, London, 1978; Snakes & Ladders 1, pp. 22-23. This is the earliest known English version of the game, with 100 cells in a spiral and 5 snakes and 5 ladders.

N. W. Bazely & P. J. Davis. Accuracy of Monte Carlo methods in computing finite Markov chains. J. of Res. of the Nat. Bureau of Standards -- Mathematics and Mathematical Physics 64B:4 (Oct-Dec 1960) 211-215. ??NYS -- cited by Davis & Chinn and Bewersdorff. Bewersdorff [email of 6 Jun 1999] brought these items to my attention and says it is an analysis based on absorbing Markov chains.

D. E. Daykin, J. E. Jeacocke & D. G. Neal. Markov chains and snakes and ladders. MG 51 (No. 378) (Dec 1967) 313-317. Shows that the game can be modelled as a Markov process and works out the expected length of play for one player (47.98 moves) or two players (27.44 moves) on a particular board with finishing rule A.

Philip J. Davis & William G. Chinn. 3.1416 and All That. S&S, 1969, ??NYS; 2nd ed, Birkhäuser, 1985, chap. 23 (by Davis): "Mr. Milton, Mr. Bradley -- meet Andrey Andreyevich Markov", pp. 164-171. Simply describes how to set up the Markov chain transition matrix for a game with 100 cells and ending B. Doesn't give any results.

Lewis Carroll. Board game for one. In: Lewis Carroll's Bedside Book; ed. by Gyles Brandreth (under the pseud. Edgar Cuthwellis); Methuen, 1979, pp. 19-21. ??look for source; not in Carroll-Wakeling, Carroll-Wakeling II or Carroll-Gardner. Board of 27 cells with pictures in the odd cells. If you land on any odd cell, except the last one, you have to return to square 1. "Sleep is almost certain to have overwhelmed the player before he reaches the final square." Ending A is probably intended. (The average duration of this game should be computable.)

S. C. Althoen, L. King & K. Schilling. How long is a game of snakes and ladders? MG 77 (No. 478) (Mar 1993) 71-76. Similar analysis to Daykin, Jeacocke & Neal, using finishing rule B, getting 39.2 moves. They also use a simulation to find the number of moves is about 39.1.

David Singmaster. Letter [on Snakes and ladders]. MG 79 (No. 485) (Jul 1995) 396-397. In response to Althoen et al. Discusses history, other ending rules and wonders how the length depends on the number of snakes and ladders.

Irving L. Finkel. Notes on two Tibetan dice games. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 24-47. Part II: The Tibetan 'Game of Liberation', pp. 34-47, discusses promotion games with many references to the literature and describes a particular game.

Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Das Leiterspiel, pp. 67-68 & Das Leiterspiel als Markow-Kette. Discusses setting up the Markov chain, citing Bazley & Davis, with the same board as in Davis & Chinn, then states that the average duration is 39.224 moves.

Jay Belloli, ed. The Universe A Convergence of Art, Music, and Science. [Catalogue for a group of exhibitions and concerts in Pasadena and San Marino, Sep 2000 - Jun 2001.] Armory Center for the Arts, Pasadena, 2001. P. 68 has a discussion of the Jain versions of the game, called 'gyanbazi', with a colour plate of a 19C example with a 9 x 9 board with three extra cells.

4.B.10. MU TORERE

This is a Maori game which can be found in several books on board games. I have included it because it has been completely analysed. There are eight (or 2n) points around a central area. Each player has four (or n) markers, originally placed on consecutive points. One can move from a point to an adjacent point or to the centre, or one can move from the centre to a point, provided the position moved to is empty. The first player who cannot move is the loser. To prevent the game becoming trivial, it is necessary to require that the first two (or one) moves of each player involve his end pieces, though other restrictions are sometimes given.

Marcia Ascher. Mu Torere: An analysis of a Maori game. MM 60 (1987) 90-100. Analyses the game with 2n points. For n = 1, there are 6 inequivalent positions (where equivalence is by rotation or reflection of the board) and play is trivially cyclic. For n = 2, there are 12 inequivalent positions, but there are no winning positions. For n = 3, there are 30 inequivalent positions, some of which are wins, but the game is a tie. Obtains the number of positions for general n. For the traditional version with n = 4, there are 92 inequivalent positions, some of which are wins, but the game is a tie, though this is not at all obvious to an inexperienced player. In 1856, it was reported that no foreigner could win against a Maori. For n = 5, there are 272 inequivalent positions, but the game is a easy win for the first player -- the constraints on first moves need to be revised. Ascher gives references to the ethnographic literature for descriptions of the game.

Marcia Ascher. Ethnomathematics. Brooks/Cole Publishing, Pacific Grove, California, 1991. Sections 4.4-4.7, pp. 95-109 & Notes 4-7, pp. 118-119. Amplified version of her MM article.

4.B.11. MASTERMIND, ETC.

There were a number of earlier guessing games of the Mastermind type before the popular version devised by Marco Meirovitz in 1973 -- see: Reddi. One of these was the English Bulls and Cows, but I haven't seen anything written on this and it doesn't appear in Bell, Falkener or Gomme. Since 1975 there have been several books on the game and a number of papers on optimal strategies. I include a few of the latter.

NOTATION. If there are h holes and c choices at each hole, then I abbreviate this as ch.

A. K. Austin. How do You play 'Master Mind'. MTg 71 (Jun 1975) 46-47. How to state the rules correctly.

S. S. Reddi. A game of permutations. JRM 8:1 (1975) 8-11. Mastermind type guessing of a permutation of 1,2,3,4 can win in 5 guesses.

Donald E. Knuth. The computer as Master Mind. JRM 9:1 (1976-77) 1-6. 64 can be won in 5 guesses.

Robert W. Irving. Towards an optimum Mastermind strategy, JRM 11:2 (1978-79) 81-87. Knuth's algorithm takes an average of 5804/1296 = 4.478 guesses. The author presents a better strategy that takes an average of 5662/1296 = 4.369 guesses, but requires six guesses in one case. A simple adaptation eliminates this, but increases the average number of guesses to 5664/1296 = 4.370. An intelligent setter will choose a pattern with a single repetition, for which the average number of guesses is 3151/720 = 4.376.

A. K. Austin. Strategies for Mastermind. G&P 71 (Winter 1978) 14-16. Presents Knuth's results and some other work.

Merrill M. Flood. Mastermind strategy. JRM 18:3 (1985-86) 194-202. Cites five earlier papers on strategy, including Knuth and Irving. He considers it as a two-person game and considers the setter's strategy. He has several further papers in JRM developing his ideas.

Antonio M. Lopez, Jr. A PROLOG Mastermind program. JRM 23:2 (1991) 81-93. Cites Knuth, Irving, Flood and two other papers on strategy.

Kenji Koyama and Tony W. Lai. An optimal Mastermind strategy. JRM 25:4 (1994) 251-256. Using exhaustive search, they find the strategy that minimizes the expected number of guesses, getting expected number 5625/1296 = 4.340. However, the worst case in this problem requires 6 guesses. By a slight adjustment, they find the optimal strategy with worst case requiring 5 guesses and its expected number of guesses is 5626/1296 = 4.341. 10 references to previous work, not including all of the above.

Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Section 2.15 Mastermind: Auf Nummer sicher, pp. 227-234 & Section 3.13 Mastermind: Farbcodes und Minimax, pp. 316-319. Surveys the work on finding optimal strategies. Then studies Mastermind as a two-person game. Finds the minimax strategy for the 32 game and describes Flood's approach.

4.B.12. RITHMOMACHIA = THE PHILOSOPHERS' GAME

I have generally not tried to include board games in any comprehensive manner, but I have recently seen some general material on this which will be useful to anyone interested in the game. The game is one of the older and more mathematical of board games, dating from c1000, but generally abandoned about the end of the 16C along with the Neo-Pythagorean number theory of Boethius on which the game was based.

Arno Borst. Das mittelalterliche Zahlenkampfspiel. Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse 5 (1986) Supplemente. Available separately: Carl Winter, Heidelberg, 1986. Edits the surviving manuscripts on the game. ??NYS -- cited by Stigter & Folkerts.

Detlef Illmer, Nora Gädeke, Elisabeth Henge, Helen Pfeiffer & Monika Spicker-Beck. Rhythmomachia. Hugendubel, Munich, 1987.

Jurgen Stigter. Emanuel Lasker: A Bibliography AND Rithmomachia, the Philosophers' Game: a reference list. Corrected, 1988 with annotations to 1989, 1 + 15 + 16pp preprint available from the author, Molslaan 168, NL-2611 CZ Delft, Netherlands. Bibliography of the game.

Jurgen Stigter. The history and rules of Rithmomachia, the Philosophers' Game. 14pp preprint available from the author, as above.

Menso Folkerts. 'Rithmimachia'. In: Die deutsche Litteratur des Mittelalters: Verfasserlexikon; 2nd ed., De Gruyter, Berlin, 1990; vol. 8, pp. 86-94. Sketches history and describes the 10 oldest texts.

Menso Folkerts. Die Rithmachia des Werinher von Tegernsee. In: Vestigia Mathematica, ed. by M. Folkerts & J. P. Hogendijk, Rodopi, Amsterdam, 1993, pp. 107-142. Discusses Werinher's work (12C), preserved in one MS of c1200, and gives an edition of it.

4.B.13. MANCALA GAMES

This is a very broad field and I will only mention a few early items. Four row mancala games are played in south and east Africa. Three row games are played in Ethiopia and adjacent parts of Somaliland. Two row games are played everywhere else in Africa, the Middle East and south and south-east Asia. See the standard books by R. C. Bell and Falkener for many examples. Many general books mention the game, but I only know a few specific books on the game -- these are listed first below.

One article says that game boards have been found in the pyramids of Khamit (-1580) and there are numerous old boards carved in rocks in several parts of Africa.

An anonymous article, by a member of the Oware Society in London, [Wanted: skill, speed, strategy; West Africa (16-22 Sep 1996) 1486-1487] lists the following names for variants of the game: Aditoe (Volta region of Ghana), Awaoley (Côte d'Ivoire), Ayo (Nigeria), Chongkak (Johore), Choro (Sudan), Congclak (Indonesia), Dakon (Philippines), Guitihi (Kenya), Kiarabu (Zanzibar), Madji (Benin), Mancala (Egypt), Mankaleh (Syria), Mbau (Angola), Mongola (Congo), Naranji (Sri Lanka), Qai (Haiti), Ware (Burkina Faso), Wari (Timbuktu), Warri (Antigua),

Stewart Culin. Mancala, The National Game of Africa. IN: US National Museum Annual Report 1894, Washington, 1896, pp. 595-607.

Chief A. O. Odeleye. Ayo A Popular Yoruba Game. University Press Ltd., Ibadan, Nigeria, 1979. No history.

Laurence Russ. Mancala Games. Reference Publications, Algonac, Michigan, 1984. Photocopy from Russ, 1995.

Kofi Tall. Oware The Abapa Version. Kofi Tall Enterprise, Kumasi, Ghana, 1991.

Salimata Doumbia & J. C. Pil. Les Jeux de Cauris. Institut de Recherches Mathématiques, 08 BP 2030, Abidjan 08, Côte d'Ivoire, 1992.

Pascal Reysset & François Pingaud. L'Awélé. Le jeu des semailles africaines. 2nd ed., Chiron, Paris, 1995 (bought in Dec 1994). Not much history.

François Pingaud. L'awélé jeu de strategie africain. Bornemann, 1996.

Alexander J. de Voogt. Mancala Board Games. British Museum Press, 1997. ??NYR.

Larry (= Laurence) Russ. The Complete Mancala Games Book How to Play the World's Oldest Board Games. Foreword by Alex de Voogt. Marlowe & Co., NY, 2000. His 1984 book is described as an earlier edition of this.

William Flinders Petrie. Objects of Daily Use. (1929); Aris & Phillips, London??, 1974. P. 55 & plate XLVII. ??NYS -- described with plate reproduced in Bell, below. Shows and describes a 3 x 14 board from Memphis, ancient Egypt, but with no date given, but Bell indicates that the context implies it is probably earlier than -1500. Petrie calls it 'The game of forty-two and pool' because of the 42 holes and a large hole on the side, apparently for storing pieces either during play or between games.

R. C. Bell. Games to Play. Michael Joseph (Penguin), 1988. Chap. 4, pp. 54-61, Mancala games. On pp. 54-55, he shows the ancient Egyptian board from Petrie and his own photo of a 3 x 6 board cut into the roof of a temple at Deir-el-Medina, probably about -87.

Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo Mancala, pp. 226-232. Have X of part of this.

R. H. Macmillan. Wari. Eureka 13 (Oct 1950) 12. 2 x 6 board with each cup having four to start. Says it is played on the Gold Coast.

Vernon A. Eagle. On some newly described mancala games from Yunnan province, China, and the definition of a genus in the family of mancala games. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 48-62. Discusses the game in general, with many references. Attempts a classification in general. Describes six forms found in Yunnan.

Ulrich Schädler. Mancala in Roman Asia Minor? Board Games Studies International Journal for the Study of Board Games 1 (1998) 10-25. Notes that mancala could have been played on a flat board of two parallel rows of squares, i.e. something like a 2 x n chessboard, but that archaeologists have tended to view such patterns as boards for race games, etc. Describes 52 examples from Asia Minor. Some general discussion of Greek and Roman games.

John Romein & Henri E. Bal (Vrije Universiteit, Amsterdam). New computer cluster solves 3500-year old game. Posted on on 29 Aug 2002. They show that Awari is a tie game. They determined all 889,063,398,406 possible positions and stored them in a 778 GByte database. They then used a 144 processor cluster to analyse the graph, which 'only' took 51 hours.

4.B.14. DOMINOES, ETC.

R. C. Bell. Games to Play. 1988. Op. cit. in 4.B.13. P. 136 gives some history. The Académie Français adopted the word for both the pieces and the game in 1790 and it was generally thought that they were an 18C invention. However, a domino was found on the Mary Rose, which sank in 1545, and a record of Henry VIII (reigned 1509-1547) losing £450 at dominoes has been found.

Bell, p. 131, describes the modern variant Tri-Ominos which are triangular pieces with values at the corners. They were marketed c1970 and marked © Pressman Toy Corporation, NY.

Hexadoms are hexagonal pieces with numbers on the edges -- opposite edges have the same numbers. These were also marketed in the early 1970s -- I have a set made by Louis Marx, Swansea, but there is no date on it.

4.B.15. SVOYI KOSIRI

Anonymous [R. S. & J. M. B[rew ?]]. Svoyi kosiri is an easy game. Eureka 16 (Oct 1953) 8-12. This is an intriguing game of pure strategy commonly played in Russia and introduced to Cambridge by Besicovitch. It translates roughly as 'One's own trumps'. There are two players and the hands are exposed, with one's spades and clubs being the same as the other's hearts and diamonds. At Cambridge, the cards below 6 are removed, leaving 36 cards in the deck. The article doesn't explain how trumps are chosen, but if one has spades as trumps, then the other has hearts as trumps! Players alternate playing to a central discard pile. A player can take the pile and start a new pile with any card, or he can 'cover' the top card and then play any card on that. 'Covering' is done by playing a higher card of the same suit or one of the player's own trumps -- if this cannot be done, e.g. if the ace of the player's own trumps has been played, the player has to take the pile. The object is to get rid of all one's cards.

5. COMBINATORIAL RECREATIONS

7.AZ is actually combinatorial rather than arithmetical and I may shift it.

5.A. THE 15 PUZZLE, ETC.

Pictorial versions: The Premier (1880), Lemon (1890), Stein (1898), King (1927).

Double-sided versions: The Premier (1880), Brown (1891).

Relation to Magic Squares: Loyd (1896), Cremer (1880), Tissandier (1880 & 1880?), Cassell's (1881), Hutchison (1891).

Making a magic square with the Fifteen Puzzle: Dudeney (1898), Anon & Dudeney (1899), Loyd (1914), Dudeney (1917), Gordon (1988). See also:  Ollerenshaw & Bondi in 7.N.

GENERAL

Peter Hajek. 1995 report of his 1992 visit to the Museum of Money, Montevideo, Uruguay, with later pictures by Jaime Poniachik. In this Museum is a metal chest made in England in 1870 for the National State Bank of Uruguay. The front has a 7 x 7 array of metal squares with bolt heads. These have to be slid in a 12 move sequence to reveal the three keyholes for opening the chest. This opens up a whole new possible background for the 15 Puzzle -- can anyone provide details of other such sliding devices?

S&B, pp. 126-129, shows several versions of the puzzle.

L. Edward Hordern. Sliding Piece Puzzles. OUP, 1986. Chap. 2: History of the sliding block puzzle, pp. 18-30. This is the most extensive survey of the history. He concludes that Loyd did not invent the general puzzle where the 15 pieces are placed at random, which became popular in 1879(?). Loyd may have invented the 14-15 version or he may have offered the $1000 prize for it, but there is no evidence of when (1881??) or where. However, see the entries for Loyd's Tit-Bits article and Dudeney's 1904 article which seem to add weight to Loyd's claims. Most of the puzzles considered here are described by Hordern and have code numbers beginning with a letter, e.g. E23, which I will give.

I contributed a note about computer techniques of solving such puzzles and hoping that programmers would attack them as computer power increased.

In 1993-1995, I produced four Sliding Block Puzzle Circulars, totalling 24 pages (since reformatted to 21), largely devoted to reporting on computer solutions of puzzles in Hordern. Since then, a large number of solution programs have appeared and many more puzzles have appeared. The best place to look is on Nick Baxter's Sliding Block Home Page: .

EARLY ALPHABETIC VERSIONS

Embossing Co. Puzzle labelled "No. 2 Patent Embossed puzzle of Fifteen and Magic Sixteen. Manufactured by the Embossing Co. Patented Oct 24 1865". Illustrated in S&B, p. 127. Examples are in the collections of Slocum and Hordern. Hordern, p. 25, says that searching has not turned up such a patent.

Edward F. [but drawing gives E.] Gilbert. US Patent 91,737 -- Alphabetical Instruction Puzzle. Patented 22 Jun 1869. 1p + 1p diagrams. Described by Hordern, p. 26. This is not really a puzzle -- it has the sliding block concept, but along several tracks and with many blank spaces. I recall a similar toy from c1950.

Ernest U. Kinsey. US Patent 207,124 -- Puzzle-Blocks. Applied: 22 Nov 1877; patented: 20 Aug 1878. 2pp + 1p diagrams. Described by Hordern, p. 27. 6 x 6 square sliding block puzzle with one vacant space and tongue & grooving to prevent falling out. Has letters to spell words. He suggests use of triangular and diamond-shaped pieces. This seems to be the most likely origin of the Fifteen Puzzle craze.

Montgomery Ward & Co. Catalogue. 1889. Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 34. Spelling Boards. Like Gilbert's idea, but a more compact layout.

LOYD

Loyd prize puzzle: One hundred pounds. Tit-Bits (14 Oct 1893) 25 & (18 Nov 1893) 111. Loyd is described as "author of "Fifteen Puzzle," ...."

Loyd. Tit-Bits 31 (24 Oct 1896) 57. Loyd asserts he developed the 15 puzzle from a 4 x 4 magic square. "[The fifteen block puzzle] had such a phenomenal run some twenty years ago. ... There was one of the periodical revivals of the ancient Hindu "magic square" problem, and it occurred to me to utilize a set of movable blocks, numbered consecutively from 1 to 16, the conditions being to remove one of them and slide the others around until a magic square was formed. The "Fifteen Block Puzzle" was at once developed and became a craze.

I give it as originally promulgated in 1872 ..." and he shows it with the 15 and 14 interchanged. "The puzzle was never patented" so someone used round blocks instead of square ones. He says he would solve such puzzles by turning over the 6 and the 9. "Sphinx" [= Dudeney] says he well remembers the sensation and hopes "Mr. Loyd is duly penitent."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Fifteen Puzzle" that in 1872 and 1873 was sold by millions, .... When this puzzle was brought out by its inventor, Mr. Sam Loyd, ... he thought so little of it that he did not even take any steps to protect his idea, and never derived a penny profit from it.... We have recently tried all over the metropolis to obtain a single example of the puzzle, without success." Dudeney says the puzzle came with 16 pieces and you removed the 16. He also says he recently could not find a single example in London.

Loyd. The 14-15 puzzle in puzzleland. Cyclopedia, 1914, pp. 235 & 371 (= MPSL1, prob. 21, pp. 19-20 & 128). He says he introduced it 'in the early seventies'. One problem asks to move from the wrong position to a magic square with sum = 30 (i.e. the blank is counted as 0). This is c= SLAHP, pp. 17-18 & 89.

G. G. Bain. Op. cit. in 1, 1907. Story of Loyd being unable to patent it.

Anonymous & Sam Loyd. Loyd's puzzles, op. cit. in 1, 1896. Loyd "owns up to the great sin of having invented the "15 block puzzle"", but doesn't refer to the patent story or the date.

W. P. Eaton. Loc. cit. in 1, 1911. Loyd refers to it as the 'Fifteen block' puzzle, but doesn't say he couldn't patent it.

Loyd Jr. SLAHP. 1928. Pp. 1-3 & 87. "It was in the early 80's, ... that the world-disturbing "14-15 Puzzle" flashed across the horizon, and the Loyds were among its earliest victims." He gives many of the stories in the Cyclopedia and two of the same problems. He doesn't mention the patent story.

THE 15 PUZZLE

W. W. Johnson. Notes on the 15-Puzzle -- I. Amer. J. Math. 2 (1879) 397-399.

W. E. Story. Notes on the 15-Puzzle -- II. Ibid., 399-404.

J. J. Sylvester. Editorial comment. Ibid., 404.

(This issue may have been delayed to early 1880?? Johnson & Story are not terribly readable, but Sylvester is interesting, asserting that this is the first time that the parity of a permutation has become a popular concept.)

Anonymous. Untitled editorial. New York Times (23 Feb 1880) 4. "... just now the chief amusement of the New York mind, ... a mental epidemic .... In a month from now, the whole population of North America will be at it, and when the 15 puzzle crosses the seas, it is sure to become an English mania."

Anonymous. EUREKA! The Popular but Perplexing Problem Solved at Last. "THIRTEEN -- FOURTEEN -- FIFTEEN" New York Herald (28 Feb 1880) 8. ""Fifteen" is a puzzle of seeming simplicity, but is constructed with diabolical cunning. At first sight the victim feels little or no interest; but if he stops for a single moment to try it, or to look at any one else who is trying it, the mania strikes him. ... As to the last two numbers, it depends entirely upon the way in which the blocks happen to fall in the first place .... Two or three enterprising gamblers took up the puzzle and for a time made an excellent living.... The subject was brought up in the Academy of Sciences by the veteran scientist Dr. P. H. Vander Weyde", who showed it could not be solved. The Herald reporter discovered that the problem is solvable if one turns the board 90o, i.e. runs the numbers down instead of across, and Vander Weyde was impressed. The article implies the puzzle had already been widely known for some time.

Mary T. Foote. US Patent 227,159 -- Game apparatus. Filed: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. The patent is for a box with sliding numbered blocks for teaching the multiplication tables. Lines 57-63: "I am aware that it is not novel to produce a game apparatus in which blocks are to be mixed and then replaced by a series of moves; also, that it is not novel to number such blocks, as in the "game of 15," so called, where the fifteen numbers are first mixed and then moved into place."

Persifor Frazer Jr. Three methods and forty-eight solutions of the Fifteen Problem. Proc. Amer. Philos. Soc. 18 (1878-1880) 505-510. Meeting of 5 Mar 1880. Rather cryptic presentation of some possible patterns. Asserts his 26 Feb article in the Bulletin (??NYS -- ??where -- Philadelphia??) was the first "solution for the 13, 15, 14 case".

J. A. Wales. 15 - 14 - 13 -- The Great Presidential Puzzle. Puck 7 (No. 158) (17 Mar 1880) back cover.

Anonymous. Editorial: "Fifteen". New York Times (22 Mar 1880) 4. "No pestilence has ever visited this or any other country which has spread with the awful celerity of what is popularly called the "Fifteen Puzzle." It is only a few months ago that it made its appearance in Boston, and it has now spread over the entire country." Asserts that an unregenerate Southern sympathiser has introduced it into the White House and thereby disrupted a meeting of President Hayes' cabinet.

Sch. [H. Schubert]. The Boss Puzzle. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 82 (6 Apr 1880) 11, with response on 87 (11 Apr 1880) 12 (Sprechsaal). Gives a fairly careful description of odd and even permutations and shows the puzzle is solvable if and only if it is in an even permutation. The response is signed X and says that when the problem is insoluble, just turn the box by 90o to see another side of the problem!

Gebr. Spiro, Hofliefer (Court supplier), Jungfernsteig 3(?--hard to read), Hamburg. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 88 (13 Apr 1880) 7. Advertises Boss Puzzles: "Kaiser-Spiel 50Pf. Bismarck-Spiel 50 Pf. Spiel der 15 u. 16, 50 Pf. Spiel der 16 separat, 15 Pf. System und Lösung, 20 Pf."

G. W. Warren. Letter: Clew to the Fifteen Puzzle. The Nation 30 (No. 774) (29 Apr 1880) 326.

Anon. Shavings. The London Figaro (1 May 1880) 12. "The "15 Puzzle," which has for some months past been making a sensation in New York equal to that aroused by "H. M. S. Pinafore" last year, has at length reached this country, and bids fair to become the rage here also." (Complete item!)

George Augustus Sala. Echoes of the Week. Illustrated London News 76 (No. 2138) (22 May 1880) 491.

Mary T. Foote. US Patent 227,159 -- Game Apparatus. Applied: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. Described in Hordern, p. 27. 3 x 12 puzzles based on multiplication tables. Refers to the "game of 15" and Kinsey.

Arthur Black. ?? Brighton Herald (22 May 1880). ??NYS -- mentioned by Black in a letter to Knowledge 1 (2 Dec 1881) 100.

Anonymous. Our latest gift to England. From the London Figaro. New York Times (11 Jun 1880) 2(?). ??page

The Premier. First (?) double-sided version, with pictures of Gladstone and Beaconsfield, apparently produced for the 1880 UK election. Described in Hordern, pp. 32-33 & plate I.

Ahrens. MUS II 227. 1918. Story of Reichstag being distracted in 1880.

P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664-665. Brief but valid analysis. Mentions Johnson & Story. First mention of the possibility of a 3D version.

T. P. Kirkman. Question 6489 and Note on the solution of the 15-puzzle in question 6489. Mathematical Questions with their Solutions from the Educational Times 34 (1880) 113-114 & 35 (1881) 29-30. The question considers the n x n problem. The note is rather cryptic. (No use??)

Messrs. Cremer (210 Regent St. and 27 New Bond St., London). Brilliant Melancholia. Albrecht Durer's Game of the Thirty Four and "Boss" Game of the Fifteen. 1880. Small booklet, 16pp + covers, apparently instructions to fit in a box with pieces numbered 1 to 16 to be used for making magic squares as well as for the 15 puzzle. Explains that only half the positions of the 15 puzzle are obtainable and describes them by examples. (Photo in The Hordern Collection of Hoffmann Puzzles, p. 74, and in Hordern, op. cit. above, plate IV.) Possibly written by "Cavendish" (Henry Jones).

H. Schubert. Theoretische Entscheidung über das Boss-Puzzle Spiel. 2nd ed., Hamburg, 1880. ??NYS (MUS, II, p. 227)

Gaston Tissandier. Les carrés magiques -- à propos du "Taquin," jeu mathématique. La Nature 8 (No. 371) (10 Jul 1880) 81-82. Simple description of the puzzle called 'Taquin' which came from America and has had a very great success for several weeks. Says it had 16 squares and was usable as a sliding piece puzzle or a magic square puzzle. Cites Frénicle's 880 magic squares of order 4.

Anon. & C. Henry. Gaz. Anecdotique Littéraire, Artistique et Bibliographique. (Pub. by G. d'Heylli, Paris) Year 5, t. II, 1880, pp. 58-59 & 87-92. ??NYS

Piarron de Mondésir. Le dernier mot du taquin. La Nature 8 (No. 382) (25 Sep 1880) 284-285. Simple description of parity decision for the 15 puzzle. Says 'la Presse illustrée' offered 500 francs for achieving the standard pattern from a random pattern, but it was impossible, or rather it was possible in only half the cases.

Jasper W. Snowdon. The "Fifteen" Puzzle. Leisure Hour 29 (1880) 493-495.

Gwen White. Antique Toys. Batsford, London, 1971; reprinted by Chancellor Press, London, nd [1982?]. On p. 118, she says: "The French game of Taquin was played in 1880, in which 15 pieces had to be moved into 16 compartments in as few moves as possible; the word 'taquin' means 'a teaser'." She gives no references.

Tissandier. Récréations Scientifiques. 1880?

2nd ed., 1881 -- unlabelled section, pp. 143-153. As: Le taquin et les carrés magiques; seen in 1883 ed., ??NX; 1888: pp. 208-215. Adapted from the 1880 La Nature articles of Tissandier and de Mondésir. 1881 says it came from America -- 'récemment une nouvelle apparition', but this is dropped in 1888 -- otherwise the two versions are the same.

Translated in Popular Scientific Recreations, nd [c1890], pp. 731-735. Text says "Mathematical games, ..., have recently obtained a new addition .... ... from America, ...." The references to contemporary reactions are deleted and the translation is confused. E.g. the newspaper is now just "a French paper" and the English says the problem is impossible in nine cases out of ten!

Lucas. Récréations scientifiques sur l'arithmétique et sur la géométrie de situation. Sixième récréation: Sur le jeu du taquin ou du casse-tête américain. Revue scientifique de France et de l'étranger (3) 27 (1881) 783-788. c= Le jeu du taquin, RM1, 1882, pp. 189-211. Revue says that Sylvester told him that it was invented 18 months ago by an American deaf-mute. RM1 says "vers la fin de 1878". Cf Schubert, 1895.

Cassell's. 1881. Pp. 96-97: American puzzles "15" and "34". = Manson, pp. 246-248. Says "articles ... have appeared in many periodicals, but no one has ... publish[ed] a solution." Then sketches the parity concept and its application.

Richard A. Proctor. The fifteen puzzle. Gentlemen's Magazine 250 (No. 1801) (1881) 30-45.

"Boss". Letter: The fifteen puzzle. Knowledge 1 (11 Nov 1881) 37-38, item 13. This magazine was edited by Proctor. The letter starts: "I am told that in a magazine article which appeared some time ago, you have attempted to show that there are positions in the Fifteen Puzzle from which the won position can never be obtained." I suspect the letter was produced by Proctor. The response is signed Ed. and begins: "I thought the Fifteen Puzzle was dead, and hoped I had had some share in killing the time-absorbing monster." Notes that many people get to the position starting blank, 1, 2, 3 and view this as a win. Sketches parity argument and suggests "Boss" work on the 3 x 3 or 3 x 2 or even the 2 x 2 version.

Editorial comment. The fifteen puzzle. Knowledge 1 (25 Nov 1881) 79. "I supposed every one knew the Fifteen Puzzle." Proceeds to explain, obviously in response to readers who didn't know it.

Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (2 Dec 1881) 100, item 80. Sketches a proof which he says he published in the Brighton Herald of 22 May 1880.

"Yawnups". Letter: The fifteen puzzle. Knowledge 1 (30 Dec 1881) 185. Solution from the 15-14 position obtained by turning the box. Editorial comment says the solution uses 102 moves and the editor gets an easy solution in 57 moves. Adds that a 60 move solution has been received.

Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (13 Jan 1882) 230. Finds a solution from the 15-14 position in 39 moves by turning the box and asserts no shorter solution is possible. Says he also gave this in the Brighton Herald in May 1880. An addition says J. Watson has provided a similar solution, which takes 38 moves??

A. B. Letter: The fifteen puzzle. Knowledge 2 (20 Oct 1882) 345, item 598. Finds a box-turning solution in 39 moves.

C. J. Malmsten. Göteborg Handl 1882, p. 75. ??NYS -- cited by Ahrens in his Encyklopadie article, op. cit. in 3.B, 1904.

Anonymous. Enquire Within upon Everything. Houlston and Sons, London. This was a popular book with editions almost every year -- I don't know when the following material was added. Section 2591: Boss; or the Fifteen Puzzle, p. 363. Place the pieces 'indifferently' in the box. Half the positions are unsolvable. Cites Cavendish for the solution by turning the box 90o but notes this only works with round pieces. Goes on to The thirty-four puzzle, citing Dürer. I found this material in the 66th ed., 862nd thousand, of 1883, but I didn't find the material in the 86th ed of 1892.

Letters received and short answers. Knowledge 4 (16 Nov 1883) 310. 'Impossible'.

P. G. Tait. Listing's Topologie. Philosophical Mag. (5) 17 (No. 103) (Jan 1884) 30-46 & plate opp. p. 80. Section 11, p. 39. Simple but cryptic solution.

Letters received and short answers. Letter from W. S. B. asks how to solve the problem when the last row has 13, 14, 15 [sic!]; Answer by Ed. points out the misprint and says the easiest solution is to remove the 15 and put it after the 14, or to invert the 6 and 9. Knowledge 6 (No. 159) (14 Nov 1884) 412 & 6 (No. 160) (21 Nov 1884) 429.

Don Lemon. Everybody's Pocket Cyclopedia .... Saxon & Co., London, (1888), revised 8th ed., 1890. P. 137: The fifteen puzzle. Brief description, with pieces placed randomly in the box -- "to get the last three into order is often a puzzle indeed".

John D. Champlin & Arthur E. Bostwick. The Young Folk's Cyclopedia of Games and Sports. 1890. ??NYS Cited in Rohrbough; Brain Resters and Testers; c1935; Fifteen Puzzle, p. 20. Describes idea of parity of number of exchanges. [Another reference provided more details of Champlin & Bostwick.]

Lemon. 1890. A trick puzzle, no. 202, pp. 31 & 105 (= Sphinx, no. 422, pp. 60 & 112). 15 puzzle with lines on the pieces to arrange as "a representation of a president with only one eye". The solution is a spelling of the word 'president'. Attributed to Golden Days -- ??. After The Premier puzzle of c1880, this is the second suggestion of using a picture and the first publication of the idea that I have seen.

G. A. Hutchison, ed. Indoor Games and Recreations. The Boy's Own Bookshelf. (1888); New ed., Religious Tract Society, London, 1891. (See M. Adams; Indoor Games for a much revised version, but which doesn't contain this material.) Chap. 19: The American Puzzles., pp. 240-241. "These puzzles, known as the 'Thirty-four Game' and the 'Fifteen Game,' on their introduction amongst us some years ago ...." "The '15' puzzle would appear to have been, on its coming to England a few years ago, strictly a new introduction ...." He sketches the parity concept. [NOTE. I have seen a reference to the editor as Hutchinson, but the book definitely omits the first n.]

Daniel V. Brown. US Patent 471,941 -- Puzzle. Applied: 23 Apr 1891; patented: 29 Mar 1892, 2pp + 1p diagrams. Double-sided 16 block puzzle to spell George Washington on one side and Benjamin Harrison on the other. No sliding involved.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. American fifteen puzzle, pp. 105-107. "The Fifteen Puzzle was introduced by a shrewd American some ten years ago, ...." Refers to Tait's 1880 paper. Says half the positions are impossible, but solves them by turning the box 90o or by inverting the 6 and the 9.

Hoffmann. 1893. Chap IV, no. 69: The "Fifteen" or "Boss" puzzles, pp. 161-162 & 217-218 = Hoffmann-Hordern, pp. 142-144, with photo of five early examples, two or three of which also are thirty-four puzzles. (Hordern Collection, p. 74, has a photo of a version by Cremer, cf above.) "This, like a good many of the best puzzles, hails from America, where, some years ago, it had an extraordinary vogue, which a little later spread to this country, the British public growing nearly as excited over the mystic "Fifteen" as they did at a later date over the less innocent "Missing Word" competitions." He distinguishes between the ordinary Fifteen where one puts the pieces in at random, and the Boss or Master puzzle which has the 14 and 15 reversed. "Notwithstanding the enormous amount of energy that has been expended over the "Fifteen" Puzzle, no absolute rule for its solution has yet been discovered and it appears to be now generally agreed by mathematicians that out of the vast number of haphazard positions ... about half admit [of solution]. To test whether ... the following rule has been suggested." He then says to count the parity of the number of transpositions.

Hoffmann. 1893. Chap. IV, no. 70: The peg-away puzzle, pp. 163 & 218 = Hoffmann-Hordern, p. 145. This is a 3 x 3 version of the Fifteen puzzle, made by Perry & Co. Start with a random pattern and get to standard form. "The possibility of success in solving this puzzle appears to be governed by precisely the same rule as the "Fifteen" Puzzle." Hoffmann-Hordern has no photo of this -- do any examples exist??

H. Schubert. Zwölf Geduldspiele. Dümmler, Berlin, 1895. [Taken from his columns in Naturwissenschaftlichen Wochenschrift, 1891-1894.] Chap. VII: Boss-Puzzle oder Fünfzehner-Spiel, pp. 75-94?? Pp. 75-77 sketches the history, saying it was called "Jeu du Taquin" (Neck-Spiel) in France and was popular in 1879-1880 in Germany. Cites Johnson & Story and his own 1880 booklet. Gives the story of a deaf and dumb American inventing it in Dec 1878, saying "Sylvester communicated this at the annual meeting of the Association Française pour l'Avancement des Sciences at Reims". Cf Lucas, 1881. [There is a second edition, Teubner??, Leipzig, 1899, ??NYS. However this material is almost identical to the beginning of Chap. 15 in Schubert's Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1909, vol. 2. The later version omits only some of the Hamburg details of 1879-1880. Hence the 2nd ed. of Zwölf Geduldspiele is probably very close to these versions.]

Dudeney. Problem 49: The Victoria Cross puzzle. Tit-Bits 32 (4 & 25 Sep 1897) 421 & 475. = AM, 1917, prob. 218, pp. 60 & 194. B7. 3 x 3 board with letters Victoria going clockwise around the edges, leaving the middle empty, and starting with V in a corner. Slide to get Victoria starting at an edge cell, in the fewest moves. Does it in 18 moves, by interchanging the i's and says there are 6 such solutions.

Dudeney. Problem 65: The Spanish dungeon. Tit-Bits 33 (1 Jan & 5 Feb 1898) 257 & 355. = AM, 1917, prob. 403, pp. 122-123 & 244. B14. Convert 15 Puzzle, with pieces in correct order, into a magic square. Does it in 37 moves.

Conrad F. Stein. US Design 29,649 -- Design for a Game-Board. Applied: 29 Sep 1898; patented: 8 Nov 1898 as No. 692,242. 1p + 1p diagrams. This appears to be a 3 x 4 puzzle with a picture of a city with a Spanish flag on a tower. Apparently the object is to move an American flag to the tower.

Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. The eight fat boys. 3 x 3 square with pieces: 1 2 3; 4 X 5; 6 7 8 to be shifted into a magic square. Two solutions in 19 moves. Cf Dudeney, 1917.

Addison Coe. US Patent 785,665 -- Puzzle or Game Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Gives a 3 x 5 flat version and a 3-dimensional version -- cf 5.A.2.

Dudeney. AM. 1917.

Prob. 401: Eight jolly gaol birds, pp. 122 & 243. E23. Same as 'The eight fat boys' (see Anon. & Dudeney, 1899) with the additional condition that one person refuses to move, which occurs in one of the two previous solutions.

Prob. 403: The Spanish dungeon, pp. 122-123 & 244. = Tit-Bits prob. 65 (1898). B14.

Prob. 404: The Siberian dungeons, pp. 123 & 244. B16. 2 x 8 array with prisoners 1, 2, ..., 8 in top row and 9, 10, ..., 16 in bottom row. Two extra rows of 4 above the right hand end (i.e. above 5, 6, 7, 8) are empty. Slide the prisoners into a magic square. Gives a solution in 14 moves, due to G. Wotherspoon, which they feel is minimal. This allows long moves -- e.g. the first move moves 8 up two and left 3.

"H. E. Licks" [pseud. of Mansfield Merriman]. Recreations in Mathematics. Van Nostrand, NY, 1917. Art. 28, pp. 20-21. 'About the year 1880 ... invented in 1878 by a deaf and dumb man....'

[From sometime in the 1980s, I suspected the author's name was a pseudonym. On pp. 132-138, he discusses the Diaphote Hoax, from a Pennsylvania daily newspaper of 10 Feb 1880, which features the following people: H. E. Licks, M. E. Kannick, A. D. A. Biatic, L. M. Niscate. The diaphote was essentially a television. He says this report was picked up by the New York Times and the New York World. An email from Col. George L. Sicherman on 5 Jun 2000 agrees that the name is false and suggested that the author was "the eminent statistician Mansfield Merriman" who wrote the article on The Cattle Problem of Archimedes in Popular Science Monthly (Nov 1905), which is abridged on pp. 33-39 of the book, but omitting the author's name. Sichermann added that Merriman was one of the authors of Pillsbury's List. William Hartston says this was an extraordinary list of some 30 words which Pillsbury, who did memory feats, was able to commit to memory quite rapidly. Sicherman continued to investigate Merriman and got Prof. Andri Lange interested and Lange corresponded with a James A. McLennan, author of a history of the physics department at Lehigh University where Merriman had been. McLennan found Merriman's obituary from the American Society of Civil Engineers which states that Merriman used H. E. Licks as a pseudonym. [Email from Sicherman on 25 Feb 2002.]]

Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 300. "But this puzzle stuff, as I say, is as old as human thought. As soon as mankind began to have brains they must have loved to exercise them for exercise' sake. The 'jig-saw' puzzles come from China where they had them four thousand years ago. So did the famous 'sixteen puzzle' (fifteen movable squares and one empty space) over which we racked our brains in the middle eighties."

G. Kowalewski. Boss-Puzzle und verwandte Spiele. K. F. Kohler Verlag, Leipzig, 1921 (reprinted 1939). Gives solution of general polygonal versions, i.e. on a graph with a Hamilton circuit and one or more diagonals.

Hummerston. Fun, Mirth & Mystery. 1924.

1 2

9 Push, pp. 22 & 25. This is played on the board

3 10 11 4 shown at the left with its orthogonal lines, like

12 13 3, 10, 11, 4, and its diagonal lines, like

5 14 15 6 1, 9, 11, 13, 6. 10, 15 and 11, 14 are not

16 connected, so this is an octagram. Take 16

7 8 numbered counters and place them at random on

the board and remove counter 16. Move the pieces

to their correct locations. He asserts that 'unlike the original ["Sixteen" Puzzle],

no position can be set up in "Push" that cannot be solved'.

The six bulls puzzle, Puzzle no. 34, pp. 90 & 177. This uses the 2 x 3 + 1 0    

board shown at the right, where the 0 is the blank space. Exchange 1 2 3

3 and 6 and 4 and 5. He does it in 20 moves. [This is Hordern's 4 5 6

B3, first known from 1977 under the name Bull Pen, but is a variant of

Hordern's B2, first known from 1973.]

Q. E. D. -- The sergeant's problem, Puzzle no. 40, pp. 106 & 178. Take a 2 x 3 board, with the centre of one long side blank. Interchange the men along one short side. He does this in 17 moves, but the blank is not in its initial position nor are the other men. [This is Hordern's B1, first known from Loyd's Cyclopedia, 1914.]

King. Best 100. 1927. No. 26, p. 15. = Foulsham's, no. 9, pp. 7 & 10. "An entertaining variation ... is to draw, and colour, if you like, a small picture; then cut it into sixteen squares and discard the lower right hand square."

G. Kowalewski. Alte und neue mathematische Spiele. Teubner, Leipzig, 1930, pp. 61-81. Gives solution of general polygonal versions.

Dudeney. PCP. 1932. The Angelica puzzle, prob. 253, pp. 76 & 167. = 435, prob. 378, pp. 136 & 340. B8. 3 x 3 problem -- convert: A C I L E G N A X to A N G E L I C A X. Requires interchanging the As. Solution in 36 moves. In the answer in 435, Gardner notes that it can be done in 30 moves.

H. V. Mallison. Note 1454: An array of squares. MG 24 (No. 259) (May 1940) 119-121. Discusses 15 Puzzle and says any legal position can be achieved in at most about 150 moves. But if one fixes cells 6, 7, 11, then a simple problem requires about 900 moves.

McKay. At Home Tonight. 1940. Prob. 44: Changing the square, pp. 73 & 88. In the usual formation, colour the pieces alternately blue and red, as on a chessboard, with the blank at the lower right position 16 being a missing red, so there are 7 reds. Move so the colours are still alternating but the blank is at the lower left, i.e. position 13. Takes 15 moves.

Sherley Ellis Stotts. US Patent 3,208,753 -- Shiftable Block Puzzle Game. Filed: 7 Oct 1963; patented: 28 Sep 1965. 4pp + 2pp diagrams. Described in Hordern, pp. 152-153, F10-12. Rectangular pieces of different sizes. One can also turn a piece.

Gardner. SA (Feb 1964) = 6th Book, chap. 7. Surveys sliding-block puzzles with non-square pieces and notes there is no theory for them. Describes a number of early versions and the minimum number of moves for solution, generally done by hand and then confirmed by computer. Pennant Puzzle, C19; L'Âne Rouge, C27d; Line Up the Quinties, C4; Ma's Puzzle, D1; a form of Stotts' Baby Tiger Puzzle, F10.

Gardner. SA (Mar & Jun 1965) c= 6th Book, chap. 20. Prob. 9: The eight-block puzzle. B5. 3 x 3 problem -- convert: 8 7 6 5 4 3 2 1 X to 1 2 3 4 5 6 7 8 X. Compares it with Dudeney's Angelica puzzle (1932, B8) but says it can be done if fewer than 36 moves. Many readers found solutions in 30 moves; two even found all 10 minimal solutions by hand! Says Schofield (see next entry) has been working on this and gives the results below, but this did not quite resolve Gardner's problem. William F. Dempster, at Lawrence Radiation Laboratory, programmed a IBM 7094 to find all solutions, getting 10 solutions in 30 moves; 112 in 32 moves and 512 in 34 moves. Notes it is unknown if any problem with the blank in a side or corner requires more than 30 moves. (The description of Schofield's work seems a bit incorrect in the SA solution, and is changed in the book.)

Peter D. A. Schofield. Complete solution of the 'Eight-Puzzle'. Machine Intelligence 1 (1967) 125-133. This is the 3 x 3 version of the 15 Puzzle, with the blank space in the centre. Works with the corner twists which take the blank around a 2 x 2 corner in four moves. Shows that the 5-puzzle, which is the 3 x 2 version, has every position reachable in at most 20 moves, from which he shows that an upper bound for the 8-puzzle is 48 moves. Since the blank is in the middle, the 8!/2 = 20160 possible positions fall into 2572 equivalence classes. He also considers having inverse permutations being equivalent, which reduces to 1439 classes, but this was too awkward to implement. An ATLAS program found that the maximum number of moves required was 30 and 60 positions of 12 classes required this maximum number, but no example is given -- but see previous entry.

A. L. Davies. Rotating the fifteen puzzle. MG 54 (No. 389) (Oct 1970) 237-240. Studies versions where the numbers are printed diagonally so one can make a 90o turn of the puzzle. Then any pattern can be brought to one of two 'natural' patterns. He then asks when this is true for an m x n board and obtains a complicated solution. For an n x n board, n must be divisible by 4.

R. M. Wilson. Graph puzzles, homotopy and the alternating group. J. Combinatorial Thy., Ser. B, 16 (1974) 86-96. Shows that a sliding block puzzle, on any graph of n + 1 points which is non-separable and not a cycle, has at least An as its group -- except for one case on 7 points.

Alan G. & Dagmar R. Henney. Systematic solutions of the famous 15-14 puzzles. Pi Mu Epsilon J. 6 (1976) 197-201. They develop a test-value which significantly prunes the search tree. Kraitchik gave a problem which took him 114 moves -- the authors show the best solution has 58 moves!

David Levy. Computer Gamesmanship. Century Publishing, London, 1983. [Most of the material appeared in Personal Computer World, 1980-1981.] Pp. 16-29 discusses 8-puzzle and uses the Henney's test-value as an evaluation function. Cites Schofield.

Nigel Landon & Charles Snape. A Way with Maths. CUP, 1984. Cube moving, pp. 23 & 46. Consider a 9-puzzle in the usual arrangement: 1 2 3, 4 5 6, 7 8 x. Move the 1 to the blank position in the minimal number of moves, ignoring what happens to the other pieces. Generalise. Their answer only says 13 is minimal for the 3 x 3 board.

My student Tom Henley asked me the m x n problem in 1993 and gave a conjectural minimum, which I have corrected to: if m = n, then it can be done in 8m - 11 moves; but if n  5, 4, 2n - 3. (He has 2n - 1 by mistake. Simple modification shows we also have 5, 4, 7; 6, 5, 9; 7, 6, 5; 8, 7, 7; n > 8, n - 1, 5.)

Ball. MRE, 1st ed., 1892, pp. 45-47, says Lucas posed the problem of minimizing x for a given n and quotes the Delannoy solution (with erroneous 2n - 1) and also gives De Fontenay's version and solution. (He spells it De Fonteney as does his French translator, though Ahrens gives De Fontenay and the famous abbey in Burgundy is Fontenay -- ??)

The Ballybunnion and Listowel Railway in County Kerry, Ireland, was a late 19C railway using the Lartigue monorail system. This had a single rail, about three feet off the ground, with a carriage hanging over both sides of the rail. The principle job of the conductor/guard to make sure the passengers and goods were equally distributed on both sides. Kerry legend asserts that a piano had to be sent on this railway and there were not enough passengers or goods to balance it. So a cow was sent on the other side. At the far end, the piano was unloaded and replaced with two large calves and the carriage sent back. The cow was then unloaded and one calf moved to the other side, so the carriage could be sent back to the far end and everyone was happy.

Hoffmann. 1893. Chap. IV, pp. 157-158 & 211-213 = Hoffmann-Hordern, pp. 136-138, with photos.

No. 56: The three travellers. Masters and servants, equivalent to missionaries and cannibals. Solution says Jaques & Son make a puzzle version with six figures, three white and three black. Photos in Hoffmann-Hordern, pp. 136 & 137 -- the latter shows Caught in the Rain, 1880-1905, where Preacher, Deacon, Janitor and their wives have to get somewhere using one umbrella.

No. 57: The wolf, the goat, and the cabbages. Photo on p. 136 of La Chevre et le Chou. with box, by Watilliaux, 1874-1895. Hordern Collection, p. 72, and S&B, p. 134, show the same puzzle.)

No. 58: The three jealous husbands.

No. 59: The captain and his company. This is Alcuin's prop. 19 with many adults.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895].

P. 7: The wolf, the goat and the cabbages. Identical to Hoffmann No. 57, with nice colour picture. No solution.

P. 9: The missionaries' and cannibals' puzzle. Usual form, with nice colour picture, but only one cannibal can row. No solution. This seems to be the first to use the context of missionaries and cannibals and the first to restrict the number of rowers.

Lucas. L'Arithmétique Amusante. 1895. Les vilains maris jaloux, pp. 125-144 & Note II, pp. 198-202.

Prob. XXXVI: La traversée des trois ménages, pp. 125-130. 3 couples. Gives Bachet's 1624 reasoning for the essentially unique solution -- but attributes it to 1613.

Prob. XXXVII: La traversée des quatre ménages, pp. 130-132. 4 couples in a 3 person boat done in 9 crossings.

L'erreur de Tartaglia, pp. 133-134. Discusses Tartaglia's error and Bachet's notice of it and gives an easy proof that 4 couples cannot be done with a 2 person boat.

Prob. XXXVIII: La station dans une île, pp. 135-140. 4 couples, 2 person boat, with an island. Gives De Fontenay's solution in 24 crossings.

Prob. XXXIX: La traversée des cinq ménages, pp. 141-143. 5 couples, 3 person boat in 11 crossings.

Énoncé général du problème des traversées, pp. 143-144. n couples, x person boat, can be done in N crossings as given by Delannoy above. He corrects 2n - 1 to 2n - 3 here.

Note II: Sur les traversées, pp. 198-202. Gives Tarry's version with an island and with n men having harems of size m, where the women are obviously unable to row. He gives solutions in various cases. For the ordinary case, i.e. m = 1, he finds a solution for 4 couples in 21 moves, using the basic ferrying technique that Pressman and Singmaster found to be optimal, but the beginning and end take longer because the women cannot row. He says this gives a solution for n couples in 4n + 5 crossings. He then considers the case of n - 1 couples and a ménage with m wives and finds a solution in 8n + 2m + 7 crossings. I now see that this solution has the same defects as those in Pressman & Singmaster, qv.

Ball. MRE, 3rd ed., 1896, pp. 61-64, repeats 1st ed., but adds that Tarry has suggested the problem for harems -- see above.

Dudeney. Problem 68: Two rural puzzles. Tit-Bits 33 (5 Feb & 5 Mar 1898) 355 & 432. Three men with sacks of treasure and a boat that will hold just two men or a man and a sack, with additional restrictions on who can be trusted with how much. Solution in 13 crossings.

Carroll-Collingwood. 1899. P. 317 (Collins: 231 or 232 (missing in my copy)) Cf Carroll-Wakeling II, prob. 10: Crossing the river, pp. 17 & 66. Four couples -- only posed, no solution. Wakeling gives a solution, but this is incorrect. After one wife is taken across, he has another couple coming across and from Bachet onward, this is considered improper as the man could get out of the boat and attack the first, undefended, wife.

E. Fourrey. Op. cit. in 4.A.1, 1899. Section 211: Les trois maîtres et les trois valets. Says a master cannot leave his valet with the other masters for fear that they will intimidate him into revealing the master's secrets. Hence this is the same as the jealous couples.

H. D. Northrop. Popular Pastimes. 1901.

No. 5: The three gentlemen and their servants, pp. 67 & 72. = The Sociable.

No. 12: The dishonest servants, pp. 68 & 73. "... the servants on either side of the river should not outnumber the masters", so this is the same as missionaries and cannibals.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:2 (Jun 1903) 140-141. A matrimonial difficulty. Three couples. No answer given.

Dudeney. Problem 523. Weekly Dispatch (15 & 29 Nov 1903), both p. 10, (= AM, prob. 375, pp. 113 & 236-237). 5 couples in a 3 person boat.

Johannes Bolte. Der Mann mit der Ziege, dem Wolf und dem Kohle. Zeitschrift des Vereins für Volkskunde 13 (1903) 95-96 & 311. The first part is unaware of Alcuin and Albert. He gives a 12C Latin solution: It capra, fertur olus, redit hec, lupus it, capra transit [from Wattenbach; Neuen Archiv für ältere deutsche Geschichtskunde 2 (1877) 402, from Vorauer MS 111, ??NYS] and a 14C solution: O natat, L sequitur, redit O, C navigat ultra, / Nauta recurrit ad O, bisque natavit ovis (= ovis, lupus, ovis, caulis, ovis) [from Mone; Anzeiger für Kunde der deutschen Vorzeit 45 (No. 105) (1838), from Reims MS 743, ??NYS]. Cites Kamp and several other versions, some using a fox, a sheep, or a lamb. The addendum cites and quotes Alcuin and Albert as well as relatively recent French and Italian versions.

H. Parker. Ancient Ceylon. Op. cit. in 4.B.1. 1909. Crossing the river, p. 623.

A King, a Queen, a washerman and a washerwoman have to cross a river in a boat that holds two. However the King and Queen cannot be left on a bank with the low caste persons, though they can be rowed by the washerperson of the same sex. Solution in 7 crossings.

Ferry-man must transport three leopards and three goats in a boat which holds himself and two others. If leopards ever outnumber goats, then the goats get eaten. So this is like missionaries and cannibals, but with a ferry-man. Solution in 9 crossings.

H. Schubert. Mathematische Mussestunde. Vol. 2, 3rd ed., Göschen, Leipzig, 1909. Pp. 160-162: Der drei Herren und der drei Sklaven. (Same as missionaries and cannibals.)

Arbiter Co. (Philadelphia). 1910. Capital and Labor Puzzle. Shown in S&B, p. 134. Equivalent to missionaries and cannibals.

Ball. MRE, 5th ed., 1911, pp. 71-73, repeats 3rd ed., but omits the details of De Fonteney's solution in 8(n-1) crossings.

Loyd. Cyclopedia, 1914.

Summer tourists, pp. 207 & 366. 3 couples, 2 person boat, with additional complications -- the women cannot row and there have been some arguments. Solution in 17 crossings.

The four elopements, pp. 266 & 375. 4 couples, 2 person boat, with an island and the stronger constraint that no man is to get into the boat alone if there is a girl alone on either the island or the other shore. "The [problem] presents so many complications that the best or shortest answer seems to have been overlooked by mathematicians and writers on the subject." "Contrary to published answers, ... the feat can be performed in 17 trips, instead of 24."

Ball. MRE, 6th ed., 1914, pp. 71-73, repeats 5th ed., but adds that 6n - 7 trips suffices for n couples with an island, though he gives no reference.

Williams. Home Entertainments. 1914. Alcuin's riddle, pp. 125-126. "This will be recognized as perhaps the most ancient British riddle in existence, though there are several others conceived on the same lines." Three jealous couples.

Clark. Mental Nuts. 1916, no. 67. The men and their wives. "... no man shall be left alone with another's wife."

Dudeney. AM. 1917. Prob. 376: The four elopements, pp. 113 & 237. 4 couples, 2 person boat, with island, can be done in 17 trips and that this cannot be improved. This is the same solution as given by Loyd. (See Pressman and Singmaster, below.)

Ball. MRE, 8th ed., 1919, pp. 71-73 repeats 6th ed. and adds a citation to Dudeney's AM prob. 376 for the solution in 6n - 7 trips for n couples.

Hummerston. Fun, Mirth & Mystery. 1924. Crossing the river puzzles, Puzzle no, 52, pp. 128 & 180. 'Puzzles of this type ... interested people who lived more than a thousand years ago'.

No. 1: The eight travellers. Six men and two boys who weigh half as much.

No. 2: White and black. = Missionaries and cannibals.

No. 3: The fox, the goose, and the corn.

No. 4: the jealous husbands.

H&S, 1927, p. 51 says missionaries and cannibals is 'a modern variant'.

King. Best 100. 1927. No. 10, pp. 10 & 40. Dog, goose and corn.

Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930.

P. 106: Der Wolf, die Ziege und der Kohlkopf. Usual wolf, goat, cabbage.

Pp. 106-107: Die 100 Pfund-Familie. Parents weigh 100 pounds; the two children weigh 100 pounds together.

P. 107: Der Landjäger and die Strolche [The policeman and the vagabonds]. Two of the vagabonds hate each other so much that they cannot be left together. As far as I recall, this formulation is novel and I was surprised to realise that it is essentially equivalent to the wolf, goat and cabbage version.

Phillips. Week-End. 1932. Time tests of intelligence, no. 41, pp. 22 & 194. Rowing explorer with 4 natives: A, B, C, D, who cannot abide their neighbours in this list. A can row. They get across in seven trips.

Abraham. 1933. Prob. 54 -- The missionaries at the ferry, pp. 18 & 54 (14 & 115). 3 missionaries and 3 cannibals. Doesn't specify boat size, but says 'only one cannibal can row'. 1933 solution says 'eight double journeys', 1964 says 'seven crossings'. This seems to assume the boat holds 3. (For a 2 man boat, it takes 11 crossings with one missionary and two cannibals who can row or 13 crossings with one missionary and one cannibal who can row.)

The Bile Beans Puzzle Book. 1933. No. 34: Missionaries & cannibals. Three of each but only one of each can row. Done in 13 crossings.

Phillips. Brush. 1936. Prob. L.2: Crossing the Limpopo, pp. 39-40 & 98. Same as in Week-End, 1932.

M. Adams. Puzzle Book. 1939. Prob. C.63: Going to the dance, pp. 139 & 178. Same as Week-End, 1932, phrased as travelling to a dance on a motorcycle which carries one passenger.

R. L. Goodstein. Note 1778: Ferry puzzle. MG 28 (No. 282) (1944) 202-204. Gives a graphical way of representing such problems and considers m soldiers and m cannibals with an n person boat, 3 jealous husbands and how many rowers are required.

David Stein. Party and Indoor Games. P. M. Productions, London, nd [c1950?]. P. 98, prob. 5: Man with cat, parrot and bag of seeds.

C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960.

A puzzle-solving machine, pp. 377-384. Describes how Paul Bezold made a logic machine from relays to solve the fox, goose, corn problem.

How to design a "Pircuit" or Puzzle circuit, pp. 388-394. On pp. 391-394, Harry Rudloe describes relay circuits for solving the three jealous couples problem, which he attributes to Tartaglia, and the missionaries and cannibals problem.

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Mentor (New American Library), NY, 1961. [John Fauvel sent some pages from a different printing which has much different page numbers than my copy.] "River crossing" problems, pp. 168-171. Discusses various forms of the problem and adds a problem with two parents weighing 160, two children weighing 80 and a dog weighing 12, with a boat holding 160.

E. A. Beyer, proposer; editorial solution. River-crossing dilemma. RMM 4 (Aug 1961) 46 & 5 (Oct 1961) 59. Explorers and natives (= missionaries and cannibals), with all the explorers and one native who can row. Solves in 13 crossings, but doesn't note that only one rowing explorer is needed. (See note at Abraham, 1933, above.)

Philip Kaplan. Posers. (Harper & Row, 1963); Macfadden Books, 1964. Prob. 36, pp. 41 & 91. 5 men and a 3 person boat on one side, 5 women on the other side. One man and one woman can row. Men are not allowed to outnumber women on either side nor in the boat. Exchange the men and the women in 7 crossings.

T. H. O'Beirne. Puzzles and Paradoxes, 1965, op. cit. in 4.A.4, chap. 1, One more river to cross, pp. 1-19. Shows 2n - 1 couples (or 2n - 1 each of missionaries and cannibals ?) can cross in a n person boat in 11 trips. 2n - 2 can cross in 9 trips. He also considers variants on Gori's second version.

Doubleday - 2. 1971. Family outing, pp. 49-50. Three couples, but one man has quarrelled with the other men and his wife has quarrelled with the other women, so this man and wife cannot go in the boat nor be left on a bank with others of their sex. Further men cannot be outnumbered by women on either bank. Gives a solution in 9 crossings, but I find the conditions unworkable -- e.g. the initial position is prohibited!

Claudia Zaslavsky. Africa Counts. Prindle, Weber & Schmidt, Boston, 1973. Pp. 109-110 says that leopard, goat and pile of cassava leaves is popular with the Kpelle children of Liberia. However, Ascher's Ethnomathematics (see below), p. 120, notes that this is based on an ambiguous description and that an earlier report of a Kpelle version has the form described below.

Ball. MRE, 12th ed., 1974, p. 119, corrects Delannoy's 2n - 1 to 2n - 3 and corrects De Fontenay's 8n - 8 to 8n - 6, but still gives the solution for n = 4 with 24 crossings.

W. Gibbs. Pebble Puzzles -- A Source Book of Simple Puzzles and Problems. Curriculum Development Unit, Solomon Islands, 1982. ??NYS, o/o??. Excerpted in: Norman K. Lowe, ed.; Games and Toys in the Teaching of Science and Technology; Science and Technology Education, Document Series No. 29, UNESCO, Paris, 1988, pp. 54-57. On pp. 56-57 is a series of river crossing problems. E.g. get people of weights 1, 2, 3 across with a boat that holds a weight of at most 3. Also people numbered 1, 2, 3, 4, 5 such that no two consecutive people can be in the boat or left together.

In about 1986, James Dalgety designed interactive puzzles for Techniquest in Cardiff. Their version has a Welshman with a dragon, a sheep and a leek!

Ian Pressman & David Singmaster. Solutions of two river crossing problems: The jealous husbands and the missionaries and the cannibals. Extended Preprint, April 1988, 14pp. MG 73 (No. 464) (Jun 1989) 73-81. (The preprint contains historical and other detail omitted from the article as well as some further information.) Observes that De Fontenay seems to be excluding bank to bank crossings and that Lucas' presentation is cryptic. Shows that De Fontenay's method should be 8n - 6 crossings for n > 3 and that this is minimal. If bank to bank crossings are permitted, as by Loyd and Dudeney, a computer search revealed a solution with 16 crossings for n = 4, using an ingenious move that Dudeney could well have ignored. For n > 4, there is a simple solution in 4n + 1 crossings, and these numbers are minimal. [When this was written, I had forgotten that Loyd had done the problem for 4 couples in 17 moves, which changes the history somewhat. However, I now see that Loyd was copying from Dudeney's Weekly Dispatch problem 270 of 23 Apr 1899 & 11 Jun 1899. Loyd states what appears to be a stronger constraint but all the methods in our article do obey the stronger constraint. However, one could make the constraints stronger -- e.g. our solutions have a husband taking the boat from bank to bank while his wife and another wife are on the island -- the solution of Loyd & Dudeney avoids this and may be minimal in this case --??.]

For the missionaries and cannibals problem, the 16 crossing solution reduces to 15 and gives a general solution in 4n - 1 crossings, which is shown to be minimal. If bank to bank crossings are not permitted, then De Fontenay's amended 8n - 6 solution is still optimal.

Marcia Ascher. A river-crossing problem in cultural perspective. MM 63 (1990) 26-28. Describes many appearances in folklore of many cultures. Discusses African variants of the wolf, goat and cabbage problem in which the man can take two of the items in the boat. This is much easier, requiring only three crossings, but some versions say that the man cannot control the items in the boat, so he cannot have the wolf and goat or the goat and cabbage in the boat with him. This still only takes three crossings. Various forms of these problems are mentioned: fox, fowl and corn; tiger, sheep and reeds; jackal, goat and hay; caged cheetah, fowl and rice; leopard, goat and leaves -- see below for more details.

She also discusses an Ila (Zambia) version with leopard, goat, rat and corn which is unsolvable!

Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Section 4.8, pp. 109-116 & Note 8, pp. 119-121. Good survey of the problem and numerous references to the folklore and ethnographic literature. Amplifies the above article. A version like the Wolf, goat and cabbage is found in the Cape Verde Islands, in Cameroon and in Ethiopia. The African version is found as far apart as Algeria and Zanzibar, but with some variations. An Algerian version with jackal, goat and hay allows one to carry any two in the boat, but an inefficient solution is presented first. A Kpelle (Liberia) version with cheetah, fowl and rice adds that the man cannot keep control while rowing so he cannot take the fowl with either the cheetah or the rice in the boat. A Zanzibar version with leopard, goat and leaves adds instead that no two items can be left on either bank together. (A similar version occurs among African-Americans on the Sea Islands of South Carolina.) Ascher notes that Zaslavsky's description is based on an ambiguous report of the Kpelle version and probably should be like the Algerian or Kpelle version just described.

Liz Allen. Brain Sharpeners. New English Library (Hodder & Stoughton), London, 1991. Crossing the river, pp. 62 & 125. Three mothers and three sons. The sons are unwilling to be left with strange mothers, so this is a rephrasing of the jealous husbands.

Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, probs. 4-6: The knights and the pages; More knights and pages; Yet more knights and pages: no man is an island, pp. 4-5 & 100-102. Equivalent to the jealous couples. Prob. 4 is three couples, solved in 11 crossings. Prob. 5 is four couples -- "There is no solution unless one of the four pages is sacrificed. (In medieval times, this was not a problem.)" Prob. 6 is four couples with an island in the river, solved in general by moving all pages to the island, then having the pages go back and accompany his knight to other side, then return to the island. After the last knight is moved, the pages then move from the island to the other side. This takes 7n - 6 steps in general. It satisfies the jealousy conditions used by Pressman & Singmaster, but not those of Loyd & Dudeney.

John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 16: The missionaries and the pirates. Politically correct rephrasing of the missionaries and the cannibals version. All the missionaries, but only one pirate, can row. Solves in 13 crossings.

Prof. Dr. Robert Weismantel, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, PSF 3120, D-39016 Magdeburg, Germany; tel: 0391/67-18745; email: weismantel@imo.math.uni-magdeburg.de; has produced a 45 min. film: "Der Wolf, die Ziege und Kohlköpfe Transportprobleme von Karl dem Grossen bis heute", suitable for the final years of school.

5.B.1. LOWERING FROM TOWER PROBLEM

The problem is for a collection of people (and objects or animals) to lower themselves from a window using a rope over a pulley, with baskets at each end. The complication is that the baskets cannot contain very different weights, i.e. there is a maximum difference in the weights, otherwise they go too fast. This is often attributed to Carroll.

Carroll-Collingwood. 1899. P. 318 (Collins: 232-233 (232 is lacking in my copy)). = Carroll-Wakeling II, prob. 4: The captive queen, pp. 8 & 65-66. 3 people of weights 195, 165, 90 and a weight of 75, with difference at most 15. He also gives a more complex form. No solutions. Although the text clearly says 165, the prevalence of the exact same problem with 165 replaced by 105 makes me wonder if this was a misprint?? Wakeling says there is no explicit evidence that Carroll invented this, and neither book assigns a date, but Carroll seems a more original source than the following and he was more active before 1890 than after.

An addition is given in both books: add three animals, weighing 60, 45, 30.

Lemon. 1890. The prisoners in the tower, no. 497, pp. 65 & 116. c= Sphinx, The escape, no. 113, pp. 19 & 100-101. Three people of weights 195, 105, 90 with a weight of 75. The difference in weights cannot be more than 15.

Hoffmann. 1893. Chap. IV, no. 28: The captives in the tower, pp. 150 & 196 = Hoffmann-Hordern, p. 123. Same as Lemon.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 3: The captives in the tower. Same as Lemon. Identical to Hoffmann. With colour picture. No solution.

Loyd. The fire escape puzzle. Cyclopedia, 1914, pp. 71 & 348. c= MPSL2, prob. 140, pp. 98-99 & 165. = SLAHP: Saving the family, pp. 59 & 108. Simplified form of Carroll's problem. Man, wife, baby & dog, weighing a total of 390.

Williams. Home Entertainments. 1914. The escaping prisoners, pp. 126-127. Same as Lemon.

Rudin. 1936. No. 92, pp. 31-32 & 94. Same as Lemon.

Haldeman-Julius. 1937. No. 150: Fairy tale, pp. 17 & 28. Same as Lemon, except the largest weight is printed as 196, possibly an error.

Kinnaird. Op. cit. in 1 -- Loyd. 1946. Pp. 388-389 & 394. Same as Lemon.

Simon Dresner. Science World Book of Brain Teasers. Scholastic Book Services, NY, 1962. Prob. 61: Escape from the tower, pp. 29 & 99-100. Same as Lemon.

Robert Harbin [pseud. of Ned Williams]. Party Lines. Oldbourne, London, 1963. Escape, p. 29. As in Lemon.

Howard P. Dinesman. Superior Mathematical Puzzles. Allen & Unwin, London, 1968. No. 60: The tower escape, pp. 78 & 118. Same as Carroll. Answer in 15 stages. He cites Carroll, noting that Carroll did not give a solution and he asks if a shorter solution can be found.

F. Geoffrey Hartswick. In: H. O. Ripley & F. G. Hartswick; Detectograms and Other Puzzles; Scholastic Book Services, NY, 1969. No. 15: Stolen treasure puzzle, pp. 54-55 & 87. Same as Lemon.

5.B.2. CROSSING A BRIDGE WITH A TORCH

New section.

Four people have to get across a bridge which is dark and needs to be lit with the torch. The torch can serve for at most two people and the gap is too wide to throw the torch across, so the torch has to be carried back and forth. The various people are of different ages and require 5, 10, 20, 25 minutes to cross and when two cross, they have to go at the speed of the slower. But the torch (= flashlight) battery will only last an hour. Can it be done? I heard this about 1997, when it was claimed to be used by Microsoft in interviewing candidates. I never found any history of it, until I recently found a discussion on Torsten Sillke's site: Crossing the bridge in an hour (mathematik.uni-bielefeld.de/~sillke/PUZZLES/crossing-bridge), starting in Jun 1997 and last updated in Sep 2001. This cites the 1981 source and the other references below. Denote the problem with speeds a, b, c, d and total time t by

(a, b, c, d; t), etc. t is sometimes given, sometimes not.

Saul X. Levmore & Elizabeth Early Cook. Super Strategies for Puzzles and Games. Doubleday, 1981, p. 3 -- ??NYS. (5, 10, 20, 25; 60), as in the introduction to this section..

Heinrich Hemme. Das Problem des Zwölf-Elfs. Vandenhoeck & Ruprecht, 1998. Prob. 81: Die Flucht, pp. 40 & 105-106, citing a web posting by Gunther Lientschnig on 4 Dec 1996. (2, 4, 8, 10; t).

Dick Hess. Puzzles from Around the World. Apr 1997. Prob. 107: The Bridge.

(1, 2, 5, 10; 17). Poses versions with more people: (1, 3, 4, 6, 8, 9; 31) and, with a three-person bridge, (1, 2, 6, 7, 8, 19, 10; 25).

Quantum (May/Jun 1997) 13. Brainteaser B 205: Family planning. Problem (1, 3, 8, 10; 20).

Karen Lingel. Email of 17 Sep 1997 to rec.puzzles. Careful analysis, showing that the 'trick' solution is better than the 'direct' solution if and only if a + c > 2b. [Indeed, a + c - 2b is the time saved by the 'trick' solution.] She cites (2, 3, 5, 8; 19) and (2, 2, 3, 3; 11) to Sillke and (1, 3, 6, 8, 12; 30), from an undated website. Expressing the solution for more people seems to remain an open question.

5.C. FALSE COINS WITH A BALANCE

See 5.D.3 for use of a weighing scale.

There are several related forms of this problem. Almost all of the items below deal with 12 coins with one false, either heavy or light, and its generalizations, but some other forms occur, including the following.

8 coins, (1 light: Schell, Dresner

26 coins, (1 light: Schell

8 coins, 1 light: Bath (1959)

9 coins, 1 light: Karapetoff, Meyer (1946), Meyer (1948), M. Adams, Rice

I have been sent an article by Jack Sieburg; Problem Solving by Computer Logic; Data Processing Magazine, but the date is cut off -- ??

E. D. Schell, proposer; M. Dernham, solver. Problem E651 -- Weighed and found wanting. AMM 52:1 (Jan 1945) 42 & 7 (Aug/Sep 1945) 397. 8 coins, at most one light -- determine the light one in two weighings.

Benjamin L. Schwartz. Letter: Truth about false coins. MM 51 (1978) 254. States that Schell told Michael Goldberg in 1945 that he had originated the problem.

Emil D. Schell. Letter of 17 Jul 1978 to Paul J. Campbell. Says he did NOT originate the problem, nor did he submit the version published. He first heard of it from Walter W. Jacobs about Thanksgiving 1944 in the form of finding at most one light coin among 26 good coins in three weighings. He submitted this to the AMM, with a note disclaiming originality. The AMM problem editor published the simpler version described above, under Schell's name. Schell says he has heard Eilenberg describe the puzzle as being earlier than Sep 1939. Campbell wrote Eilenberg, but had no response.

Schell's letter is making it appear that the problem derives from the use of 1, 3, 9, ... as weights. This usage leads one to discover that a light coin can be found in 3n coins using n weighings. This is the problem mentioned by Karapetoff. If there is at most one light coin, then n weighings will determine it among 3n - 1 coins, which is the form described by Schell. The problem seems to have been almost immediately converted into the case with one false coin, either heavy or light.

Walter W. Jacobs. Letter of 15 Aug 1978 to Paul J. Campbell. Says he heard of the problem in 1943 (not 1944) and will try to contact the two people who might have told it to him. However, Campbell has had no further word.

V. Karapetoff. The nine coin problem and the mathematics of sorting. SM 11 (1945) 186-187. Discusses 9 coins, one light, and asks for a mathematical approach to the general problem. (?? -- Cites AMM 52, p. 314, but I cannot find anything relevant in the whole volume, except the Schell problem. Try again??)

Dwight A. Stewart, proposer; D. B. Parkinson & Lester H. Green, solvers. The counterfeit coin. In: L. A. Graham, ed.; Ingenious Mathematical Problems and Methods; Dover, 1959; pp. 37-38 & 196-198. 12 coins. First appeared in Oct 1945. Original only asks for the counterfeit, but second solver shows how to tell if it is heavy or light.

R. L. Goodstein. Note 1845: Find the penny. MG 29 (No. 287) (Dec 1945) 227-229. Non-optimal solution of general problem.

Editorial Note. Note 1930: Addenda to Note 1845. Ibid. 30 (No. 291) (Oct 1946) 231. Comments on how to extend to optimal solution.

Howard D. Grossman. The twelve-coin problem. SM 11:3/4 (Sep/Dec 1945) 360-361. Finds counterfeit and extends to 36 coins.

Lothrop Withington, Jr. Another solution of the 12-coin problem. Ibid., 361-362. Finds also whether heavy or light.

Donald Eves, proposer; E. D. Schell & Joseph Rosenbaum, solvers. Problem E712 -- The extended coin problem. AMM 53:3 (Mar 1946) 156 & 54:1 (Jan 1947) 46-48. 12 coins.

Jerome S. Meyer. Puzzle Paradise. Crown, NY, 1946. Prob. 132: The nine pearls, pp. 94 & 132. Nine pearls, one light, in two weighings.

N. J. Fine, proposer & solver. Problem 4203 -- The generalized coin problem. AMM 53:5 (May 1946) 278 & 54:8 (Oct 1947) 489-491. General problem.

H. D. Grossman. Generalization of the twelve-coin problem. SM 12 (1946) 291-292. Discusses Goodstein's results.

F. J. Dyson. Note 1931: The Problem of the Pennies. MG 30 (No. 291) (Oct 1946) 231-234. General solution.

C. A. B. Smith. The Counterfeit Coin Problem. MG 31 (No. 293) (Feb 1947) 31-39.

C. W. Raine. Another approach to the twelve-coin problem. SM 14 (1948) 66-67. 12 coins only.

K. Itkin. A generalization of the twelve-coin problem. SM 14 (1948) 67-68. General solution.

Howard D. Grossman. Ternary epitaph on coin problems. SM 14 (1948) 69-71. Ternary solution of Dyson & Smith.

Jerome S. Meyer. Fun-to-do. A Book of Home Entertainment. Dutton, NY, 1948. Prob. 40: Nine pearls, pp. 41 & 188. Nine pearls, one light, in two weighings.

Blanche Descartes [pseud. of Cedric A. B. Smith]. The twelve coin problem. Eureka 13 (Oct 1950) 7 & 20. Proposal and solution in verse.

J. S. Robertson. Those twelve coins again. SM 16 (1950) 111-115. Article indicates there will be a continuation, but Schaaf I 32 doesn't cite it and I haven't found it yet.

E. V. Newberry. Note 2342: The penny problem. MG 37 (No. 320) (May 1953) 130. Says he has made a rug showing the 120 coins problems and makes comments similar to Littlewood's, below.

J. E. Littlewood. A Mathematician's Miscellany. Methuen, London, 1953; reprinted with minor corrections, 1957 (& 1960). [All the material cited is also in the later version: Littlewood's Miscellany, ed. by B. Bollobás, CUP, 1986, but on different pages. Since the 1953 ed. is scarce, I will also cite the 1986 pages in (  ).] Pp. 9 & 135 (31 & 114). "It was said that the 'weighing-pennies' problem wasted 10,000 scientist-hours of war-work, and that there was a proposal to drop it over Germany."

John Paul Adams. We Dare You to Solve This! Berkley Publishing, NY, nd [1957?]. [This is apparently a collection of problems used in newspapers. The copyright is given as 1955, 1956, 1957.] Prob. 18: Weighty problem, pp. 13 & 46. 9 equal diamonds but one is light, to be found in 2 weighings.

Hubert Phillips. Something to Think About. Revised ed., Max Parrish, London, 1958. Foreword, p. 6 & prob. 115: Twelve coins, pp. 81 & 127-128. Foreword says prob. 115 has been added to this edition and "was in oral circulation during the war. So far as I know, it has only appeared in print in the Law Journal, where I published both the problem and its solution." This may be an early appearance, so I should try and track this down. ??NYS

Dan Pedoe. The Gentle Art of Mathematics. (English Universities Press, 1958); Pelican (Penguin), 1963. P. 30: "We now come to a problem which is said to have been planted over here during the war by enemy agents, since Operational Research spent so many man-hours on its solution."

Philip E. Bath. Fun with Figures. The Epworth Press, London, 1959. No. 7: No weights -- no guessing, pp. 8 & 40. 8 balls, including one light, to be determined in two weighings. Method actually works for ( 1 light.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 9. Five boxes of sugar, but some has been taken from one box and put in another. Determine which in least number of weighings. Does by weighing each division of A, B, C, D into two pairs.

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. The "False Coin" problem, pp. 178-182. Sketches history and solution.

Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 46: Dud reckoning, pp. 21 & 94. Find one light among eight in two weighings.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 55, pp. 57 & 98. Six identical appearing coins, three of which are identically heavy. In two weighings, identify two of the heavy coins.

Charlie Rice. Challenge! Hallmark Editions, Kansas City, Missouri, 1968. Prob. 7, pp. 22 & 54-55. 9 pearls, one light.

Jonathan Always. Puzzling You Again. Tandem, London, 1969. Prob. 86: Light-weight contest, pp. 51-52 & 106-107. 27 weights of sizes 1, 2, ..., 27, except one is light. Find it in 3 weighings. He divides into 9 sets of three having equal weights. Using two weighings, one locates the light weight in a set of three and then weighing two of these with good weights reveals the light one. [3 weights 1, 2, 3 cannot be done in one weighing, but 9 weights 1, 2, ..., 9 can be done in two weighings.]

Robert H. Thouless. The 12-balls problem as an illustration of the application of information theory. MG 54 (No. 389) (Oct 1970) 246-249. Uses information theory to show that the solution process is essentially determined.

Ron Denyer. Letter. G&P, No. 37 (Jun 1975) 23. Asks for a mnemonic for the 12 coins puzzles. He notes that one can use three predetermined weighings and find the coin from the three answers.

Basil Mager & E. Asher. Letters: Coining a mnemonic. G&P, No. 40 (Sep 1975) 26. One mnemonic for a variable method, another for a predetermined method.

N. J. Maclean. Letter: The twelve coins. G&P, No. 45 (Feb 1976) 28-29. Exposits a ternary method for predetermined weighings for (3n-3)/2 in n weighings. Each weighing determines one ternary digit and the resulting ternary number gives both the coin and whether it is heavy or light.

Tim Sole. The Ticket to Heaven and Other Superior Puzzles. Penguin, 1988. Weighty problems -- (iii), pp. 124 & 147. Nine equal pies, except someone has removed some filling from one and inserted it in a pie, possibly the same one. Determine which, if any, are the heavy and light ones in 4 balancings.

Calvin T. Long. Magic in base 3. MG 76 (No. 477) (Nov 1992) 371-376. Good exposition of the base 3 method for 12 coins.

Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Problems for an equal-arm balance, pp. 137-141.

1. Six balls, two of each of three colours. One of each colour is lighter than normal and all light weights are equal. Determine the light balls in three weighings.

2. Five balls, three normal, one heavy, one light, with the differences being equal, i.e. the heavy and the light weigh as much as two normals. Determine the heavy and light in three weighings.

3. Same problem with nine balls and seven normals, done in four weighings.

5.C.1 RANKING COINS WITH A BALANCE

If one weighs only one coin against another, this is the problem of sorting except that we don't actually put the objects in order. If one weighs pairs, etc., this is a more complex problem.

J. Schreier. Mathesis Polska 7 (1932) 154-160. ??NYS -- cited by Steinhaus.

Hugo Steinhaus. Mathematical Snapshots. Not in Stechert, NY, 1938, ed. OUP, NY: 1950: pp. 36-40 & 258; 1960: pp. 51-55 & 322; 1969 (1983): pp. 53-56 & 300. Shows n objects can be ranked in M(n) = 1 + kn - 2k steps where k = 1 + [log2 n]. Gets M(5) = 8.

Lester R. Ford Jr. & Selmer M. Johnson. A tournament problem. AMM 66:5 (May 1959) 387-389. Note that (log2 n!( = L(n) is a lower bound from information theory. Obtain a better upper bound than Steinhaus, denoted U(n), which is too complex to state here. For convenience, I give the table of these values here.

n 1 2 3 4 5 6 7 8 9 10 11 12 13

M(n) 0 1 3 5 8 11 14 17 21 25 29 33 37

U(n) 0 1 3 5 7 10 13 16 19 22 26 30 34

L(n) 0 1 3 5 7 10 13 16 19 22 26 29 33

U(n) = L(n) also holds at n = 20 and 21.

Roland Sprague. Unterhaltsame Mathematik. Op. cit. in 4.A.1. 1961. Prob. 22: Ein noch ungelöstes Problem, pp. 16 & 42-43. (= A still unsolved problem, pp. 17 & 48-49.) Sketches Steinhaus's method, then does 5 objects in 7 steps. Gives the lower bound L(n) and says the case n = 12 is still unsolved.

Kobon Fujimura, proposer; editorial comment. Another balance scale problem. RMM 10 (Aug 1962) 34 & 11 (Oct 1962) 42. Eight coins of different weights and a balance. How many weighings are needed to rank the coins? In No. 11, it says the solution will appear in No. 13, but it doesn't appear there or in the last issue, No. 14. It also doesn't appear in the proposer's Tokyo Puzzles.

Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 6: In the balance, pp. 18 & 85-86. Rank five balls in order in seven weighings.

John Cameron. Establishing a pecking order. MG 55 (No. 394) (Dec 1971) 391-395. Reduces Steinhaus's M(n) by 1 for n ( 5, but this is not as good as Ford & Johnson.

W. Antony Broomhead. Letter: Progress in congress? MG 56 (No. 398) (Dec 1972) 331. Comments on Cameron's article and says Cameron can be improved. States the values U(9) and U(10), but says he doesn't know how to do 9 in 19 steps. Cites Sprague for numerical values, but these don't appear in Sprague -- so Broomhead presumably computed L(9) and L(10). He gets 10 in 23 steps, which is better than Cameron.

Stanley Collings. Letter: More progress in congress. MG 57 (No. 401) (Oct 1973) 212-213. Notes the ambiguity in Broomhead's reference to Sprague. Improves Cameron by 1 (or more??) for n ( 10, but still not as good as Ford & Johnson.

L. J. Upton, proposer; Leroy J. Myers, solver. Problem 1138. CM 12 (1986) 79 & 13 (1987) 230-231. Rank coins weighing 1, 2, 3, 4 with a balance in four weighings.

5.D. MEASURING PROBLEMS

5.D.1. JUGS & BOTTLES

See MUS I 105-124, Tropfke 659.

NOTATION: I-(a, b, c) means we have three jugs of sizes a, b, c with a full and we want to divide a in half using b and c. We normally assume a ( b ( c and GCD(a, b, c) = 1. Halving a is clearly impossible if GCD(b, c) does not divide a/2 or if b+c  a and c > a/2 is unsolvable.

From about the mid 19C, I have not recorded simple problems.

I-( 8, 5, 3): almost all the entries below

I-(10, 6, 4): Pacioli, Court

I-(10, 7, 3): Yoshida

I-(12, 7, 5): Pacioli, van Etten/Henrion, Ozanam, Bestelmeier, Jackson, Manuel des Sorciers, Boy's Own Conjuring Book

I-(12, 8, 4): Pacioli

I-(12, 8, 5): Bachet, Arago

I-(16, 9, 7): Bachet-Labosne

I-(16,11, 6): Bachet-Labosne

I-(16,12, 7): Bachet-Labosne

I-(20,13, 9): Bachet-Labosne

I-(42,27,12): Bachet-Labosne

II-(10,3,2;6) Leacock

= II(10,3,2;4)

II-(11,4,3;9): McKay

= II(11,4,3;2)

II-( (,5,3;1): Wood, Serebriakoff, Diagram Group

II-( (,5,3;4): Chuquet, Wood, Fireside Amusements,

II-( (,7,4;5): Meyer, Stein, Brandes

II-( (,8,5;11): Young World,

III-(20;19,13,7;10): Devi

General problem, usually form I, sometimes form II: Bachet-Labosne, Schubert, Ahrens, Cowley, Tweedie, Grossman, Buker, Goodstein, Browne, Scott, Currie, Sawyer, Court, O'Beirne, Lawrence, McDiarmid & Alfonsin.

Versions with 4 or more jugs: Tartaglia, Anon: Problems drive (1958), Anon (1961), O'Beirne.

Impossible versions: Pacioli, Bachet, Anon: Problems drive (1958).

Abbot Albert. c1240. Prob. 4, p. 333. I-(8,5,3) -- one solution.

Columbia Algorism. c1350. Chap. 123: I-(8,5,3). Cowley 402-403 & plate opposite 403. The plate shows the text and three jars. I have a colour slide of the three jars from the MS.

Munich 14684. 14C. Prob. XVIII & XXIX, pp. 80 & 83. I-(8,5,3).

Folkerts. Aufgabensammlungen. 13-15C. 16 sources with I-(8,5,3).

Pseudo-dell'Abbaco. c1440. Prob. 66, p.62. I-(8,5,3) -- one solution. "This problem is of little utility ...." I have a colour slide of this.

Chuquet. 1484. Prob. 165. Measure 4 from a cask using 5 and 3. You can pour back into the cask, i.e. this is II-((,5,3;4). FHM 233 calls this the tavern-keeper's problem.

HB.XI.22. 1488. P. 55 (= Rath 248). Same as Abbot Albert.

Pacioli. De Viribus. c1500.

Ff. 97r - 97v. LIII. C(apitolo). apartire una botte de vino fra doi (To divide a bottle of wine between two). = Peirani 137-138. I-(8,5,3). One solution.

Ff. 97v - 98v. LIIII. C(apitolo). a partire unaltra botte fra doi (to divide another bottle between two). = Peirani 138-139. I-(12,7,5). Dario Uri points out that the solution is confused and he repeats himself so it takes him 18 pourings instead of the usual 11. He then says one can divide 18 among three brothers who have containers of sizes 5, 6, 7, which he does by filling the 6 and then the problem is reduced to the previous problem. [He could do it rather more easily by pouring the 6 into the 7 and then refilling the 6!]

Ff. 98v - 99r. LV. (Capitolo) de doi altri sotili divisioni. de botti co'me se dira (Of two other subtle divisions of bottles as described). = Peirani 139-140. I-(10,6,4) and I-(12,8,4). Pacioli suggests giving these to idiots.

Ghaligai. Practica D'Arithmetica. 1521. Prob. 20, ff. 64v-65r. I-(8,5,3). One solution.

Cardan. Practica Arithmetice. 1539. Chap. 66, section 33, f. DD.iiii.v (p. 145). I-(8,5,3). Gives one solution and says one can go the other way.

H&S 51 says I-(8,5,3) case is also in Trenchant (1566). ??NYS

Tartaglia. General Trattato, 1556, art. 132 & 133, p. 255v-256r.

Art. 132: I-(8,5,3).

Art. 133: divide 24 in thirds, using 5, 11, 13.

Buteo. Logistica. 1559. Prob. 73, pp. 282-283. I-(8,5,3).

Gori. Libro di arimetricha. 1571. Ff. 71r-71v (p. 76). I-(8,5,3).

Bachet. Problemes. 1612. Addl. prob. III: Deux bons compagnons ont 8 pintes de vin à partager entre eux également, ..., 1612: 134-139; 1624: 206-211; 1884: 138-147. I-(8,5,3)  -- both solutions; I-(12,8,5) (omitted by Labosne). Labosne adds I-(16,9,7); I-(16,11,6); I-(42,27,12); I-(20,13,9); I-(16,12,7) (an impossible case!) and discusses general case. (This seems to be the first discussion of the general case.)

van Etten. 1624. Prob. 9 (9), pp. 11 & fig. opp. p. 1 (pp. 22-23). I-(8,5,3) -- one solution. Henrion's Nottes, 1630, pp. 11-13, gives the second solution and poses and solves I-(12,7,5).

Hunt. 1631 (1651). P. 270 (262). I-(8,5,3). One solution.

Yoshida (Shichibei) Kōyū (= Mitsuyoshi Yoshida) (1598-1672). Jinkō-ki. 2nd ed., 1634 or 1641??. ??NYS The recreational problems are discussed in Kazuo Shimodaira; Recreative Problems on "Jingōki", a 15 pp booklet sent by Shigeo Takagi. [This has no details, but Takagi says it is a paper that Shimodaira read at the 15th International Conference for the History of Science, Edinburgh, Aug 1977 and that it appeared in Japanese Studies in the History of Science 16 (1977) 95-103. I suspect this is a copy of a preprint.] This gives both Jingōki and Jinkōki as English versions of the title and says the recreational problems did not appear in the first edition, 4 vols., 1627, but did appear in the second edition of 5 vols. (which may be the first use of coloured wood cuts in Japan), with the recreational problems occurring in vol. 5. He doesn't give a date, but Mikami, p. 179, indicates that it is 1634, with further editions in 1641, 1675, though an earlier work by Mikami (1910) says 2nd ed. is 1641. Yoshida (or Suminokura) is the family name. Shimodaira refers to the current year as the 350th anniversary of the edition and says copies of it were published then. I have a recent transcription of some of Yoshida into modern Japanese and a more recent translation into English, ??NYR, but I don't know if it is the work mentioned by Shimodaira.

Shimodaira discusses a jug problem on p. 14: I-(10,7,3) -- solution in 10 moves. Shimodaira thinks Yoshida heard about such puzzles from European contacts, but without numerical values, then made up the numbers. I certainly can see no other example of these numbers. The recent transcription includes this material as prob. 7 on pp. 69-70.

Wingate/Kersey. 1678?. Prob. 7, pp. 543-544. I-(8,5,3). Says there is a second way to do it.

Witgeest. Het Natuurlyk Tover-Boek. 1686. Prob. 38, p. 308. I-(8,5,3).

Ozanam. 1694.

Prob. 36, 1696: 91-92; 1708: 82-83. Prob. 42, 1725: 238-240. Prob. 21, 1778: 175-177; 1803: 174-176; 1814: 153-154. Prob. 20, 1840: 79. I-(8,5,3) -- both solutions.

Prob. 43, 1725: 240-241. Prob. 22, 1778: 177-178; 1803: 176-177; 1814: 154-155. Prob. 21, 1840: 79-80. I-(12,7,5) -- one solution.

Dilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant and diverting Questions, p. 168. Problem 8. I-(8,5,3). (Dilworth cites Wingate for this -- cf in 5.B.) = D. Adams; Scholar's Arithmetic; 1801, p. 200, no. 10.

Les Amusemens. 1749. Prob. 17, p. 139: Partages égaux avec des Vases inégaux. I-(8,5,3) -- both solutions.

Bestelmeier. 1801. Item 416: Die 3 Maas-Gefäss. I-(12,7,5).

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 48-49, no. 75: How to part an eight gallon bottle of wine, equally between two persons, using only two other bottles, one of five gallons, and the other of three. Gives both solutions.

Jackson. Rational Amusement. 1821. Arithmetical Puzzles.

No. 14, pp. 4 & 54. I-( 8,5,3). One solution.

No. 52, pp. 12 & 67. I-(12,7,5). One solution.

Rational Recreations. 1824. Exer. 10, p. 55. I-(8,5,3) one way.

Manuel des Sorciers. 1825. ??NX

Pp. 55-56, art. 27-28. I-(8,5,3) two ways.

P. 56, art. 29. I-(12,7,5).

Endless Amusement II. 1826? Prob. 7, pp. 193-194. I-(8,5,3). One solution. = New Sphinx, c1840, p. 133.

Nuts to Crack III (1834), no. 212. I-(8,5,3). 8 gallons of spirits.

Young Man's Book. 1839. Pp. 43-44. I-(8,5,3). Identical to Wingate/Kersey.

The New Sphinx. c1840. P. 133. I-(8,5,3). One solution.

Boy's Own Book. 1843 (Paris): 436 & 441, no. 7. The can of ale: 1855: 395; 1868: 432. I-(8,5,3). One solution. The 1843 (Paris) reads as though the owners of the 3 and 5 kegs both want to get 4, which would be a problem for the owner of the 3. = Boy's Treasury, 1844, pp. 425 & 429.

Fireside Amusements. 1850. Prob. 9, pp. 132 & 184. II-((,5,3;4). One solution.

Arago. [Biographie de] Poisson (16 Dec 1850). Oeuvres, Gide & Baudry, Paris, vol. 2, 1854, pp. 593-??? P. 596 gives the story of Poisson's being fascinated by the problem I-(12,8,5). "Poisson résolut à l'instant cette question et d'autres dont on lui donna l'énoncé. Il venait de trouver sa véritable vocation." No solution given by Arago.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Arithmetical puzzles, no. 8, pp. 174-175 (1868: 185-186). I-(8,5,3). Milkmaid with eight quarts of milk.

Magician's Own Book. 1857.

P. 223-224: Dividing the beer: I-(8,5,3).

P. 224: The difficult case of wine: I-(12,7,5).

Pp. 235-236: The two travellers: I-(8,5,3) posed in verse.

Each problem gives just one solution.

Boy's Own Conjuring Book. 1860.

P. 193: Dividing the beer: I-(8,5,3).

P. 194: The difficult case of wine: I-(12,7,5).

Pp. 202-203: The two travellers: I-(8,5,3) posed in verse.

Each problem gives just one solution.

Illustrated Boy's Own Treasury. 1860. Prob. 21, pp. 428-429 & 433. I-(8,5,3). "A man coming from the Lochrin distillery with an 8-pint jar full of spirits, ...."

Vinot. 1860. Art. XXXVIII: Les cadeaux difficiles, pp. 57-58. I-(8,5,3). Two solutions.

The Secret Out (UK). c1860. To divide equally eight pints of wine ..., pp. 12-13.

Bachet-Labosne. 1874. For details, see Bachet, 1612. Labosne adds a consideration of the general case which seems to be the first such.

Kamp. Op. cit. in 5.B. 1877. No. 17, p. 326: I-(8,5,3).

Mittenzwey. 1880. Prob. 106, pp. 22 & 73-74; 1895?: 123, pp. 26 & 75-76; 1917: 123, pp. 24 & 73-74. I-(8,5,3). One solution.

Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 135, no. 1. I-(8,5,3). No solution.

Loyd. Problem 11: "Two thieves of Damascus". Tit-Bits 31 (19 Dec 1896 & 16 Jan 1897) 211 & 287. Thieves found with 2 & 2 quarts in pails of size 3 & 5. They claim the merchant measured the amounts out from a fresh hogshead. Solution is that this could only be done if the merchant drained the hogshead, which is unreasonable!

Loyd. Problem 13: The Oriental problem. Tit-Bits 31 (19 Jan, 30 Jan & 6 Feb 1897) 269, 325 & 343. = Cyclopedia, 1914, pp. 188 & 364: The merchant of Bagdad. Complex problem with hogshead of water, barrel of honey, three 10 gallon jugs to be filled with 3 gallons of water, of honey and of half and half honey & water. There are a 2 and a 4 gallon measure and also 13 camels to receive 3 gallons of water each. Solution takes 521 steps. 6 Feb reports solutions in 516 and 513 steps. Cyclopedia gives solution in 506 steps.

Dudeney. The host's puzzle. London Magazine 8 (No. 46) (May 1902) 370 & 8 (No. 47) (Jun 1902) 481-482 (= CP, prob. 6, pp. 28-29 & 166-167). Use 5 and 3 to obtain 1 and 1 from a cask. One must drink some!

H. Schubert. Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1907. Vol. 1, chap. 6, Umfüllungs-Aufgaben, pp. 48-56. Studies general case and obtains some results. (The material appeared earlier in Zwölf Geduldspiele, 1895, op. cit. in 5.A, Chap. IX, pp. 110-119. The 13th ed. (De Gruyter, Berlin, 1967), Chap. 9, pp. 62-70, seems to be a bit more general (??re-read).)

Ahrens. MUS I, 1910, chap. 4, Umfüllungsaufgaben, pp. 105-124. Pp. 106-107 is Arago's story of Poisson and this problem. He also extends and corrects Schubert's work.

Dudeney. Perplexities: No. 141: New measuring puzzle. Strand Magazine 45 (Jun 1913) 710 & 46 (Jul 1913) 110. (= AM, prob. 365, pp. 110 & 235.) Two 10 quart vessels of wine with 5 and 4 quart measures. He wants 3 quarts in each measure. (Dudeney gives numerous other versions in AM.)

Loyd. Cyclopedia. 1914. Milkman's puzzle, pp. 52 & 345. (= MPSL2, prob. 23, pp. 17 & 127-128 = SLAHP: Honest John, the milkman, pp. 21 & 90.) Milkman has two full 40 quart containers and two customers with 5 and 4 quart pails, but both want 2 quarts. (Loyd Jr. says "I first published [this] in 1900...")

Williams. Home Entertainments. 1914. The measures puzzle, p. 125. I-(8,5,3).

Hummerston. Fun, Mirth & Mystery. 1924. A shortage of milk, Puzzle no. 75, pp. 164 & 183. I-(8,5,3), one solution.

Elizabeth B. Cowley. Note on a linear diophantine equation. AMM 33 (1926) 379-381. Presents a technique for resolving I-(a,b,c), which gives the result when a = b+c. If a < b+c, she only seems to determine whether the method gets to a point with A empty and neither B nor C full and it is not clear to me that this implies impossibility. She mentions a graphical method of Laisant (Assoc. Franç. Avance. Sci, 1887, pp. 218-235) ??NYS.

Wood. Oddities. 1927.

Prob. 15: A problem in pints, pp. 16-17. Small cask and measures of size 5 and 3, measure out 1 in each measure. Starts by filling the 5 and the 3 and then emptying the cask, so this becomes a variant of II-((,5,3;1).

Prob. 26: The water-boy's problem, pp. 28-29. II-((;,5,3;4).

Ernest K. Chapin. Scientific Problems and Puzzles. In: S. Loyd Jr.; Tricks and Puzzles, Vol. 1 (only volume to appear); Experimenter Publishing Co., NY, nd [1927] and Answers to Sam Loyd's Tricks and Puzzles, nd [1927]. [This book is a selection of pages from the Cyclopedia, supplemented with about 20 pages by Chapin and some other material.] P. 89 & Answers p. 8. You have a tablet that has to be dissolved in 7½ quarts of water, though you only need 5 quarts of the resulting mixture. You have 3 and 5 quart measures and a tap.

Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 298. "He's trying to think how a farmer with a ten-gallon can and a three-gallon can and a two-gallon can, manages to measure out six gallons of milk." II-(10,3,2;6) = II-(10,3,2;4).

M. C. K. Tweedie. A graphical method of solving Tartaglian measuring puzzles. MG 23 (1939) 278-282. The elegant solution method using triangular coordinates.

H. D. Grossman. A generalization of the water-fetching puzzle. AMM 47 (1940) 374-375. Shows II-((,b,c;d) with GCD(b,c) = 1 is solvable.

McKay. Party Night. 1940.

No. 18, p. 179. II-(11,4,3;9).

No. 19, pp. 179-180. I-(8,5,3).

Meyer. Big Fun Book. 1940. No. 10, pp. 165 & 753. II-((,7,4,5).

W. E. Buker, proposer. Problem E451. AMM 48 (1941) 65. ??NX. General problem of what amounts are obtainable using three jugs, one full to start with, i.e. I-(a,b,c). See Browne, Scott, Currie below.

Eric Goodstein. Note 153: The measuring problem. MG 25 (No. 263) (Feb 1941) 49-51. Shows II-((,b,c;d) with GCD(b,c) = 1 is solvable.

D. H. Browne & Editors. Partial solution of Problem E451. AMM 49 (1942) 125-127.

W. Scott. Partial solution of E451 -- The generalized water-fetching puzzle. AMM 51 (1944) 592. Counterexample to conjecture in previous entry.

J. C. Currie. Partial solution of Problem E451. AMM 53 (1946) 36-40. Technical and not complete.

W. W. Sawyer. On a well known puzzle. SM 16 (1950) 107-110. Shows that I-(b+c,b,c) is solvable if b & c are relatively prime.

David Stein. Party and Indoor Games. Op. cit. in 5.B. c1950. Prob. 13, pp. 79-80. Obtain 5 from a spring using measures 7 and 4, i.e. II-((,7,4,5).

Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 8. Given an infinite source, use: 6, 10, 15 to obtain 1, 6, 7 simultaneously; 4, 6, 9, 12 to obtain 1, 2, 3, 4 simultaneously; 6, 9, 12, 15, 21 to obtain 1, 3, 6, 8, 9 simultaneously. Answer simply says the first two are possible (the second being easy) and the third is impossible.

Young World. c1960. P. 58: The 11 pint problem. II-((,8,5;11). This is the same as II-(13,8,5;11) or II-(13,8,5,2).

Anonymous. Moonshine sharing. RMM 2 (Apr 1961) 31 & 3 (Jun 1961) 46. Divide 24 in thirds using cylindrical containers holding 10, 11, 13. Solution in No. 3 uses the cylindricity of a container to get it half full.

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. "Pouring" problems -- The "robot" method. General description of the problem. Attributes Tweedie's triangular 'bouncing ball' method to Perelman, with no reference. Does I-(8,5,3) two ways, also I-(12,7,5) and I-(16,9,7), then considers type II questions. Considers the problem with II-(10,6,4;d) and extends to II-(a,6,4;d) for a > 10, leaving it to the reader to "try to formulate some rule about the results." He then considers II-(7,6,4;d), noting that the parallelogram has a corner trimmed off. Then considers II-(12,9,7;d) and II-(9,6,3;d).

Lloyd Jim Steiger. Letter. RMM 4 (Aug 1961) 62. Solves the RMM 2 problem by putting the 10 inside the 13 to measure 3.

Irving & Peggy Adler. The Adler Book of Puzzles and Riddles. Or Sam Loyd Up-To-Date. John Day, NY, 1962. Pp. 32 & 46. Farmer has two full 10-gallon cans. Girls come with 5-quart and 4-quart cans and each wants 2 quarts.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 80, pp. 81 & 109. Tavern has a barrel with 15 pints of beer. Two customers, with 3 pint and 5 pint jugs appear and ask for 1 pint in each jug. Bartender finds it necessary to drink the other 13 pints!

T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 4: Jug and bottle department, pp. 49-75. This gives an extensive discussion of Tweedie's method and various extensions to four containers, a barrel of unknown size, etc.

P. M. Lawrence. An algebraic approach to some pouring problems. MG 56 (No. 395) (Feb 1972) 13-14. Shows II-((,b,c,d) with d ( b+c and GCD(b,c) = 1 is possible and extends to more jugs.

Louis Grant Brandes. The Math. Wizard. revised ed., J. Weston Walch, Portland, Maine, 1975. Prob. 5: Getting five gallons of water: II-((,7,4,5).

Shakuntala Devi. Puzzles to Puzzle You. Orient Paperbacks (Vision Press), Delhi, 1976.

Prob. 53: The three containers, pp. 57 & 110. III-(20;19,13,7;10). Solution in 15 steps. Looking at the triangular coordinates diagram of this, one sees that it is actually isomorphic to II-(19,13,7;10) and this can be seen by considering the amounts of empty space in the containers.

Prob. 132: Mr. Portchester's problem, pp. 82 & 132. Same as Dudeney (1913).

Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London, 1991.) Problem T.16: Pouring puttonos, part b, pp. 19-20 (1991: 37-38) & Answer 19, pp. 102-103 (1991: 118-119). II-( (,5,3;1).

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 161, with Solution at the back of the book. II-((,5,3;1), which can be done as II-(8,5,3;1).

D. St. P. Barnard. 50 Daily Telegraph Brain Twisters. 1985. Op. cit. in 4.A.4. Prob. 4: Measure for measure, pp. 15, 79-80, 103. Given 10 pints of milk, an 8 pint bowl, a jug and a flask. He describes how he divides the milk in halves and you must deduce the size of the jug and the flask.

Colin J. H. McDiarmid & Jorge Ramirez Alfonsin. Sharing jugs of wine. Discrete Mathematics 125 (1994) 279-287. Solves I-(b+c,b,c) and discusses the problem of getting from one state of the problem to another in a given number of steps, showing that GCD(b,c) = 1 guarantees the graph is connected. indeed essentially cyclic. Considers GCD(b,c) ( 1. Notes that the work done easily extends to a > b + c. Says the second author's PhD at Oxford, 1993, deals with more cases.

John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 11: The spoon and the bottle. Given a 160 ml bottle and a 30 ml spoon, measure 230 ml into a bucket.

5.D.2. RULER WITH MINIMAL NUMBER OF MARKS

Dudeney. Problem 518: The damaged measure. Strand Mag. (Sep 1920) ??NX. Wants a minimal ruler for 33 inches total length. (=? MP 180)

Dudeney. Problem 530: The six cottagers. Strand Mag. (Jan 1921) ??NX. Wants 6 points on a circle to give all arc distances 1, 2, ..., 20. (=? MP 181)

Percy Alexander MacMahon. The prime numbers of measurement on a scale. Proc. Camb. Philos. Soc. 21 (1922-23) 651-654. He considers the infinite case, i.e. a(0) = 0, a(i+1) = a(i) + least integer which is not yet measurable. This gives: 0, 1, 3, 7, 12, 20, 30, 44, ....

Dudeney. MP. 1926.

Prob. 180: The damaged measure, pp. 77 & 167. (= 536, prob. 453, pp. 173, 383-384.) Mark a ruler of length 33 with 8 (internal) marks. Gives 16 solutions.

Prob. 181: The six cottagers, pp. 77-78 & 167. = 536, prob. 454, pp. 174 & 384.

A. Brauer. A problem of additive number theory and its application in electrical engineering. J. Elisha Mitchell Sci. Soc. 61 (1945) 55-56. Problem arises in designing a resistance box.

Л. Редеи & А. Реньи [L. Redei & A. Ren'i (Rényi)]. О представленин чисел 1, 2, ..., N лосредством разностей [O predstavlenin chisel 1, 2, ... , N losredstvom raznosteĭ (On the representation of 1, 2, ..., N by differences)]. Мат. Сборник [Mat. Sbornik] 66 (NS 24) (1949) 385-389.

Anonymous. An unsolved problem. Eureka 11 (Jan 1949) 11 & 30. Place as few marks as possible to permit measuring integers up to n. For n = 13, an example is: 0, 1, 2, 6, 10, 13. Mentions some general results for a circle.

John Leech. On the representation of 1, 2, ..., n by differences. J. London Math. Soc. 31 (1956) 160-169. Improves Redei & Rényi's results. Gives best examples for small n.

Anon. Puzzle column: What's your potential? MTg 19 (1962) 35 & 20 (1962) 43. Problem posed in terms of transformer outputs -- can we arrange 6 outputs to give every integral voltage up through 15? Problem also asks for the general case. Solution asserts, without real proof, that the optimum occurs with 0, 1, 4, 7, 10, ..., n-11, n-8, n-5, n-2 or its complement.

T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 6 discusses several versions of the problem.

Gardner. SA (Jan 1965) c= Magic Numbers, chap. 6. Describes 1, 2, 6, 10 on a ruler 13 long. Says 3 marks are sufficient on 9 and 4 marks on 12 and asks for proof of the latter and for the maximum number of distances that 3 marks on 12 can produce. How can you mark a ruler 36 long? Says Dudeney, MP prob. 180, believed that 9 marks were needed for a ruler longer than 33, but Leech managed to show 8 was sufficient up to 36.

C. J. Cooke. Differences. MTg 47 (1969) 16. Says the problem in MTg 19 (1962) appears in H. L. Dorwart's The Geometry of Incidence (1966) related to perfect difference sets but with an erroneous definition which is corrected by references to H. J. Ryser's Combinatorial Mathematics. However, this doesn't prove the assertions made in MTg 20.

Jonathan Always. Puzzles for Puzzlers. Tandem, 1971. Prob. 22: Starting and stopping, pp. 18 & 66. Circular track, 1900 yards around. How can one place marker posts so every multiple of 100 yards up to 1900 can be run. Answer: at 0, 1, 3, 9, 15.

Gardner. SA (Mar 1972) = Wheels, Chap. 15.

5.D.3 FALSE COINS WITH A WEIGHING SCALE

H. S. Shapiro, proposer; N. J. Fine, solver. Problem E1399 -- Counterfeit coins. AMM 67 (1960) 82 & 697-698. Genuines weigh 10, counterfeits weigh 9. Given 5 coins and a scale, how many weighings are needed to find the counterfeits? Answer is 4. Fine conjectures that the ratio of weighings to coins decreases to 0.

Kobon Fujimura & J. A. H. Hunter, proposers; editorial solution. There's always a way. RMM 6 (Dec 1961) 47 & 7 (Feb 1962) 53. (c= Fujimura's The Tokyo Puzzles (Muller, London, 1979), prob. 29: Pachinko balls, pp. 35 & 131.) Six coins, one false. Determine which is false and whether it is heavy or light in three weighings on a scale. In fact one also finds the actual weights.

K. Fujimura, proposer; editorial solution. The 15-coin puzzle. RMM 9 (Jun 1962) & 10 (Aug 1962) 40-41. Same problem with fifteen coins and four weighings.

5.D.4. TIMING WITH HOURGLASSES

I have just started these and they are undoubtedly older than the examples here. I don't recall ever seeing a general approach to these problems.

Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 17: Two-minute eggs, pp. 9 & 87. Time 2 minutes with 3 & 5 minute timers.

Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 21: The sands of time, pp. 35 & 93. Time 9 minutes with 4 & 7 minute timers.

David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 73-74. Time 9 minutes with 4 & 7 minute timers.

Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, prob. 8: Grandfather's breakfast, pp. 6 & 102. Time 15 minutes with 7 & 11 minute timers.

5.D.5. MEASURE HALF A BARREL

I have just started this and there must be much older examples.

Benson. 1904. The water-glass puzzle, p. 254.

Dudeney. AM. 1917. Prob. 364: The barrel puzzle, pp. 109-110 & 235.

King. Best 100. 1927. No. 1, pp. 7 & 38.

Collins. Fun with Figures. 1928. The dairymaid's problem, pp. 29-30.

William A. Bagley. Puzzle Pie. Vawser & Wiles, London, nd [BMC gives 1944]. [There is a revised edition, but it only affects material on angle trisection.] No. 14: 'Arf an' 'arf, p. 15.

Anon. The Little Puzzle Book. Peter Pauper Press, Mount Vernon, NY, 1955. P. 52: The cider barrel.

Jonathan Always. Puzzles for Puzzlers. Tandem, London, 1971. Prob. 87: But me no butts, pp. 42 & 88.

Richard I. Hess. Email Christmas message to NOBNET, 24 Nov 2000. Solution sent by Nick Baxter on the same day. You have aquaria (assumed cuboidal) which hold 7 and 12 gallons and a water supply. The 12 gallon aquarium has dots accurately placed in the centre of each side face. How many steps are required to get 8 gallons into the 12 gallon aquarium? Fill the 12 gallon aquarium and tilt it on one corner so the water level passes through the centres of the two opposite faces. This leaves 8 gallons! Nick says this is two steps.

5.E. EULER CIRCUITS AND MAZES

Euler circuits have been used in primitive art, often as symbols of the passage of the soul to the land of the dead. [MTg 110 (Mar 1985) 55] shows examples from Angola and New Hebrides. See Ascher (1988 & 1991) for many other examples from other cultures.

┌────┬─────────┬────┐

│ │ │ │

├────┴────┬────┴────┤

│ │ │

└─────────┴─────────┘

Above is the 'five-brick pattern'. See: Clausen, Listing, Kamp, White, Dudeney, Loyd Jr, Ripley, Meyer, Leeming, Adams, Anon., Ascher. Prior to Loyd Jr, the problem asked for the edges to be drawn in three paths, but about 1920 the problem changed to drawing a path across every wall.

Trick solutions: Tom Tit, Dudeney (1913), Houdini, Loyd Jr, Ripley, Meyer, Leeming, Adams, Gibson, Anon. (1986).

Non-crossing Euler circuits: Endless Amusement II, Bellew, Carroll 1869, Mittenzwey, Bile Beans, Meyer, Gardner (1964), Willson, Scott, Singmaster.

Kn denotes the complete graph on n vertices.

Matthäus Merian the Elder. Engraved map of Königsberg. Bernhard Wiezorke has sent me a coloured reproduction of this, dated as 1641. He used an B&W version in his article: Puzzles und Brainteasers; OR News, Ausgabe 13 (Nov 2001) 52-54. BLW use a B&W version on their dust jacket and on p. 2 which they attribute to M. Zeiller; Topographia Prussiae et Pomerelliae; Frankfurt, c1650. I have seen this in a facsimile of the Cosmographica due to Merian in the volume on Brandenburg and Pomerania, but it was not coloured. There seem to be at least two versions of this picture --??CHECK.

L. Euler. Solutio problematis ad geometriam situs pertinentis. (Comm. Acad. Sci. Petropol. 8 (1736(1741)) 128-140.) = Opera Omnia (1) 7 (1923) 1-10. English version: Seven Bridges of Königsberg is in: BLW, 3-8; SA 189 (Jul 1953) 66-70; World of Mathematics, vol. 1, 573-580; Struik, Source Book, 183-187.

My late colleague Jeremy Wyndham became interested in the seven bridges problem and made inquiries which turned up several maps of Königsberg and a list of all the bridges and their dates of construction (though there is some ambiguity about one bridge). The first bridge was built in 1286 and until the seventh bridge of 1542, an Euler path was always possible. No further bridge was built until a railway bridge in 1865 which led to Saalschütz's 1876 paper -- see below. In 1905 and later, several more bridges were added, reaching a maximum of ten bridges in 1926 (with 4512 paths from the island), then one was removed in 1933. Then a road bridge was added, but it is so far out that it does not show on any map I have seen. Bombing and fighting in 1944-1945 apparently destroyed all the bridges and the Russians have rebuilt six or seven of them. I have computed the number of paths in each case -- from 1865 until 1935 or 1944, there were always Euler paths.

L. Poinsot. Sur les polygones et les polyèdres. J. École Polytech. 4 (Cah. 10) (1810) 16-48. Pp. 28-33 give Euler paths on K2n+1 and Euler's criterion. Discusses square with diagonals.

Endless Amusement II. 1826? Prob. 34, p. 211. Pattern of two overlapping squares has a non-crossing Euler circuit.

Th. Clausen. De linearum tertii ordinis propietatibus. Astronomische Nachrichten 21 (No. 494) (1844), col. 209-216. At the very end, he gives the five-brick pattern and says that its edges cannot be drawn in three paths.

J. B. Listing. Vorstudien zur Topologie. Göttinger Studien 1 (1847) 811-875. ??NYR. Gives five brick pattern as in Clausen.

?? Nouv. Ann. Math. 8 (1849?) 74. ??NYS. Lucas says this poses the problem of finding the number of linear arrangements of a set of dominoes. [For a double N set, N = 2n, this is (2n+1)(n+1) times the number of circular arrangements, which is n2n+1 times the number of Euler circuits on K2n+1.]

É. Coupy. Solution d'un problème appartenant a la géométrie de situation, par Euler. Nouv. Ann. Math. 10 (1851) 106-119. Translation of Euler. Translator's note on p. 119 applies it to the bridges of Paris.

The Sociable. 1858. Prob. 7: Puzzle pleasure garden, pp. 288 & 303. Large maze-like garden and one is to pass over every path just once -- phrased in verse. = Book of 500 Puzzles, 1859, prob. 7, pp. 6 & 21. = Illustrated Boy's Own Treasury, 1860, prob. 49, pp. 405 & 443. In fact, if one goes straight across every intersection, one finds the path, so this is really almost a unicursal problem.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 587, pp. 297 & 410: Ariadnerätsel. Three diagrams to trace with single lines. No attempt to avoid crossings.

Frank Bellew. The Art of Amusing. Carleton, NY (& Sampson Low & Co., London), 1866 [C&B list a 1871]; John Camden Hotten, London, nd [BMC & NUC say 1870] and John Grant, Edinburgh, nd [c1870 or 1866?], with slightly different pagination. 1866: pp. 269-270; 1870: p. 266. Two overlapping squares have a non-crossing Euler circuit.

Lewis Carroll. Letter of 22 Aug 1869 to Isabel Standen. Taken from: Stuart Dodgson Collingwood; The Life and Letters of Lewis Carroll; T. Fisher Unwin, London, (Dec 1898), 2nd ed., Jan 1899, p. 370: "Have you succeeded in drawing the three squares?" On pp. 369-370, the recipient is identified as Isabel Standen and she is writing Collingwood, apparently sending him the letter. Collingwood interpolates: "This puzzle was, by the way, a great favourite of his; the problem is to draw three interlaced squares without going over the same lines twice, or taking the pen off the paper". But no diagram is given.

Dudeney; Some much-discussed puzzles; op. cit. in 2; 1908, quotes Collingwood, gives the diagram and continues: "This is sometimes ascribed to him [i.e. Carroll] as its originator, but I have found it in a little book published in 1835." This was probably a printing of Endless Amusement II, qv above and in Common References, though this has two interlaced squares. John Fisher; The Magic of Lewis Carroll; op. cit. in 1; pp. 58-59, says Carroll would ask for a non-crossing Euler circuit, but this is not clearly stated in Collingwood. Cf Carroll-Wakeling, prob. 29: The three squares, pp. 38 & 72, which clearly states that a non-crossing circuit is wanted and notes that there is more than one solution. Cf Gardner (1964). Carroll-Gardner, pp. 52-53.

Mittenzwey. 1880. Prob. 269-279, pp. 47-48 & 98-100; 1895?: 298-308, pp. 51-52 & 100-102; 1917: 298-308, pp. 46-48 & 95-97. Straightforward unicursal patterns. The first is K5, but one of the diagonals was missing in my copy of the 1st ed. -- the path is not to use two consecutive outer edges. The third is the 'envelope' pattern. The fourth is three overlapping squares, where the two outer squares just touch in the middle. The last is a simple maze with no dead ends and the path is not to cross itself. See also the entry for Mittenzwey in 5.E.1, below.

M. Reiss. Évaluation du nombre de combinaisons desquelles les 28 dés d'un jeu de dominos sont susceptibles d'après la règle de ce jeu. Annali di Matematica Pura ed Applicata (2) 5 (1871) 63-120. Determines the number of linear arrangements of a double-6 set of dominoes, which gives the number of Euler circuits on K7.

L. Saalschütz. [Report of a lecture.] Schriften der Physikalisch-Ökonomischen Gesellschaft zu Königsberg 16 (1876) 23-24. Sketches Euler's work, listing the seven bridges. Says that a recent railway bridge, of 1865, connecting regions B and C on Euler's diagram, can be considered within the walkable region. He shows there are 48 x 2 x 4 = 384 possible paths -- the 48 are the lists of regions visited starting with A; the 2 corresponds to reversing these lists; the 4 (= 2 x 2) corresponds to taking each of the two pairs of bridges connecting the same regions in either order, He lists the 48 sequences of regions which start at A. I wrote a program to compute Euler paths and I tested it on this situation. I find that Saalschütz has omitted two cases, leading to four sequences or 16 paths starting at A or 32 paths considering both directions. That is, his 48 should be 52 and his 384 should be 416.

Kamp. Op. cit. in 5.B. 1877. Pp. 322-327 show several unicursal problems.

No. 8 is the five-brick pattern as in Clausen.

No. 10 is two overlapping squares.

No. 11 is a diagram from which one must remove some lines to leave an Eulerian figure.

C. Hierholzer. Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math. Annalen 6 (1873) 30-32. (English is in BLW, 11-12.)

G. Tarry. Géométrie de situation: Nombre de manières distinctes de parcourir en une seule course toutes les allées d'un labyrinthe rentrant, en ne passant qu'une seule fois par chacune des allées. Comptes Rendus Assoc. Franç. Avance. Sci. 15, part 2 (1886) 49-53 & Plates I & II. General technique for the number of Euler circuits.

Lucas. RM2. 1883. Le jeu de dominos -- Dispositions rectilignes, pp. 63-77 & Note 1: Sur le jeu de dominos, p. 229.

RM4. 1894. La géométrie des réseaux et le problème des dominos, pp. 123-151.

Cites Reiss's work and says (in RM4) that it has been confirmed by Jolivald. The note in RM2 is expanded in RM4 to explain the connection between dominoes and K2n+1. There are obviously 2 Euler circuits on K3. He sketches Tarry's method and uses it to compute that K5 has 88 Euler circuits and K7 has 1299 76320. [This gives 28 42582 11840 domino rings for the double-6 set.] He says Tarry has found that K9 has 911 52005 70212 35200.

Tom Tit, vol. 3. 1893. Le rectangle et ses diagonales, pp. 155-156. = K, no. 16: The rectangle and its diagonals, pp. 46-48. = R&A, The secret of the rectangle, p. 100. Trick solutions by folding the paper and making an arc on the back.

Hoffmann. 1893. Chap. X, no. 9: Single-stroke figures, pp. 338 & 375 = Hoffmann-Hordern, pp. 230-231. Three figures, including the double crescent 'Seal of Mahomet'. Answer states Euler's condition.

Dudeney. The shipman's puzzle. London Mag. 9 (No. 49) (Aug 1902) 88-89 & 9 (No. 50) (Sep 1902) 219 (= CP, prob. 18, pp. 40-41 & 173). Number of Euler circuits on K5.

Benson. 1904. A geometrical problem, p. 255. Seal of Mahomet.

William F. White. A Scrap-Book of Elementary Mathematics. Open Court, 1908. [The 4th ed., 1942, is identical in content and pagination, omitting only the Frontispiece and the publisher's catalogue.] Bridges and isles, figure tracing, unicursal signatures, labyrinths, pp. 170-179. On p. 174, he gives the five-brick puzzle, asking for a route along its edges.

Dudeney. Perplexities: No. 147: An old three-line puzzle. Strand Magazine 46 (Jul 1913) 110 & (Aug 1913) 221. c= AM, prob. 239: A juvenile puzzle, pp. 68-69 & 197. Five-brick form to be drawn or rubbed out on a board in three strokes. Either way requires doing two lines at once, either by folding the paper as you draw or using two fingers to rub out two lines at once. "I believe Houdin, the conjurer, was fond of showing this to his child friends, but it was invented before his time -- perhaps in the Stone Age."

Loyd. Problem of the bridges. Cyclopedia, 1914, pp. 155 & 359-360. = MPSL1, prob. 28, pp. 26-27 & 130-131. Eight bridges. Asks for number of routes.

Loyd. Puzzle of the letter carrier's route. Cyclopedia, 1914, pp 243 & 372. Asks for a circuit on a 3 x 4 array with a minimal length of repeated path.

Dudeney. AM. 1917.

Prob. 242: The tube inspector's puzzle, pp. 69 & 198. Minimal route on a 3 x 4 array.

Prob. 261: The monk and the bridges, pp. 75-76 & 202-203. River with one island. Four bridges from island, two to each side of the river, and another bridge over the river. How many Euler paths from a given side of the river to the other? Answer:  16.

Collins. Book of Puzzles. 1927. The fly on the octahedron, pp. 105-108. Asserts there are 1488 Euler circuits on the edges of an octahedron. He counts the reverse as a separate circuit.

Harry Houdini [pseud. of Ehrich Weiss] Houdini's Book of Magic. 1927 (??NYS); Pinnacle Books, NY, 1976, p. 19: Can you draw this? Take a square inscribed in a circle and draw both diagonals. "The idea is to draw the figure without taking your pencil off the paper and without retracing or crossing a line. There is a trick to it, but it can be done. The trick in drawing the figure is to fold the paper once and draw a straight line between the folded halves; then, not removing your pencil, unfold the paper. You will find that you have drawn two straight lines with one stroke. The rest is simple." This perplexed me for some time, but I believe the idea is that holding the pencil between the two parts of the folded sheet and moving the pencil parallel to the fold, one can draw a line, parallel to the fold, on each part.

Loyd Jr. SLAHP. 1928. Pp. 7-8. Discusses what he calls the "Five-brick puzzle", the common pattern of five rectangles in a rectangle. He says that the object was to draw the lines in four strokes -- which is easily done -- but that it was commonly misprinted as three strokes, which he managed to do by folding the paper. He says "a similar puzzle ... some ten or fifteen years ago" asked for a path crossing each of the 16 walls once, which is also impossible.

The Bile Beans Puzzle Book. 1933.

No. 32. Draw the triangular array of three on an edge without crossing.

No. 36. Draw the five-brick pattern in three lines. Folds paper and draws two lines at once.

R. Ripley. Believe It Or Not! Book 2. (Simon & Schuster, 1931); Pocket Books, NY, 1948, pp. 70-71. = Omnibus Believe It Or Not! Stanley Paul, London, nd [c1935?], p. 270. Gives the five-brick problem of drawing a path crossing each wall once, with the trick solution having the path going along a wall. Asserts "This unicursal problem was solved thus by the great Euler himself." and cites the Euler paper above!!

Meyer. Big Fun Book. 1940.

Tryangle, pp. 98 & 731. Triangle subdivided into triangles, with three small triangles along each edge. Draw an Euler circuit without crossings.

Cutting the walls, pp. 637 & 794. Five-brick problem. Solution has line crossing through a vertex.

Ern Shaw. The Pocket Brains Trust - No. 2. W. H. Allen, London, nd but inscribed 1944. Prob. 29: Five bricks teaser, pp. 10 & 39.

Leeming. 1946. Chap. 6, prob. 2: Through the walls, pp. 70 & 184. Five-brick puzzle, with trick solution having the path go through an intersection.

John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957? Prob. 49: In just one line, pp. 30 & 48-49. Five-brick puzzle, with answer having the path going along a wall, as in Ripley. Asserts Euler invented this solution.

Gibson. Op. cit. in 4.A.1.a. 1963. Pp. 70 & 75: The "impossible" diagram. Same as Tom Tit.

Gardner. SA (Apr 1964) = 6th Book, chap. 10. Says Carroll knew that a planar Eulerian graph could be drawn without crossings. Gives a method of O'Beirne for doing this -- two colour the regions and then make a path which separates the colours into simply connected regions.

Ripley's Puzzles and Games. 1966. P. 39. Euler paths on the 'envelope', i.e. a rectangle with its diagonals drawn and an extra connection between the top corners, looking like an unfolded envelope. Asserts the envelope has 50 solutions, but it is not clear if the central crossing is a further vertex. I did this by hand but did not get 50, so I wrote a program to count Euler paths. If the central crossing is not a vertex, then I find 44 paths from one of the odd vertices to the other, and of course 44 going the other way -- and I had found this number by hand. However, if the central crossing is a vertex, then my hand solution omitted some cases and the computer found 120 paths from one odd vertex to the other.

Pp. 40-43 give many problems of drawing non-crossing Euler paths or circuits.

W. Wynne Willson. How to abolish cross-roads. MTg 42 (Spring 1968) 56-59. Euler circuit of a planar graph can be made without crossings.

[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. Tempo (Grosset & Dunlap), NY, 1973 (& 1978?? -- both dates are given -- I'm presuming the 1978 is a 2nd ptg or a reissue under a different imprint??). One line/no crossing, pp. 85-86. Non-crossing Euler circuits on the triangular array of side 3 and non-crossing Euler paths on the 'envelope' -- cf under Ripley's, above. Asserts the envelope has 50 solutions. I adapted the program mentioned above to count the number of non-crossing Euler paths -- one must rearrange the first case as a planar graph -- and there are 16 in the first case and 26 in the second case. Taking the reversals doubles these numbers so it is possible that the Scotts meant the second case and missed one path and its reversal.

David Singmaster, proposer; Jerrold W. Grossman & E. M. Reingold, solvers. Problem E2897 -- An Eulerian circuit with no crossings. AMM 88:7 (Aug 1981) 537-538 & 90:4 (Apr 1983) 287-288. A planar Eulerian graph can be drawn with no crossings. Solution cites some previous work.

Anon. [probably Will Shortz ??check with Shortz]. The impossible file. No. 2: In just one line. Games (Apr 1986) 34 & 64 & (Jul 1986) 64. Five brick pattern -- draw a line crossing each wall once. Says it appeared in a 1921 newspaper [perhaps by Loyd Jr??]. Gives the 1921 solution where the path crosses a corner, hence two walls at once. Also gives a solution with the path going along a wall. In the July issue, Mark Kantrowitz gives a solution by folding over a corner and also a solution on a torus.

Marcia Ascher. Graphs in cultures: A study in ethnomathematics. HM 15 (1988) 201-227. Discusses the history of Eulerian circuits and non-crossing versions and then exposits many forms of the idea in many cultures.

Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Chapter Two: Tracing graphs in the sand, pp. 30-65. Sketches the history of Eulerian graphs with some interesting references -- ??NYS. Describes graph tracing in three cultures: the Bushoong and the Tshokwe of central Africa and the Malekula of Vanuatu (ex-New Hebrides). Extensive references to the ethnographic literature.

5.E.1. MAZES

This section is mainly concerned with the theory. The history of mazes is sketched first, with references to more detailed sources. There is even a journal, Caerdroia (53 Thundersley Grove, Thundersley, Essex, SS7 3EB, England), devoted to mazes and labyrinths, mostly concentrating on the history. It is an annual, began in 1980 and issue 31 appeared in 2000.

Mazes are considered under Euler Circuits, since the method of Euler Circuits is often used to find an algorithm. However, some mazes are better treated as Hamiltonian Circuits -- see 5.F.2.

A maze can be considered as a graph formed by the nodes and paths -- the path graph. For the usual planar maze, one can also look at the graph formed by the walls -- the wall graph, which is a kind of dual to the path graph. In later mazes, the walls do not form a connected whole, and an isolated part of the wall appears as a region or 'face' in the path graph. Such isolated bits of walling are sometimes called islands, but they are the same as the components of the wall graph, with the outer wall being one component, so the number of components is one more than the number of islands. The 'hand-on-wall' method will solve a maze if and only if the goals are adjacent to walls in the component of the outer wall.

A 'ring maze' is a plate with holes and raised areas with an open ring which must be removed by moving it from hole to hole. I have put these in 11.K.5 as they are a kind of mechanical or topological puzzle, though there are versions with a simple two legged spacer.

HISTORICAL SOURCES

W. H. Matthews. Mazes & Labyrinths: A General Account of Their History and Developments. Longmans, Green and Co., London, 1922. = Mazes and Labyrinths: Their History and Development. Dover, 1970. (21 pages of references.) [For more about the book and the author, see: Zeta Estes; My Father, W. H. Matthews; Caerdroia (1990) 6-8.]

Walter Shepherd. For Amazement Only. Penguin, 1942; Let's go amazing, pp. 5-12. Revised as: Mazes and Labyrinths -- A Book of Puzzles. Dover, 1961; Let's go a-mazing, pp. v-xi. (Only a few minor changes are made in the text.) Sketch of the history.

Sven Bergling invented the rolling ball labyrinth puzzle/game and they began being produced in 1946. [Kenneth Wells; Wooden Puzzles and Games; David & Charles, Newton Abbot, 1983, p. 114.]

Walter Shepherd. Big Book of Mazes and Labyrinths. Dover, 1973, More amazement, pp. vii-x. Extends the historical sketch in his previous book, arguing that mazes with multiple choices perhaps derive from Iron Age hill forts whose entrances were designed to confuse an enemy.

Janet Bord. Mazes and Labyrinths of the World. Latimer, London, 1976. (Extensively illustrated.)

Nigel Pennick. Mazes and Labyrinths. Robert Hale, London, 1990.

Adrian Fisher [& Georg Gerster (photographer)]. The Art of the Maze. Weidenfeld and Nicolson, London, 1990. (Also as: Labyrinth; Solving the Riddle of the Maze; Harmony (Crown Publishers), NY, 1990.) Origins and History occupies pp. 11-56, but he also describes many recent developments and innovations. He has convenient tables of early examples.

Adrian Fisher & Diana Kingham. Mazes. Shire Album 264. Shire, Aylesbury, 1991.

Adrian Fisher & Jeff Saward. The British Maze Guide. Minotaur Designs, St. Alban's, 1991.

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HISTORICAL SKETCH

Up to about the 16C, all mazes were unicursal, i.e. with no decision points. The word labyrinth is sometimes used to distinguish unicursal mazes from others, but this distinction is not made consistently. Until about 1000, all mazes were of the classical 'Cretan' seven-ring type shown above. (However, see Shepherd's point in his 1973 book, above.) The oldest examples are rock carvings, the earliest being perhaps that in the Tomba del Labirinto at Luzzanas, Sardinia, c-2000 [Fisher, pp. 12, 25, 26, with photo on p. 12]. (In fact, Luzzanas is a local name for an uninhabited area of fields, so does not appear on any ordinary map. It is near Benetutti. See my A Mathematical Gazetteer or Mazing in Sardinia (Caerdroia 30 (1999) 17-21). Jeff Saward writes that current archaeological feeling is that the maze is Roman, though the cave is probably c-2000.) On pottery, there are labyrinths on fragments, c-1300, from Tell Rif'at, Syria [the first photos of this appeared in [Caerdroia 30 (2000) 54-55]), and on tablets, c-1200, from Pylos. Fisher [p. 26] lists the early examples. Staffen Lundán; The labyrinth in the Mediterranean; Caerdroia 27 (1996) 28-54, catalogues all known 'Cretan' labyrinths from prehistory to the end of antiquity, c250, excluding the Roman 'spoked' form. All these probably had some mystical significance about the difficulty of reaching a goal, often with substantial mythology -- e.g. Theseus in the Labyrinth or, later, the Route to Jerusalem.

Roman mosaics were unicursal but essentially used the Cretan form four times over in the four corners. Lundán, above, calls these 'spoked'. Most of the extant examples are 2C-4C, but some BC examples are known -- the earliest seems to be c-110 at Selinunte, Sicily. Fisher [pp. 36-37] lists all surviving examples. Saward says the earliest Roman example is at Pompeii, so ( 79.

In the medieval period, the Christians developed a quite different unicursal maze. See Fisher [pp. 60-67] for detailed comparison of this form with the Roman and Cretan forms. The earliest large Christian example is the Chemin de Jerusalem of 1235 on the floor of Chartres Cathedral. Fisher [pp. 41 & 48] lists early and later Christian examples.

The legendary Rosamund's Bower was located in Woodstock Park, Oxfordshire, and its purported site is marked by a well and fountain. It was some sort of maze to conceal Rosamund Clifford, the mistress of Henry II (1133-1189), from the Queen, Eleanor of Aquitaine. Legend says that about 1176, Eleanor managed to solve the maze and confronted Rosamund with the choice of a dagger or poison -- she drank the poison and Henry never smiled again. [Fisher, p. 105]. Historically, Henry had imprisoned Eleanor for fomenting rebellion by her sons and Rosamund was his acknowledged mistress. Rosamund probably spent her last days at a nunnery in Godstow, near Oxford. The legend of the bower dates from the 14C and her murder is a later addition [Collins, Book of Puzzles, 1927, p. 121.] In the 19C, many puzzle collections had a maze called Rosamund's Bower.

The earliest record of a hedge maze is of one destroyed in a siege of Paris in 1431.

Non-unicursal mazes and islands in the wall graph start to appear in the late 16C. Matthews [p. 96] says that: "A simple "interrupted-circle" type of labyrinth was adopted as a heraldic device by Gonzalo Perez, a Spanish ecclesiastic ... and published ... in 1566 ..." in his translation of the Odyssey. Matthews doesn't show this, but he then [pp. 96-97] describes and illustrates a simple maze used as a device by Bois-dofin de Laval, Archbishop of Embrum. He copies it from Claude Paradin; Devises Héroiques et Emblèmes of the early 17C. It has four entrances and possibly three goals, with walls having 8 components, two being part of the outer wall. The central goals is accessible from two of the entrances, but the two minor goals are each accessible from just one of the other entrances. Presumably this sort of thing is what Matthews meant as an "interrupted circle".

However, Saward has found a mid 15C anonymous English poem, The Assembly of Ladies, which describes the efforts of a group of ladies to reach the centre of a maze, which, as he observes, implies there must be some choices involved.

[Matthews, p. 114] has three examples from a book by Androuet du Cerceau; Les Plus Excellents Bastiments de France of 1576. Fig. 82 was in the gardens at Charleval and has four entrances, only one of which goes to the central goal. There are four minor goals. The N entrance connects to the NE and SE goals, with several dead ends. The E entrance is a dead end. The S entrance goes to the SW goal. The W entrance goes to the central goal, but the NW goal is on an island, though 'left-hand-on-wall' goes past it. Figs. 83 and 84 are essentially identical and seem to be corruptions of unicursal examples so that most of the maze is bypassed. In fig. 84, one has to walk around to the back of the maze to find the correct entrance to get to the central goal, which is an interesting idea. A small internal change in both cases and moving the entrances converts them to a standard unicursal pattern.

Matthews' Chap. XIII [pp. 100-109] is on floral mazes and reproduces some from Jan Vredeman De Vries; Hortorum Viridariorumque Formae; Antwerp, 1583. Fig. 74 is one of these and has two components and a short dead-end, but the 'hand-on-wall' rule solves it. Fig. 73 is another of De Vries's, but it is not all shown. It appears to have two entrances and there is certainly a decision point by the far gate, but one route goes to the apparent exit at the bottom of the page. There is a small dead end near the central goal. Fig. 78 shows a maze from a 17C manuscript book in the Harley Manuscripts at the BL, identified on p. 224 as BM Harl. 5308 (71, a, 12). This has two components with the central goal in the inner component, so the 'hand-on-wall' rule fails, but it brings you within sight of the centre and Matthews describes it as unicursal! Fig. 79 is from Adam Islip; The Orchard and the Garden, compiled from continental sources and published in 1602. It has 5 components, but four of these are small enclosures which could be considered as minor goals, especially if they had seats in them. The 'hand-on-wall' rule gets to the central goal. There is a lengthy dead end which goes to two of the inner islands. Fig. 80 is from a Dutch book: J. Commelyn; Nederlantze Hesperides of 1676. It has two components, a central goal and four minor goals. The 'hand-on-wall' gets you to the centre and passes two minor goals. One minor goal is on a dead end so 'left-hand-on-wall' gets to it, but 'right-hand-on-wall' does not. The fourth minor goal is on the island.

At Versailles, c1675, André Le Nôtre built a Garden Maze, but the objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one fountain. Some fountains were not at path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian problem, except that one fountain was located at the end of a short dead end. [Fisher, pp. 49, 79, 130 & 144-145, with contemporary map on p. 144. Fisher says there are 39 fountains, and the map has 40. Close examination shows that the map counts two statues at the entrance but omits to count a fountain between numbers 37 and 38. Matthews, pp. 117-121, says it was built by J. Hardouin-Mansart and his map has 39 fountains.] It has a main entrance and exit but there is another exit, so the perimeter wall already has three components, and there are 14 other components. Sadly, it was destroyed in 1775.

Several other mazes, of increasing complexity, occur in the second half of the 17C [Matthews, figs. 93-109, opp. p. 120 - p. 127]. Several of these could be from 20C maze books. Fig. 94, designed for Chantilly by Le Nôtre, is surprisingly modern in that there are eight paths spiralling to the centre. The entrance path takes you directly to the centre, so the real problem is getting back out! One of the mazes presently at Longleat has this same feature.

The Hampton Court Maze, planted c1690, is the oldest extant hedge maze and one of the earliest puzzle mazes. ([Christopher Turner; Hampton Court, Richmond and Kew Step by Step; (As part of: Outer London Step by Step, Faber, 1986); Revised and published in sections, Faber, 1987, p. 16] says the present shape was laid out in 1714, replacing an earlier circular shape, but I haven't seen this stated elsewhere.) Matthews [p. 128] says it probably replaced an older maze. It has dead ends and one island, i.e. the graph has two components, though the 'hand on wall' rule will solve it.

The second Earl Stanhope (1714-1786) is believed to be the first to design mazes with the goal (at the centre) surrounded by an island, so that the 'hand on wall' rule will not solve it. It has seven components and only a few short dead ends.. The fourth Earl planted one of these at Chevening, Kent, in c1820 and it is extant though not open to the public. [Fisher, p. 71, with photo on p. 72 and diagram on p. 73.] However, investigation in Matthews revealed the earlier examples above. Further Bernhard Wiezorke (below at 2001) has found a hedge maze in Germany, dating from c1730, which is not solved by the 'hand on wall' rule. This maze has 12 components.

In 1973, Stuart Landsborough, an Englishman settled at Wanaka, South Island, New Zealand, began building his Great Maze. This was the first of the board mazes designed by Landsborough which were immensely popular in Japan. Over 200 were built in 1984-1987, with 20 designed by Landsborough. Many of these were three dimensional -- see below. About 60 have been demolished since then. [Fisher, pp. 78-79 & 118-121 has 6 colour photos, pp. 156-157 lists Landsborough's designs.]

If Minos' labyrinth ever really existed, it may have been three dimensional and there may have been garden examples with overbridges, but I don't know of any evidence for such early three dimensional mazes. Lewis Carroll drew mazes which had paths that crossed over others making a simple three dimensional maze, in his Mischmasch of c1860, see below. John Fisher [The Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19-20] gives this and another example. Are there earlier examples? Boothroyd & Conway, 1959, seems to be the earliest cubical maze. Much more complex versions were developed by Larry Evans from about 1970 and published in a series of books, starting with 3-Dimensional Mazes (Troubador Press, San Francisco, 1976). His 3-Dimensional Maze Art (Troubador, 1980) sketches some general history of the maze and describes his development of pictures of three dimensional mazes. The first actual three dimensional maze seems to be Greg Bright's 1978 maze at Longleat House, Warminster. [Fisher, pp. 74, 76, 94-95 & 152-153, with colour photos on pp. 94-95.] Since then, Greg Bright, Adrian Fisher, Randoll Coate, Stuart Landsborough and others have made many innovations. Bright seems to have originated the use of colour in mazes c1980 and Fisher has extensively developed the idea. [Fisher, pp. 73-79.]

Abu‘l-Rayhan Al-Biruni (= ’Abû-alraihân [the h should have an underdot] Muhammad ibn ’Ahmad [the h should have an underdot] Albêrûnî). India. c1030. Chapter XXX. IN: Al-Beruni's India, trans. by E. C. Sachau, 2 vols., London, 1888, vol. 1, pp. 306-307 (= p. 158 of the Arabic ed., ??NYS). In describing a story from the fifth and sixth books of the Ramayana, he says that the demon Ravana made a labyrinthine fortress, which in Muslim countries "is called Yâvana-koti, which has been frequently explained as Rome." He then gives "the plan of the labyrinthine fortress", which is the classical Cretan seven-ring form. Sachau's notes do not indicate whether this plan is actually in the Ramayana, which dates from perhaps -300.

Pliny. Natural History. c77. Book 36, chap. 19. This gives a brief description of boys playing on a pavement where a thousand steps are contained in a small space. This has generally been interpreted as referring to a maze, but it is obviously pretty vague. See: Michael Behrend; Julian and Troy names; Caerdroia 27 (1996) 18-22, esp. note 5 on p. 22.

Pacioli. De Viribus. c1500. Part II: Cap. (C)XVII. Do(cumento). de saper fare illa berinto con diligentia secondo Vergilio, f. 223v = Peirani 307-308. A sheet (or page) of the MS has been lost. Cites Vergil, Æneid, part six, for the story of Pasiphæ and the Minotaur, but the rest is then lost.

Sebastiano Serlio. Architettura, 5 books, 1537-1547. The separate books had several editions before they were first published together in 1584. The material of interest is in Book IV which shows two unicursal mazes for gardens. I have seen the following.

Tutte l'Opere d'Architetture et Prospetiva, .... Giacomo de'Franceschi, Venice, 1619; facsimile by Gregg Press, Ridgewood, New Jersey, 1964. F. 199r shows the designs and f. 197v has some text, partly illegible in my photocopy. [Cf Caerdroia 30 (1999) 15.]

Sebastiano Serlio on Architecture Volume One Books I-V of 'Tutte l'Opere d'Architettura et Prospetiva'. Translated and edited by Vaughan Hart and Peter Hicks. Yale Univ. Press, New Haven, 1996. P. 388 shows the designs and p. 389 has the text, saying these 'are for the compartition of gardens'. The sidenotes state that these pages are ff. LXXVr and LXXIIIIr of the 3rd ed. of 1544 and ff. 198v-199r and 197v-198r of the 1618/19 ed.

William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98-100: "The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green For lack of tread are undistinguishable." Fiske 126 opines that the latter two lines may indicate that the board was made in the turf, though he admits that they may refer just to dancers' tracks, but to me it clearly refers to turf mazes.

John Cooke. Greene's Tu Quoque; or the Cittie Gallant; a Play of Much Humour. 1614. ??NYS -- quoted by Matthews, p. 135. A challenge to a duel is given by Spendall to Staines.

Staines. I accept it ; the meeting place?

Spendall. Beyond the maze in Tuttle.

This refers to a maze in Tothill Fields, close to Westminster Abbey.

Lewis Carroll. Untitled maze. In: Mischmasch, the last of his youthful MS magazines, with entries from 1855 to 1862. Transcribed version in: The Rectory Umbrella and Mischmasch; Cassell, 1932; Dover, 1971; p. 165 of the Dover ed. John Fisher [The Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19-20] gives this and another example. Cf Carroll-Wakeling, prob. 35: An amazing maze, pp. 46-47 & 75 and Carroll-Gardner, pp. 80-81 for the Mischmasch example. I don't find the other example elsewhere, but it was for Georgina "Ina" Watson, so probably c1870.

Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53-54 & 102; 1917: 310, pp. 49 & 97. The garden of a French place has a maze with 31 points to see. Find a path past all of them with no repeated edges and no crossings. The pattern is clearly based on the Versailles maze of c1675 mentioned in the Historical Sketch above, but I don't recall the additional feature of no crossings occurring before.

C. Wiener. Ueber eine Aufgabe aus der Geometria situs. Math. Annalen 6 (1873) 29-30. An algorithm for solving a maze. BLW asserts this is very complicated, but it doesn't look too bad.

M. Trémaux. Algorithm. Described in Lucas, RM1, 1891, pp. 47-51. ??check 1882 ed. BLW assert Lucas' description is faulty. Also described in MRE, 1st ed., 1892, pp. 130-131; 3rd ed., 1896, pp. 155-156; 4th ed., 1905, pp. 175-176 is vague; 5th-10th ed., 1911-1922, 183; 11th ed., 1939, pp. 255-256 (taken from Lucas); (12th ed. describes Tarry's algorithm instead) and in Dudeney, AM, p. 135 (= Mazes, and how to thread them, Strand Mag. 37 (No. 220) (Apr 1909) 442-448, esp. 446-447).

G. Tarry. Le problème des labyrinthes. Nouv. Annales de Math. (3) 4 (1895) 187-190. ??NYR

Collins. Book of Puzzles. 1927. How to thread any maze, pp. 122-124. Discusses right hand rule and its failure, then Trémaux's method.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 2. 5 x 5 x 5 cubical maze. Get from a corner to an antipodal corner in a minimal number of steps.

Anneke Treep. Mazes... How to get out! (part I). CFF 37 (Jun 1995) 18-21. Based on her MSc thesis at Univ. of Twente. Notes that there has been very little systematic study. Surveys the algorithms of Tarry, Trémaux, Rosenstiehl. Rosenstiehl is greedy on new edges, Trémaux is greedy on new nodes and Trémaux is a hybrid of these. ??-oops-check. Studies probabilities of various routes and the expected traversal time. When the maze graph is a tree, the methods are equivalent and the expected traversal time is the number of edges.

Bernhard Wiezorke. Puzzles und Brainteasers. OR News, Ausgabe 13 (Nov 2001) 52-54. This reports his discovery of a hedge maze in Germany -- the first he knew of. It is in Altjessnitz, near Dessau in Sachsen-Anhalt. (My atlas doesn't show such a place, but Jessnitz is about 10km south of Dessau.) This maze dates from 1720 and has 12 components, with the goal completely separated from the outside so that the 'hand on wall' rule does not solve it. Torsten Silke later told Wiezorke of two other hedge mazes in Germany. One, in Probststeierhagen, Schleswig-Holstein, about 12km NE of Kiel, is in the grounds of the restaurant Zum Irrgarten (At the Labyrinth) and is an early 20C copy of the Altjessnitz example. The other, in Kleinwelka, Sachsen, about 50km NE of Dresden, was made in 1992 and is private. Though it has 17 components, the 'hand on wall' method will solve it. He gives plans of both mazes. He discusses the Seven Bridges of Königsberg, giving a B&W print of the 1641 plan of the city mentioned at the beginning of Section 5.E -- he has sent me a colour version of it. He also describes Tremaux's solution method.

5.E.2. MEMORY WHEELS = CHAIN CODES

These are cycles of 2n 0s and 1s such that each n-tuple of 0s and 1s appears just once. They are sometimes called De Bruijn sequences, but they have now been traced back to the late 19C. An example for n = 3 is 00010111.

Émile Baudot. 1884. Used the code for 25 in telegraphy. ??NYS -- mentioned by Stein.

A. de Rivière, proposer; C. Flye Sainte-Marie, solver. Question no. 58. L'Intermédiare des Mathématiciens 1 (1894) 19-20 & 107-110. ??NYS -- described in Ralston and Fredricksen (but he gives no. 48 at one point). Deals with the general problem of a cycle of kn symbols such that every n-tuple of the k basic symbols occurs just once. Gives the graphical method and shows that such cycles always exist and there are k!g(n)/ kn of them, where g(n) = kn-1. This work was unknown to the following authors until about 1975.

N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49 (1946) 758-764. ??NYS -- described in Ralston and Fredricksen. Gives the graphical method for finding examples and finds there are 2f(n) solutions, where f(n) = 2n-1 - n.

I. J. Good. Normal recurring decimals. J. London Math. Soc. 21 (1946) 167-169. ??NYS -- described in Ralston and Fredricksen. Shows there are solutions but doesn't get the number.

R. L. Goodstein. Note 2590: A permutation problem. MG 40 (No. 331) (Feb 1956) 46-47. Obtains a kind of recurrence for consecutive n-tuples.

Sherman K. Stein. Mathematics: The Man-made Universe. Freeman, 1963. Chap. 9: Memory wheels. c= The mathematician as explorer, SA (May 1961) 149-158. Surveys the topic. Cites the c1000 Sanskrit word: yamátárájabhánasalagám used as the mnemonic for 01110100(01) giving all triples of short and long beats in Sanskrit poetry and music. Describes the many reinventions, including Baudot (1882), ??NYS, and the work of Good (1946), ??NYS, and de Bruijn (1946), ??NYS. 15 references.

R. L. Goodstein. A generalized permutation problem. MG 54 (No. 389) (Oct 1970) 266-267. Extends his 1956 note to find a cycle of an symbols such that the n-tuples are distinct.

Anthony Ralston. De Bruijn sequences -- A model example of the interaction of discrete mathematics and computer science. MM 55 (1982) 131-143 & cover. Deals with the general problem of cycles of kn symbols such that every n-tuple of the k basic symbols occurs just once. Discusses the history and various proofs and algorithms which show that such cycles always exist. 27 references.

Harold Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review 24:2 (Apr 1982) 195-221. Mostly about their properties and their generation, but includes a discussion of the door lock connection, a mention of using the 23 case as a switch for three lights, and gives a good history. The door lock connection is that certain push button door locks will open when a four digit code is entered, but they open if the last four buttons pressed are the correct code, so using a chain code reduces the number of button pushes required by a burglar to 1/4 of the number required if he tries all four digit combinations. 58 references.

At G4G2, 1996, Persi Diaconis spoke about applications of the chain code in magic and mentioned uses in repeated measurement designs, random number generators, robot location, door locks, DNA comparison.

They were first used in card tricks by Charles T. Jordan in 1910. Diaconis' example had a deck of cards which were cut and then five consecutive cards were dealt to five people in a row. He then said he would determine what cards they had, but first he needed some help so he asked those with red cards to step forward. The position of the red cards gives the location of the five cards in a cycle of 32 (which was the size of the deck)! Further, there are simple recurrences for the sequence so it is fairly easy to determine the location. One can code the binary quintuples to give the suit and value of the first card and then use the succeeding quintuples for the succeeding cards.

Long versions of the chain code are printed on factory floors so that a robot can read it and locate itself.

In Jan 2000, I discussed the Sanskrit chain code with a Sanskrit scholar, Dominik Wujastyk, who said that there is no known Sanskrit source for it. He has asked numerous pandits who did not know of it and he said there is is a forthcoming paper on it, but that it did not locate any Sanskrit source.

5.E.2.a. PANTACTIC SQUARES

Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include this.

B. Astle. Pantactic squares. MG 49 (No. 368) (May 1965) 144-152. This is a two-dimensional version of the memory wheel. Take a 5 x 5 array of cells marked 0 or 1 (or Black or White). There are 16 ways to take a 2 x 2 subarray from the 5 x 5 array. If these give all 16 2 x 2 binary patterns, the array is called pantactic. The author shows a number of properties and some types of such squares.

C. J. Bouwkamp, P. Janssen & A. Koene. Note on pantactic squares. MG 54 (No. 390) (Dec 1970) 348-351. They find 800 such squares, forming 50 classes of 16 forms.

[Surprisingly, neither paper considers a 4 x 4 array viewed toroidally, which is the natural generalization of the memory wheel. Precisely two of the fifty classes, namely nos. 25 & 41, give such a solution and these are the same pattern on the torus. One can also look at the 4 x 4 subarrays of a 131 x 131 or a 128 x 128 array, etc., as well as 3 and higher dimensional arrays. I submitted the question of the existence and numbers of these as a problem for CM, but it was considered too technical.]

Ivan Moscovich. US Patent 3,677,549 -- Board Game Apparatus. Applied: 14 Jun 1971; patented: 18 Jul 1972. Front page, 1p diagrams, 2pp text. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams. This uses the 16 2 x 2 binary patterns as game pieces. He allows the pieces to be rotated, scoring different values according to the orientation. No mention of reversing pieces or of the use of the pieces as a puzzle.

John Humphries. Review of Q-Bits. G&P 54 (Nov 1976) 28. This is Moscovich's game idea, produced by Orda. Though he mentions changing the rules to having non-matching, there is no mention of two-sidedness.

Pieter van Delft & Jack Botermans. Creative Puzzles of the World. (As: Puzzels uit de hele wereld; Spectrum Hobby, 1978); Harry N. Abrams, NY, 1978. The colormatch square, p. 165. See Haubrich,1994, for description.

Jacques Haubrich. Pantactic patterns and puzzles. CFF 34 (Oct 1994) 19-21. Notes the toroidal property just mentioned. Says Bouwkamp had the idea of making the 16 basic squares in coloured card and using them as a MacMahon-type puzzle, with the pieces double-sided and such that when one side had MacMahon matching, the other side had non-matching. There are two different bijections between matching patterns and non-matching patterns, so there are also 800 solutions in 50 classes for the non-matching problem. Bouwkamp's puzzle appeared in van Delft & Botermans, though they did not know about and hence did not mention the double-sidedness. [In an email of 22 Aug 2000, Haubrich says he believes Bouwkamp did tell van Delft and Botermans about this, but somehow it did not get into their book.] The idea was copied by two manufacturers (Set Squares by Peter Pan Playthings and Regev Magnetics) who did not understand Bouwkamp's ideas -- i.e. they permitted pieces to rotate. Describes Verbakel's puzzle of 5.H.2.

Jacques Haubrich. Letter: Pantactic Puzzles = Q-Bits. CFF 37 (Jun 1995) 4. Says that Ivan Moscovich has responded that he invented the version called "Q-Bits" in 1960-1964, having the same tiles as Bouwkamp's (but only one-sided [clarified by Haubrich in above mentioned email]). His US Patent 3,677,549 (see above) is for a game version of he idea. The version produced by Orda Ltd. was reviewed in G&P 54 (Nov 1976) (above). So it seems clear that Moscovich had the idea of the pieces before Bouwkamp's version was published, but Moscovich's application was to use them in a game where the orientations could be varied.

5.F. HAMILTONIAN CIRCUITS

For queen's, bishop's and rook's tours, see 6.AK.

A tour is a closed path or circuit.

A path has end points and is sometimes called an open tour.

5.F.1. KNIGHT'S TOURS AND PATHS

GENERAL REFERENCES

Antonius van der Linde. Geschichte und Literatur des Schachspiels. (2 vols., Springer, Berlin, 1874); one vol. reprint, Olms, Zürich, 1981. [There are two other van der Linde books: Quellenstudien zur Geschichte des Schachspiels, Berlin, 1881, ??NYS; and Das Erste Jartausend [sic] der Schachlitteratur (850-1880), (Berlin, 1880); reprinted with some notes and corrections, Caissa Limited Editions, Delaware, 1979, which is basically a bibliography of little use here.]

Baron Tassilo von Heydebrand und von der Lasa. Zur Geschichte und Literatur des Schachspiels. Forschungen. Leipzig, 1897. ??NYS.

Ahrens. MUS I. 1910. Pp. 319-398.

Harold James Ruthven Murray. A History of Chess. OUP, 1913; reprinted by Benjamin Press, Northampton, Massachusetts, nd [c1986]. This has many references to the problem, which are detailed below.

Reinhard Wieber. Das Schachspiel in der arabischen Literatur von den Anfängen bis zur zweiten Hälfte des 16.Jahrhunderts. Verlag für Orientkunde Dr. H. Vorndran, Walldorf-Hessen, 1972.

George P. Jelliss.

Special Issue: Notes on the Knight's Tour. Chessics 22 (Summer 1985) 61-72.

Further notes on the knight's tour. Chessics 25 (Spring 1986) 106-107.

Notes on Chessics 22 continued. Chessics 29 & 30 (1987) 160.

This is a progress report on his forthcoming book on the knight's tour. I will record some of his comments at the appropriate points below. He also studies the 3 x n board extensively.

Two problems with knights on a 3 x 3 board are generally treated here, but cf 5.R.6.

The 4 knights problem has two W and two B knights at the corners (same colours at adjacent corners) and the problem is to exchange them in 16 moves. The graph of knight's connections is an 8-cycle with the pieces at alternate nodes. [Putting same colours at opposite corners allows a solution in 8 moves.]

The 7 knights problem is to place 7 knights on a 3 x 3 board in the 4 corners and 3 of the sides so each is a knight's move from the previously placed one. This is equivalent to the octagram puzzle of 5.R.6.

4 knights problem -- see: at-Tilimsâni, 1446; Civis Bononiae, c1475;

7 knights problem -- see: King's Library MS.13, A.xviii, c1275; "Bonus Socius", c1275; at-Tilimsâni, 1446;

Al-Adli (c840) and as-Suli (c880-946) are the first two great Arabic chess players. Although none of their works survive, they are referred to by many later writers who claim to have used their material.

Rudraţa: Kāvyālaʼnkāra [NOTE: ţ denotes a t with an underdot and ʼn denotes an n with an overdot.]. c900. ??NYS -- described in Murray 53-55, from an 1896 paper by Jacobi, ??NYS. The poet speaks of verses which have the shapes of "wheel, sword, club, bow, spear, trident, and plough, which are to be read according to the chessboard squares of the chariot [= rook], horse [= knight], elephant [c= bishop], &c." According to Jacobi, the poet placed syllables in the cells of a half chessboard so that it reads the same straight across as when following a piece's path. With help from the commentator Nami, of 1069, the rook's and knight's path's are reconstructed, and are given on Murray 54. Both are readily extended to full board paths, but not tours. The elephant's path is confused.

Kitâb ash-shatranj mimma’l-lafahu’l-‘Adli waş-Şûlî wa ghair-huma [Book of the Chess; extracts from the works of al-'Adlî, aş-Şûlî and others]. [NOTE: ş, Ş denote s, S with underdot.] Copied by Abû Ishâq [the h should have an underdot] Ibrâhîm ibn al-Mubârak ibn ‘Alî al-Mudhahhab al-Baghdâdî. Murray 171-172 says it is MS ‘Abd-al-Hamid [the H should have an underdot] I, no. 560, of 1140, and denotes it AH. Wieber 12-15 says it is now MS Lala Ismail Efendi 560, dates it July-August 1141, and denotes it L. Both cite van der Linde, Quellenstudien, no. xviii, p. 331+, ??NYS. The author is unknown. This MS was discovered in 1880. Catalogues in Istanbul listed it as Risâla fi’sh-shaţranj by Abû’l-‘Abbâs Ahmad [the h should have an underdot] al-‘Adlî. It is sometimes attributed to al-Lajlâj who wrote one short section of this book. Murray, van der Linde and Wieber (p. 41) cite another version: MS Khedivial Lib., Cairo, Mustafa Pasha, no. 8201, copied c1370, which Murray denotes as C and Wieber lists as unseen.

Murray 336 gives two distinct tours: AH91 & AH92. The solution of AH91 is a numbered diagram, but AH92 is 'solved' four times by acrostic poems, where the initial letters of the lines give the tour in an algebraic notation. Wieber 479-480 gives 2 tours from ff. 74a-75b: L74a = AH91 and L74b = reflection of AH92. [Since the 'solutions' of AH92 are poetic, it is not unreasonable to consider the reflection as different.] Also AH94 = L75b is a knight/bishop tour, where moves of the two types alternate. These tours may be due to as-Suli. AH196 is a knight/queen tour.

Arabic MS Atif Efendi 2234 (formerly Vefa (‘Atîq Efendî) 2234), Eyyub, Istanbul. Copied by Muhammad [the h should have an underdot] ibn Hawâ (or Rahwâr -- the MS is obscure) ibn ‘Othmân al-Mu’addib in 1221. Murray 174-175 describes it as mostly taken from the above book and denotes it V. A tour is shown on p. 336 as V93 = AH92. Wieber 20-24 denotes it A. On p. 479, he shows the tour from f. 68b which is the same as L74b, the reflection of AH92.

King's Library MS.13, A.xviii, British Museum, in French, c1275. Described in van der Linde I 305-306. Described and transcribed in Murray 579-582 & 588-600, where it is denoted as K. Van der Linde discusses the knight's path on I 295, with diagram no. 244 on p. 245. Murray 589 gives the text and a numbered diagram of a knight's path as K1. The path splits into two half board paths: a1 to d1 and e3 to h1, so the first half and the whole are corner to corner. The first half is also shown as diagram K2 with the half board covered with pieces and the path described by taking of pieces. K3 is the 7 knights problem

"Bonus Socius" [perhaps Nicolas de Nicolaï]. This is the common name of a collection of chess problems, assembled c1275, which was copied and translated many times. See Murray 618-642 for about 11 MSS. Some of these are given below. Fiske 104 & 110-111 discusses some MSS of this collection.

MS Lat. 10286, Nat. Lib., Paris. c1350. Van der Linde I 293-295 describes this but gives the number as 10287 (formerly 7390). Murray 621 describes it and denotes it PL. Van der Linde describes a half board knight's path, with a diagram no. 243 shown on p. 245. The description indicates a gap in the path which can only be filled in one way. This is a path from a8 to h8 which cannot be extended to the full board. Murray 641 says that PL275 is the same as problems in two similar MSS and as CB244, diagrammed on p. 674. However, this is not the same as van der Linde's no. 243, though cells 1-19 and 31-32 are the same in both paths, so this is also an a8 to h8 path which does not extend to a full board.

Murray 620 mentions a path in a late Italian MS version of c1530 (Florence, Nat. Lib. XIX.7.51, which he denotes It) which may be the MS described by van der Linde I 284 as no. 4 and the half board path described on I 295 with diagram no. 245 on I 245. Fiske 210-211 describes this and says von der Lasa 163-165 (??NYS) describes it as early 16C, but Murray does not mention von der Lasa. Fiske says it contains a tour on f. 28b, which von der Lasa claims is "das älteste beispiel eines vollkommenen rösselsprunges", but Murray does not detail the problems so I cannot compare these citations. Fiske also says it also contains the 7 knights problem.

Dresden MS 0/59, in French, c1400. Murray describes this on pp. 607-613 and denotes it D. On p. 609, Murray describes D57 which asks for a knight's path on a 4 x 4 board. No solution is given -- indeed this is impossible, cf Persian MS 211 in the RAS. Ibid. is D62 which asks for a half board tour, but no answer is provided.

Persian MS 211 in Royal Asiatic Society. Early 15C. ??NYS.

Extensively described as MS 250 bequeathed by Major David Price in: N. Bland; On the Persian game of chess; J. Royal Asiatic Soc. 13 (1852) 1-70. He dates it as 'at least 500 years old' and doesn't mention the knight's tour.

Described, as MS No. 260, and partially translated in Duncan Forbes; The History of Chess; Wm. H. Allen, London, 1860. Forbes says Bland's description is "very detailed but unsatisfactory". On p. 82 is the end of the translation of the preface: '"Finally I will show you how to move a Knight from any individual square on the board, so that he may cover each of the remaining squares in as many moves and finally come to rest on that square whence he started. I will also show how the same thing may be done by limiting yourself only to one half, or even to one quarter (1) of the board." -- Here the preface abruptly terminates, the following leaf being lost.' Forbes's footnote (1) correctly doubts that a knight's tour (or even a knight's path) is possible on the 4 x 4 board.

Murray 177 cites it as MS no. 211 and denotes it RAS. He says that it has been suggested that this MS may be the work of ‘Alâ'addîn Tabrîzî = ‘Alî ash-Shatranjî, late 14C, described on Murray 171. Murray mentions the knight's tour passage on p. 335. This may be in van der Linde, ??NX. Wieber 45 mentions the MS.

Abû Zakarîyâ Yahya [the h should have an underdot] ibn Ibrâhîm al-Hakîm[the H should have an underdot]. Nuzhat al-arbâb al-‘aqûl fî’sh-shaţranj [NOTE: ţ denotes a t with an underdot] al-manqûl (The delight of the intelligent, a description of chess). Arabic MS 766, John Rylands Library, Manchester.

Bland, loc. cit., pp. 27-28, describes this as no. 146 of Dr. Lee's catalogue and no. 76 of the new catalogue. Forbes, loc. cit., says that Dr. Lee had loaned his two MSS to someone who had not yet returned them, so Forbes copies Bland's descriptions (on pp. 27-31) as his Appendix C, with some clarifying notes. (The other of Dr. Lee's MSS is described below.) Van der Linde I 107ff (??NX) seems to copy Bland & Forbes.

Murray 175-176 describes it as Arab. 59 at John Rylands Library and denotes it H. He says it was Bland who had borrowed the MSS from Dr. Lee and Murray traces their route to Dr. Lee and to Manchester. Murray says it is late 15C, is based on al-Adli and as-Suli and he also describes a later version, denoted Z, late 18C. Wieber 32-35 cites it as MS 766(86) at John Rylands, dates it 1430 and denotes it Y1.

Murray 336 gives three paths. H73 = H75 are the same tour, but with different keys, one poetic as in Rudraţa [NOTE: ţ denotes a t with an underdot.], one numeric. H74 is a path attributed to Ali Mani with similar poetic solution. Wieber 480 shows two diagrams. Y1-39a, Y1-39b, Y1-41b are the same tour as H73, but with different descriptions, the latter two being attributed to al-Adli. Y1-39a (second diagram) = H74 is attributed to ‘Ali ibn Mani.

Shihâbaddîn Abû’l-‘Abbâs Ahmad [the h should have an underdot] ibn Yahya [the h should have an underdot] ibn Abî Hajala [the H should have an underdot] at-Tilimsâni alH-anbalî [the H should have an underdot]. Kitâb ’anmûdhaj al-qitâl fi la‘b ash-shaţranj [NOTE: ţ denotes a t with an underdot] (Book of the examples of warfare in the game of chess). Copied by Muhammed ibn ‘Ali ibn Muhammed al-Arzagî in 1446.

Bland, loc. cit., pp. 28-31, describes this as the second of Dr. Lee's MSS, old no. 147, new no. 77. Forbes copies this and adds notes. Van der Linde I 105-107 seems to copy from Bland and Forbes. Murray 176-177 says the author died in 1375, so this might be c1370. He says it is Dr. Lee's on 175-176, that it is MS Arab. 93 at the John Rylands Library and denotes it Man. Wieber 29-32 cites it as MS 767(59) at the Rylands Library and denotes it H. On p. 481, he shows a half-board path which cannot be extended to the full board.

This MS also gives the 4 knights and 7 knights problems. Murray 337, 673 (CB236) & 690 and Wieber 481 show these problems.

Risâlahĭ Shatranj. Persian poem of unknown date and authorship. A copy was sent to Bland by Dr. Sprenger of Delhi. See Bland, loc. cit., pp. 43-44. [Bland uses á for â.] Bland says it has the problem of the knight's tour or path. [I think this is the poem mentioned on Murray 182-183 and hence on Wieber 42.]

Şifat mal ‘ûb al-faras fî gamî abyât aš-šaţranğ [NOTE: Ş, ţ. denote S, t with underdot.] MS Gotha 10, Teil 6; ar. 366; Stz. Hal. 408. Date unknown. Wieber 37 & 480 describes this and gives a path from h8 to e4 which occurs on ff. 70 & 68.

Civis Bononiae [Citizen of Bologna]. Like Bonus Socius, this is a collection of chess problems, from c1475, which exists in several MSS and printings. All are in Latin, from Italy, and give essentially the same 288 problems. See Murray 643-703 for description of about 10 texts and transcription of the problems. Many of the texts are not in van der Linde. Murray 643 cites MS Lasa, in the library of Baron von der Lasa, c1475, as the most accurate and complete of the texts. Two other well known versions are described below.

Paulo Guarino (di Forli) (= Paulus Guarinus). No real title, but the end has 'Explicit liber de partitis scacorum' with the writer's name and the date 4 Jan 1512. This MS was in the Franz Collection and is now (1913) in the John G. White Collection in Cleveland, Ohio. This version only contains 76 problems. Van der Linde I 295-297 describes the MS and on p. 294 he describes a half board path and says Guarino's 74 is a reflection of his no. 243. Murray 645 describes the MS but doesn't list the individual problems. He implies that CB244, on p. 674, is the tour that appears in all of the Civis Bononiae texts, but this is not the same as van der Linde's no. 243. CB236, pp. 673 & 690, is the 4 knights problem, which is Guarino's 42 [according to Lucas, RM4, p. 207], but I don't have a copy of van der Linde's no. 215 to check this, ??NX.

Anon. Sensuit Jeux Partis des eschez: composez nouvellement Pour recreer tous nobles cueurs et pour eviter oysivete a ceulx qui ont voulente: desir et affection de le scavoir et apprendre et est appelle ce Livre le jeu des princes et damoiselles. Published by Denis Janot, Paris, c1535, 12 ff. ??NYS. (This is the item described by von der Lasa as 'bei Janot gedrucktes Quartbändchen' (MUS #195).) This a late text of 21 problems, mostly taken from Civis Bononiae. Only one copy is known, now (1913) in Vienna. See van der Linde I 306-307 and Murray 707-708 which identify no. 18 as van der Linde's no. 243 and with CB244, as with the Guarino work. I can't tell but van der Linde may identify no. 11 as the 4 knights problem (??NX).

Murray 730 gives another half board path, C92, of c1500 which goes from a8 to g5. Murray 732 notes that a small rearrangement makes it extendable to the whole board.

Horatio Gianutio della Mantia. Libro nel quale si tratta della Maniera di giuocar' à Scacchi, Con alcuni sottilissimi Partiti. Antonio de' Bianchi, Torino, 1597. ??NYS. Gives half board tours which can be assembled into to a full tour. (Not in the English translation: The Works of Gianutio and Gustavus Selenus, on the game of Chess, Translated and arranged by J. H. Sarratt; J. Ebers, London, 1817, vol. 1. -- though the copy I saw didn't say vol. 1. Van der Linde, Erste Jartausend ... says there are two volumes.)

Bhaţţa Nīlakaņţha. [NOTE: ţ, ņ denote t, n with underdot.] Bhagavantabhāskara. 17C. End of 5th book. ??NYS, described by Murray 63-66. The author gives three tours, in the poetic form of Rudraţa [NOTE: ţ denotes a t with an underdot.], which are the same tour starting at different points. The tour has 180 degree rotational symmetry.

Ozanam. 1725. Prob. 52, 1725: 260-269. Gives solutions due to Pierre Rémond de Montmort, Abraham de Moivre, Jean-Jacques d'Ortous de Mairan (1678-1771). Surprisingly, these are all distinct and different from the earlier examples. Ozanam says he had the problem and the solution from de Mairan in 1722. Says the de Moivre is the simplest. Kraitchik, Math. des Jeux, op. cit. in 4.A.2, p. 359, dates the de Montmort as 1708 and the de Moivre as 1722, but gives no source for these. Montmort died in 1719. Ozanam died in 1717 and this edition was edited by Grandin. Van der Linde and Ahrens say they can find no trace of these solutions prior to Ozanam (1725). See Ozanam-Montucla, 1778.

Ball, MRE, 1st ed., 1892, p. 139, says the earliest examples he knows are the De Montmort & De Moivre of the late 17C, but he only cites them from Ozanam-Hutton, 1803, & Ozanam-Riddle, 1840. In the 5th ed., 1911, p. 123, he adds that "They were sent by their authors to Brook Taylor who seems to have previously suggested the problem." He gives no reference for the connection to Taylor and I have not seen it mentioned elsewhere. This note is never changed and may be the source of the common misconception that knight's tours originated c1700!

Les Amusemens. 1749. Prob. 181, p. 354. Gives de Moivre's tour. Says one can imagine other methods, but this is the simplest and most interesting.

L. Euler. Letter to C. Goldbach, 26 Apr 1757. In: P.-H. Fuss, ed.; Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIème Siècle; (Acad. Imp. des Sciences, St. Pétersbourg, 1843) = Johnson Reprint, NY, 1968, vol. 1, pp. 654-655. Gives a 180o symmetric tour.

L. Euler. Solution d'une question curieuse qui ne paroit soumise à aucune analyse. (Mém. de l'Académie des Sciences de Berlin, 15 (1759 (1766)), 310-337.) = Opera Omnia (1) 7 (1923) 26-56. (= Comm. Arithm. Coll., 1849, vol. 1, pp. 337-355.) Produces many solutions; studies 180o symmetry, two halves, and other size boards.

[Petronio dalla Volpe]. Corsa del Cavallo per tutt'i scacchi dello scacchiere. Lelio della Volpe, Bologna, 1766. 12pp, of which 2 and 12 are blanks. [Lelio della Volpe is sometimes given as the author, but he died c1749 and was succeeded by his son Petronio.] Photographed and printed by Dario Uri from the example in the Libreria Comunale Archiginnasio di Bologna, no. 17 CAPS XVI 13. The booklet is briefly described in: Adriano Chicco; Note bibliografiche su gli studi di matematica applicata agli scacchi, publicati in Italia; Atti del Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981, ed. by Roberto Magari; Tipografia Senese for GIOCREA (Società Italiana Giochi Creativi), 1981; p. 155.

The Introduction by the publisher cites Ozanam as the originator of this 'most ingenious' idea and says he gives examples due to Montmort, Moivre and Mairan. He also says this material has 'come to hand' but doesn't give any source, so it is generally thought he was the author. He gives ten paths, starting from each of the 10 essentially distinct cells. He then gives the three cited paths from Ozanam. He then gives six tours. Each path is given as a numbered board and a line diagram of the path, which led Chicco to say there were 38 paths. The line drawing of the first tour is also reproduced on the cover/title page.

Ozanam-Montucla. 1778. Prob. 23, 1778: 178-182; 1803: 177-180; 1814: 155-157. Prob. 22, 1840: 80-81. Drops the reference to de Mairan as the source of the problem and adds a fourth tour due to "M. de W***, capitaine au régiment de Kinski". All of these have a misprint of 22 for 42 in the right hand column of De Moivre's solution.

H. C. von Warnsdorff. Des Rösselsprunges einfachste und allgemeinste Lösung. Th. G. Fr. Varnhagenschen Buchhandlung, Schmalkalden, 1823, 68pp. ??NYS -- details from Walker. Rule to make the next move to the cell with the fewest remaining neighbours. Lucas, L'Arithmétique Amusante, p. 241, gives the place of publication as Berlin.

Boy's Own Book. Not in 1828. 1828-2: 318 states a knight's tour can be made.

George Walker. The Art of Chess-Play: A New Treatise on the Game of Chess. (1832, 80pp. 2nd ed., Sherwood & Co, London, 1833, 160pp. 3rd ed., Sherwood & Co., London, 1841, 300pp. All ??NYS -- details from 4th ed.) 4th ed., Sherwood, Gilbert & Piper, London, 1846, 375pp. Chap. V -- section: On the knight, p. 37. "The problem respecting the Knight's covering each square of the board consecutively, has attracted, in all ages, the attention of the first mathematicians." States Warnsdorff's rule, without credit, but gives the book in his bibliography on p. 375, and asserts the rule will always give a tour. No diagram.

Family Friend 2 (1850) 88 & 119, with note on 209. Practical Puzzle -- No. III. Find a knight's path. Gives one answer. Note says it has been studied since 'an early period' and cites Hutton, who copies some from Montucla, an article by Walker in Frasers Magazine (??NYS) which gives Warnsdorff's rule and an article by Roget in Philosophical Magazine (??NYS) which shows one can start and end on any two squares of opposite colours. Describes using a pegged board and a string to make pretty patterns.

Boy's Own Book. Moving the knight over all the squares alternately. 1855: 511-512; 1868: 573; 1881 (NY): 346-347. 1855 says the problem interested Euler, Ozanam, De Montmart [sic], De Moivre, De Majron [sic] and then gives Warnsdorff's rule, citing George Walker's 'Treatise on Chess' for it -- presumably 'A New Treatise', London, 1832, with 2nd ed., 1833 & 3rd ed., 1841, ??NYS. Walker also wrote On Moving the Knight, London, 1840, ??NYS. 1868 drops all the names, but the NY ed. of 1881 is the same as the 1855. Gives a circuit due to Euler.

Magician's Own Book. 1857. Art. 46: Moving the knight over all the squares alternately, pp. 283-287. Identical to Boy's Own Book, 1855, but adds Another Method. = Book of 500 Puzzles; 1859, art. 46, pp. 97-101. = Boy's Own Conjuring Book, 1860, prob. 45, pp. 246-251.

Landells. Boy's Own Toy-Maker. 1858. Moving the knight over all the squares alternately, p. 143. This is the Another Method of Magician's Own Book, 1857. Cf Illustrated Boy's Own Treasury, 1860.

Illustrated Boy's Own Treasury. 1860. Prob. 47: Practical chess puzzle, pp. 404 & 443. Knight's tour. This is the Another Method of Magician's Own Book.

C. F. de Jaenisch. Traité des Applications de l'Analyse Mathématiques au Jeu des Échecs. 3 vols., no publisher, Saint-Pétersbourg. 1862-1863. Vol. 1: Livre I: Section III: De la marche du cavalier, pp. 186-259 & Plate III. Vol. 2: Livre II: Problème du Cavalier, pp. 1-296 & 31 plates (some parts ??NYS). Vol. 3: Addition au Livre II, pp. 239-243 (This Addition ??NYS). This contains a vast amount of miscellaneous material and I have not yet read it carefully. ??NYR

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 323, pp. 153-154 & 393: Rösselsprung-Aufgaben. Three arrays of syllables and one must find a poetic riddle by following a knight's tour. Arrays are 8 x 8, 8 x 8, 6 x 4.

C. Flye Sainte-Marie. Bull. Soc. Math. de France (1876) 144-150. ??NYS -- described by Jelliss. Shows there is no tour on a 4 x n board and describes what a path must look like.

Mittenzwey. 1880. Prob. 222-223, pp. 40 & 91; 1895?: 247-248, pp. 44 & 93; 1917: 247-248, pp. 40-41 & 89. First is a knight's path. Second is a board with word fragments and one has to make a poem, which uses the same path as in the first problem.

Paul de Hijo [= Abbé Jolivald]. Le Problème du Cavalier des Échecs. Metz, 1882. ??NYS -- described by Jelliss and quoted by Lucas. Jelliss notes the BL copy of de Hijo was destroyed in the war, but he has since told me there are copies in The Hague and Nijmegen. First determination of the five 6 x 6 tours with 4-fold rotational symmetry, the 150 ways to cover the 8 x 8 with two circuits of length 32 giving a pattern with 2-fold rotational symmetry, the 378 ways giving reflectional symmetry in a median, the 140 ways with four circuits giving 4-fold rotational symmetry and the 301 ways giving symmetry in both medians (quoted in Lucas, L'Arithmétique Amusante, pp. 238-241).

Lucas. Nouveaux jeux scientifiques ..., 1889, op. cit. in 4.B.3. (Described on p. 302, figure on p. 301.) 'La Fasioulette' is an 8 x 8 board with 64 links of length (5 to form knight's tours.

Knight's move puzzles. The Boy's Own Paper 11 (Nos. 557 & 558) (14 & 21 Sep 1889) 799 & 814. Four Shakespearean quotations concealed as knight's tours on a 8 x 8 board. Beginnings not indicated!

Hoffmann. 1893. Chap. X, no. 6: The knight's tour, pp. 335-336 & 367-373 = Hoffmann-Hordern, pp. 225-229. Gives knight's paths due to Euler and Du Malabare, a knight's tour due to Monneron, and four other unattributed tours. Gives Warnsdorff's rule, citing Walker's A New Treatise on Chess, 1832.

Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904, pp. 1080-1093. Pp. 1084-1086 gives many references to 19C work, including estimates of the number of tours and results on 'semi-magic tours'.

C. Planck. Chess Amateur (Dec 1908) 83. ??NYS -- described by Jelliss. Shows there are 1728 paths on the 5 x 5 board. Jelliss notes that this counts each path in both directions and there are only 112 inequivalent tours.

Ahrens. 1910. MUS I 325. Use of knight's tours as a secret code.

Dudeney. AM. 1917. Prob. 339: The four knight's tours, pp. 103 & 229. Quadrisect the board into four congruent pieces such that there is a knight's tour on the piece. Jelliss asserts that the solution is unique and says this may be what Persian MS 260 (i.e. 211) intended. He notes that the four tours can be joined to give a tour with four fold rotational symmetry.

W. H. Cozens. Cyclically symmetric knight's tours. MG 24 (No. 262) (Dec 1940) 315-323. Finds symmetric tours on various odd-shaped boards.

H. J. R. Murray. The Knight's Tour. ??NYS. MS of 1942 described by G. P. Jelliss, G&PJ 2 (No. 17) (Oct 1999) 315. Observes that a knight can move from the (0, 0) cell to the (2, 1) and (1, 2) cella and that the angle between these lines is the smaller angle of a 3, 4, 5 triangle. One can see this by extending the lines to (8, 4) and (5, 10) and seeing these points form a 3, 4, 5 triangle with (0, 0).

W. H. Cozens. Note 2761: On note 2592. MG 42 (No. 340) (May 1958) 124-125. Note 2592 tried to find the cyclically symmetric tours on the 6 x 6 board and found 4. Cozens notes two are reflections of the other two and that three such tours were omitted. He found all these in his 1940 paper.

R. C. Read. Constructing open knight's tours blindfold! Eureka 22 (Oct 1959) 5-9. Describes how to construct easily a tour between given cells of opposite colours, correcting a method of Roget described by Ball (MRE 11th ed, p. 181). Says he can do it blindfold.

W. H. Cozens. Note 2884: On note 2592. MG 44 (No. 348) (May 1960) 117. Estimates there are 200,000 cyclically symmetric tours on the 10 x 10 board.

Roger F. Wheeler. Note 3059: The KNIGHT's tour on 42 and other boards. MG 47 (No. 360) (May 1963) 136-141. KNIGHT means a knight on a toroidal board. He finds 2688 tours of 19 types on the 42 toroid. (Cf Tylor, 1982??)

J. J. Duby. Un algorithme graphique trouvant tous les circuits Hamiltoniens d'un graphe. Etude No. 8, IBM France, Paris, 22 Oct 1964. [In English with French title and summary.] Finds there are 9862 knight's tours on the 6 x 6 board, where the tours all start at a fixed corner and then go to a fixed one of the two cells reachable from the corner. He also finds 75,000 tours on the 8 x 8 board which have the same first 35 moves. He believes there may be over a million tours.

Karl Fabel. Wanderungen von Schachfiguren. IN: Eero Bonsdorff, Karl Fabel & Olavi Riihimaa; Schach und Zahl; Walter Rau Verlag, Düsseldorf, 1966, pp. 40-50. On p. 50, he says that there are 122,802,512 tours where the knight does two joined half-board paths. He also says there are upper bounds, determined by several authors, and he gives 1.5 x 1026 as an example.

Gardner. SA (Oct 1967) = Magic Show, chap. 14. Surveys results of which boards have tours or paths.

D. J. W. Stone. On the Knight's Tour Problem and Its Solution by Graph-Theoretic and Other Methods. M.Sc. Thesis, Dept. of Computing Science, Univ. of Glasgow, Jan. 1969. Confirms Duby's 9862 tours on the 6 x 6 board.

David Singmaster. Enumerating unlabelled Hamiltonian circuits. International Series on Numerical Mathematics, No. 29. Birkhäuser, Basel, 1975, pp. 117-130. Discusses the work of Duby and Stone and gives an estimate, which Stone endorses, that there are 1023±3 tours on the 8 x 8 board.

C. M. B. Tylor. 2-by-2 tours. Chessics 14 (Jul-Dec 1982) 14. Says there are 17 knight's tours on a 2 x 2 torus and gives them. Doesn't mention Wheeler, 1963.

Robert Cannon & Stan Dolan. The knight's tour. MG 70 (No. 452) (Jun 1986) 91-100. A rectangular board is tourable if it has a knight's path between any two cells of opposite colours. They prove that m x n is tourable if and only if mn is even and m ( 6, n ( 6. They also prove that m x n has a knight's tour if and only if mn is even and [(m ( 5, n ( 5) or (m = 3, n ( 10)] and that when mn is even, m x n has a knight's path if and only if m ( 3, n ( 3, except for the 3 x 6 and 4 x 4 boards. (These later results are well known -- see Gardner. The authors only cite Ball's MRE.)

George Jelliss. Figured tours. MS 25:1 (1992/93) 16-20. Exposition of paths and tours where certain stages of the path form an interesting geometric figure. E.g. Euler's first paper has a path on the 5 x 5 such that the points on one diagonal are in arithmetic progression: 1, 7, 13, 19, 25.

Martin Loebbing & Ingo Wegener. The number of knight's tours equals 33,439,123,484,294 --- Counting with binary decision diagrams. Electronic Journal of Combinatorics 3 (1996) article R5. A somewhat vague description of a method for counting knight's tours -- they speak of directed knight's tours, but it is not clear if they have properly accounted for the symmetries of a tour or of the board. Several people immediately pointed out that the number is incorrect because it has to be divisible by four. Two comments have appeared, ibid. On 15 May 1996, the authors admitted this and said they would redo the problem, but they have submitted no further comment as of Jan 2001. On 18 Feb 1997, Brendan McKay announced that he had done the computation another way and found 13,267,364,410,532.

In view of the difference between this and my 1975 estimate of 1023±3 tours, it might be worth explaining my reasoning. In 1964, Duby found 75,000 tours with the same first 35 moves. The average valence for a knight on an 8 x 8 board is 5.25, but one cannot exit from a cell in the same direction as one entered, so we might estimate the number of ways that the first 35 moves can be made as 4.2535 = 9.9 x 1021. Multiplying by 75,000 then gives 7.4 x 1026. I think I assumed that some of the first moves had already been made, e.g. we only allow one move from the starting cell, and factored by 8 for the symmetries of the square, to get 2.2 x 1025. I can't find my original calculations, and I find the estimate 1025 in later papers, so I suppose I tried to reduce the effect of the 4.2535 some more. In retrospect, I had no knowledge of how many of these had already been tried. If about half of all moves from a cell had already been tried before any circuit was found, then the estimate would be more like 2.2534 x 75,000 = 7.1 x 1016. If we divide the given number of circuits by 75,000 and take the 34th root, we get an average valence of 1.78 remaining, far less than I would have guessed.

I am grateful to Don Knuth for this reference. Neither he nor I expected to ever see this number calculated!

5.F.2. OTHER HAMILTONIAN CIRCUITS

For circuits on the n-cube, see also 5.F.4 and 7.M.1,2,3.

For circuits on the chessboard, see also 6.AK.

Le Nôtre. Le Labyrinte de Versailles, c1675. This was a hedge or garden maze, but the objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one fountain. Some fountains were not at path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian problem, except that one fountain was located at the end of a short dead end. [Fisher, op. cit. in 5.E.1, pp. 49, 79, 130 & 144-145, with contemporary diagram on p. 144. He says there are 39 fountains, but the diagram has 40.]

T. P. Kirkman. On the partitions of the R-pyramid, being the first class of R-gonous X-edra. Philos. Trans. Roy. Soc. 148 (1858) 145-161.

W. R. Hamilton. The Icosian Game. 4pp instructions for the board game. J. Jaques and Son, London, 1859. (Reproduced in BLW, pp. 32-35, with frontispiece photo of the board at the Royal Irish Academy.)

For a long time, the only known example of the game, produced by Jaques, was at the Royal Irish Academy in Dublin. This example is inscribed on the back as a present from Hamilton to his friend, J. T. Graves. It is complete, with pegs and instructions. None of the obvious museums have an example. Diligent searching in the antique trade failed to turn up an example in twenty years, but in Feb 1996, James Dalgety found and acquired an example of the board -- sadly the pegs and instructions were lacking. Dalgety obtained another board in 1998, again without the pegs and instructions, but in 1999 he obtained another example, with the pegs.

Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53-54 & 102; 1917: 310, pp. 49 & 97. The garden of a French palace has a maze with 31 points to see. Find a path past all of them with no repeated edges and no crossings. The pattern is clearly based on the Versailles maze of c1675 mentioned above, but I don't recall the additional feature of no crossings occurring before.

T. P. Kirkman. Solution of problem 6610, proposed by himself in verse. Math. Quest. Educ. Times 35 (1881) 112-116. On p. 115, he says Hamilton told him, upon occasion of Hamilton presenting him 'with his handsomest copy of the puzzle', that Hamilton got the idea for the Icosian Game from p. 160 of Kirkman's 1858 article,

Lucas. RM2, 1883, pp. 208-210. First? mention of the solid version. The 2nd ed., 1893, has a footnote referring to Kirkman, 1858.

John Jaques & Son. The Traveller's Dodecahedron; or, A Voyage Round the World. A New Puzzle. "This amusing puzzle, exercising considerable skill in its solution, forms a popular illustration of Sir William Hamilton's Icosian Game. A wood dodecahedron with the base pentagon stretched so that when it sits on the base, all vertices are visible. With ivory? pegs at the vertices, a handle that screws into the base, a string with rings at the ends and one page of instructions, all in a box. No date. The only known example was obtained by James Dalgety in 2002.

Pearson. 1907. Part III, no. 60: The open door, pp. 60 & 130. Prisoner in one corner of an 8 x 8 array is allowed to exit from from the other corner provided he visits every cell once. This requires him to enter and leave a cell by the same door.

Ahrens. Mathematische Spiele. 2nd ed., Teubner, Leipzig, 1911. P. 44, note, says that a Dodekaederspiel is available from Firma Paul Joschkowitz -- Magdeburg for .65 mark. This is not in the 1st ed. of 1907 and the whole Chapter is dropped in the 3rd ed. of 1916 and the later editions.

Anonymous. The problems drive. Eureka 12 (Oct 1949) 7-8 & 15. No. 3. How many Hamiltonian circuits are there on a cube, starting from a given point? Reflections and reversals count as different tours. Answer is 12, but this assumes also that rotations are different. See Singmaster, 1975, for careful definitions of how to count. There are 96 labelled circuits, of which 12 start at a given vertex. But if one takes all the 48 symmetries of the cube as equivalences (six of which fix the given vertex), there are just 2 circuits from a given starting point. However, these are actually the same circuit started at different points. Presumably Kirkman and Hamilton knew of this.

C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp. 24-25. Roman knobbed dodecahedra -- an ancient solid version??

R. E. Ingram. Appendix 2: The Icosian Calculus. In: The Mathematical Papers of Sir William Rowan Hamilton. Vol. III: Algebra. Ed. by H. Halberstam & R. E. Ingram. CUP, 1967, pp. 645-647. [Halberstam told me that this Appendix is due to Ingram.] Discusses the method and asserts that the tetrahedron, cube and dodecahedron have only one unlabelled circuit, the octahedron has two and the icosahedron has 17.

David Singmaster. Hamiltonian circuits on the regular polyhedra. Notices Amer. Math. Soc. 20 (1973) A-476, no. 73T-A199. Confirms Ingram's results and gives the number of labelled circuits.

David Singmaster. Op. cit. in 5.F.1. 1975. Carefully defines labelled and unlabelled circuits. Discusses results on regular polyhedra in 3 and higher dimensions.

David Singmaster. Hamiltonian circuits on the n-dimensional octahedron. J. Combinatorial Theory (B) 18 (1975) 1-4. Obtains an explicit formula for the number of labelled circuits on the n-dimensional octahedron and shows it is ( (2n)!/e. Gives numbers for n ( 8. In unpublished work, it is shown that the number of unlabelled circuits is asymptotic to the above divided by n!2n(4n.

Angus Lavery. The Puzzle Box. G&P 2 (May 1994) 34-35. Alternative solitaire, p. 34. Asks for a knight's tour on the 33-hole solitaire board. Says he hasn't been able to do it and offers a prize for a solution. In Solutions, G&P 3 (Jun 1994) 44, he says it cannot be done and the proof will be given in a future issue, but I never saw it.

5.F.3. KNIGHT'S TOURS IN HIGHER DIMENSIONS

A.-T. Vandermonde. Remarques sur les problèmes de situation. Hist. de l'Acad. des Sci. avec les Mémoires (Paris) (1771 (1774)) Mémoires: pp. 566-574 & Plates I & II. ??NYS. First? mention of cubical problem. (Not given in BLW excerpt.)

F. Maack. Mitt. über Raumschak. 1909, No. 2, p. 31. ??NYS -- cited by Gibbins, below. Knight's tour on 4 x 4 x 4 board.

Dudeney. AM. 1917. Prob. 340: The cubic knight's tour, pp. 103 & 229. Says Vandermonde asked for a tour on the faces of a 8 x 8 x 8 cube. He gives it as a problem with a solution.

N. M. Gibbins. Chess in three and four dimensions. MG 28 (No. 279) (1944) 46-50. Gives knight's tour on 3 x 3 x 4 board -- an unpublished result due to E. Hubar-Stockar of Geneva. This is the smallest 3-D board with a tour. Gives Maack's tour on 4 x 4 x 4 board.

Ian Stewart. Solid knight's tours. JRM 4:1 (Jan 1971) 1. Cites Dudeney. Gives a tour through the entire 8 x 8 x 8 cube by stacking 8 knight's paths.

T. W. Marlow. Closed knight tour of a 4 x 4 x 4 board. Chessics 29 & 30 (1987) 162. Inspired by Stewart.

5.F.4. OTHER CIRCUITS IN AND ON A CUBE

The number of Hamiltonian Circuits on the n-dimensional cube is the same as the number of Gray codes (see 7.M.3) and has been the subject of considerable research. I will not try to cover this in detail.

D. W. Crowe. The n-dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956) 29-30.

E. N. Gilbert. Gray codes and paths on the n-cube. Bell System Technical Journal 37 (1958) 815-826. Shows there are 9 inequivalent circuits on the 4-cube and 1 on the n-cube for n = 1, 2, 3. The latter cases are sufficiently easy that they may have been known before this.

Allen F. Dreyer. US Patent 3,222,072 -- Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec 1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal so that three consecutive cubes are either in a line or form a right angle. A solution is a Hamiltonian path through the 27 cells. Such puzzles were made in Germany and I was given one about 1980 (see Singmaster and Haubrich & Bordewijk below). Dreyer gives two forms.

Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2. Notes that the number of circuits on the n-cube, n > 4, is not known. SA (Apr 1973) reports that three (or four) groups had found the number of circuits on the 4-cube -- this material is included in the Addendum in Knotted, chap. 2, but none of the groups ever seem to have published their results elsewhere. Unfortunately, none of these found the number of inequivalent circuits since they failed to take all the equivalences into account -- e.g. for n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the numbers of labelled circuits. Gardner's Addendum describes some further work including some statistical work which estimates the number on the 6-cube is about 2.4 x 1025.

David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but never published. For the 3 x 3 x 3 problem, the number, S, of straight through pieces (ignoring the ends) satisfies 2 ( S ( 11.

Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into a cube in just one way. But cf the next article which gives a different number??

Jacques Haubrich & Nanco Bordewijk. Cube chains. CFF 34 (Oct 1994) 12-15. Erratum, CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to the idea and that they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's patent. Says his first version has a unique solution, but the second has 38 solutions. They have redone previous work and get new numbers. First, they consider all possible strings of 27 cubes with at most three in a line (i.e. with at most a single 'straight' piece between two 'bend' pieces and they find there are 98,515 of these. Only 11,487 of these can be folded into a 3 x 3 x 3 cube. Of these, 3654 can be folded up in only one way. The chain with the most solutions had 142 different solutions. They refer to Mel Scott's tables and indicate that the results correspond -- perhaps I miscounted Scott's solutions??

5.G. CONNECTION PROBLEMS

5.G.1. GAS, WATER AND ELECTRICITY

Dudeney. Problem 146 -- Water, gas, and electricity. Strand Mag. 46 (No. 271) (Jul 1913) 110 & (No. 272) (Aug 1913) 221 (c= AM, prob. 251, pp. 73 & 200-201). Earlier version is slightly more interesting, saying the problem 'that I have called "Water, Gas, and Electricity" ... is as old as the hills'. Gives trick solution with pipe under one house.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 96: The "three houses" problem, pp. 89-90 & 114. "Were all the houses connected up with all three supplies or not?" Answer is no -- one connection cannot be made.

Loyd, Jr. SLAHP. 1928. The three houses and three wells, pp. 6 & 87-88. "A puzzle ... which I first brought out in 1900 ..." The drawing is much less polished than Dudeney's. Trick solution with a pipe under one house, a bit differently laid out than Dudeney.

The Bile Beans Puzzle Book. 1933. No. 46: Water, gas & electric light. Trick solution almost identical to Dudeney.

Philip Franklin. The four color problem. In: Galois Lectures; Scripta Mathematica Library No. 5; Scripta Mathematica, Yeshiva College, NY, 1941, pp. 49-85. On p. 74, he refers to the graph as "the basis of a familiar puzzle, to join each of three houses with each of three wells (or in a modern version to a gas, water, and electricity plant)".

Leeming. 1946. Chap. 6, prob. 4: Water, gas and electricity, pp. 71 & 185. Dudeney's trick solution.

H. ApSimon. Note 2312: All modern conveniences. MG 36 (No. 318) (Dec 1952) 287-288. Given m houses and n utilities, the maximum number of non-crossing connections is 2(m+n-2) and this occurs when all the resulting regions are 4-sided. He extends to p-partite graphs in general and a special case.

John Paul Adams. We Dare You to Solve This! Op. cit. in 5.C. 1957? Prob. 50: Another enduring favorite appears below, pp. 30 & 49. Electricity, gas, water. Dudeney's trick solution.

Young World. c1960. P. 4: Crossed lines. Electricity, TV and public address lines. Trick solution with a line passing under a house.

T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751-753. Shows you can join 4 utilities to 4 houses on a torus without crossing.

5.H. COLOURED SQUARES AND CUBES, ETC.

5.H.1. INSTANT INSANITY = THE TANTALIZER

Note. Often the diagrams do not show all sides of the pieces so I cannot tell if one version is the same as another.

Frederick A. Schossow. US Patent 646,463 -- Puzzle. Applied: 19 May 1899; patented: 3 Apr 1900. 1p + 1p diagrams. Described in S&B, p. 38, which also says it is described in O'Beirne, but I don't find it there?? Four cubes with suit patterns. The net of each cube is shown. The fourth cube has three clubs.

George Duncan Moffat. UK Patent 9810 -- Improvements in or relating to Puzzle-apparatus. Applied: 28 May 1900; accepted: 30 Jun 1900. 2pp + 1p diagrams. For a six cube version with "letters R, K, B, W, F and B-P, the initials of the names of General Officers of the South African Field Force."

Joseph Meek. UK Patent 2775 -- Improved Puzzle Game. Applied: 5 Feb 1909; complete specification: 16 Jun 1909; accepted: 3 Feb 1910. 2pp + 1p diagrams. A four cube version with suit patterns. His discussion seems to describe the pieces drawn by Schossow.

Slocum. Compendium. Shows: The Great Four Ace Puzzle (Gamage's, 1913); Allies Flag Puzzle (Gamage's, c1915); Katzenjammer Puzzle (Johnson Smith, 1919).

Edwin F. Silkman. US Patent 2,024,541 -- Puzzle. Applied: 9 Sep 1932; patented: 17 Dec 1935. 2pp + 1 p diagrams. Four cubes marked with suits. The net of each cube is shown. The third cube has three hearts. This is just a relabelling of Schossow's pattern, though two cubes have to be reflected which makes no difference to the solution process.

E. M. Wyatt. The bewitching cubes. Puzzles in Wood. (Bruce Publishing, Co., Milwaukee, 1928) = Woodcraft Supply Corp., Woburn, Mass., 1980, p. 13. A six cube, six way version.

Abraham. 1933. Prob. 303 -- The four cubes, p. 141 (100). 4 cube version "sold ... in 1932".

A. S. Filipiak. Four ace cube puzzle. 100 Puzzles, How do Make and How to Solve Them. A. S. Barnes, NY, (1942) = Mathematical Puzzles, and Other Brain Twisters; A. S. Barnes, NY, 1966; Bell, NY, 1978; p. 108.

Leeming. 1946. Chap. 10, prob. 9: The six cube puzzle, pp. 128-129 & 212. Identical to Wyatt.

F. de Carteblanche [pseud. of Cedric A. B. Smith]. The coloured cubes problem. Eureka 9 (1947) 9-11. General graphical solution method, now the standard method.

T. H. O'Beirne. Note 2736: Coloured cubes: A new "Tantalizer". MG 41 (No. 338) (Dec 1957) 292-293. Cites Carteblanche, but says the current version is different. Gives a nicer version.

T. H. O'Beirne. Note 2787: Coloured cubes: a correction to Note 2736. MG 42 (No. 342) (Dec 1958) 284. Finds more solutions than he had previously stated.

Norman T. Gridgeman. The 23 colored cubes. MM 44:5 (Nov 1971) 243-252. The 23 colored cubes are the equivalence classes of ways of coloring the faces with 1 to 6 colors. He cites and describes some later methods for attacking Instant Insanity problems.

Jozsef Bognár. UK Patent Application 2,076,663 A -- Spatial Logical Puzzle. Filed 28 May 1981; published 9 Dec 1981. Cover page + 8pp + 3pp diagrams. Not clear if the patent was ever granted. Describes Bognár's Planets, which is a four piece instant insanity where the pieces are spherical and held in a plastic tube. This was called Bolygok in Hungarian and there is a reference to an earlier Hungarian patent. Also describes his version with eight pieces held at the corners of a plastic cube.

5.H.2. MACMAHON PIECES

Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include MacMahon puzzles as one class.

I have just added the Carroll result that there are 30 six-coloured cubes, but this must be older??

Frank H. Richards. US Patent 331,652 -- Domino. Applied: 13 Jun 1885; patented: 1 Dec 1885. 2pp + 2pp diagrams. Cited by Gardner in Magic Show, but with date 1895. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. For triangular matching games, specifically showing the MacMahon 5-coloured triangles, but considering reflections as equivalences, so he has 35 pieces. [One of the colours is blank and hence Gardner said it was a 4-colouring.]

Carroll-Wakeling. c1890? Prob. 15: Painting cubes, pp. 18-19 & 67. This is one of the problems on undated sheets of paper that Carroll sent to Bartholomew Price. How many ways can one six-colour a cube? Wakeling gives a solution, but this apparently is not on Carroll's MS.

Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 3927 A.D. 1892 -- Appliances to be used in Playing a New Class of Games. Applied: 29 Feb 1892; Complete Specification Left: 28 Nov 1892; Accepted: 28 Jan 1893. 5pp + 2pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. Describes the 24 triangles with four types of edge and mentions other numbers of edge types. Describes various games and puzzles.

Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 8275 A.D. 1892 -- Appliances for New Games of Puzzles. Applied: 2 May 1892; Complete Specification Left: 31 Jan 1893; Accepted: 4 Mar 1893. 2pp. 27 cubes with three colours, opposite faces having the same colour. Similar sets of 8, 27, 64, etc. cubes. Various matching games suggested. Using six colours and all six on each cube gives 30 cubes -- the MacMahon Cubes. Gives a complex matching problem of making two 2 x 2 x 2 cubes. Paul Garcia (email of 15 Nov 2002) commented: "8275 describes 2 different sets of blocks, using either three colours or six colours. The three colour blocks form a set of 27 that can be assembled into a large cube with single coloured faces and internal contact faces matching. For the six colour cubes, the puzzle suggested is to pick out two associated cubes, and find the sixteen cubes that can be assembled to make a copy of each. Not quite Mayblox, although using the same colouring system."

James Dalgety. R. Journet & Company A Brief History of the Company & its Puzzles. Published by the author, North Barrow, Somerset, 1989. On p. 13, he says Mayblox was patented in 1892. In an email on 12 Nov 2002, he cited UK Patent 8275.

Anon. Report: "Mathematical Society, February 9". Nature 47 (No. 1217) (23 Feb 1893) 406. Report of MacMahon's talk: The group of thirty cubes composed by six differently coloured squares.

See: Au Bon Marché, 1907, in 5.P.2, for a puzzle of hexagons with matching edges.

Manson. 1911. Likoh, pp. 171-172. MacMahon's 24 four-coloured isosceles right triangles, attributed to MacMahon.

"Toymaker". The Cubes of Mahomet Puzzle. Work, No. 1447 (9 Dec 1916) 168. 8 six-coloured cubes to be assembled into a cube with singly-coloured faces and internal faces to have matching colours.

P. A. MacMahon. New Mathematical Pastimes. CUP, 1921. The whole book deals with variations of the problem and calculates the numbers of pieces of various types. In particular, he describes the 24 4-coloured triangles, the 24 3-coloured squares, the MacMahon cubes, some right-triangular and hexagonal sets and various subsets of these. With n colours, there are n(n2+2)/3 triangles, n(n+1)(n2-n+2)/4 squares and n(n+1)(n4-n3+n2+2)/6 hexagons. [If one allows reflectional equivalence, one gets n(n+1)(n+2)/6 triangles, n(n+1)(n2+n+2)/8 squares and n(n+1)(n4-n3+4n2+2)/12 hexagons. Problem -- is there an easy proof that the number of triangles is BC(n+2, 3)?] On p. 44, he says that Col. Julian R. Jocelyn told him some years ago that one could duplicate any cube with 8 other cubes such that the internal faces matched.

Slocum. Compendium. Shows Mayblox made by R. Journet from Will Goldston's 1928 catalogue.

F. Winter. Das Spiel der 30 bunten Würfel MacMahon's Problem. Teubner, Leipzig, 1934, 128pp. ??NYR.

Clifford Montrose. Games to play by Yourself. Universal Publications, London, nd [1930s?]. The coloured squares, pp. 78-80. Makes 16 squares with four-coloured edges, using five colours, but there is no pattern to the choice. Uses them to make a 4 x 4 array with matching edges, but seems to require the orientations to be fixed.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 6. There are twelve ways to colour the edges of a pentagon, when rotations and reflections are considered as equivalences. Can you colour the edges of a dodecahedron so each of these pentagonal colourings occurs once? [If one uses tiles, one has to have reversible tiles.] Solution says there are three distinct solutions and describes them by describing contacts between 10 pentagons forming a ring around the equator.

Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Pp. 101-104 discusses MacMahon triangles, squares and hexagons.

T. H. O'Beirne. Puzzles and paradoxes 5: MacMahon's three-colour set of squares. New Scientist 9 (No. 220) (2 Feb 1961) 288-289. Finds 18 of the 20 possible monochrome border patterns.

Gardner. SA (Mar 1961) = New MD, Chap. 16. MacMahon's 3-coloured squares and his cubes. Addendum in New MD cites Feldman, below.

Gary Feldman. Documentation of the MacMahon Squares Problem. Stanford Artificial Intelligence Project Memo No. 12, Stanford Computation Center, 16 Jan 1964. ??NYS Finds 12,261 solutions for the 6 x 4 rectangle with monochrome border -- but see Philpott, 1982, for 13,328 solutions!!

Gardner. SA (Oct 1968) = Magic Show, Chap. 16. MacMahon's four-coloured triangles and numerous variants.

Wade E. Philpott. MacMahon's three-color squares. JRM 2:2 (1969) 67-78. Surveys the topic and repeats Feldman's result.

N. T. Gridgeman, loc. cit. in 5.H.1, 1971, covers some ideas on the MacMahon cubes.

J. J. M. Verbakel. Digitale tegels (Digital tiles). Niet piekeren maar puzzelen (name of a puzzle column). Trouw (a Dutch newspaper) (1 Feb 1975). ??NYS -- described by Jacques Haubrich; Pantactic patterns and puzzles; CFF 34 (Oct 1994) 19-21. There are 16 ways to 2-colour the edges of a square if one does not allow them to rotate. Assemble these into a 4 x 4 square with matching edges. There are 2,765,440 solutions in 172,840 classes of 16. One can add further constraints to yield fewer solutions -- e.g. assume the 4 x 4 square is on a torus and make all internal lines have a single colour.

Gardner. Puzzling over a problem-solving matrix, cubes of many colours and three-dimensional dominoes. SA 239:3 (Sep 1978) 20-30 & 242 c= Fractal, chap. 11. Good review of MacMahon (photo) and his coloured cubes. Bibliography cites recent work on Mayblox, etc.

Wade E. Philpott. Instructions for Multimatch. Kadon Enterprises, Pasadena, Maryland, 1982. Multimatch is just the 24 MacMahon 3-coloured squares. This surveys the history, citing several articles ??NYS, up to the determination of the 13,328 solutions for the 6 x 4 rectangle with monochrome border, by Hilario Fernández Long (1977) and John W. Harris (1978).

Torsten Sillke. Three 3 x 3 matching puzzles. CFF 34 (Oct 1994) 22-23. He has wanted an interesting 9 element subset of the MacMahon pieces and finds that of the 24 MacMahon 3-coloured squares, just 9 of them contain all three colours. He considers both the corner and the edge versions. The editor notes that a 3 x 3 puzzle has 36 x 32/2 = 576 possible edge contacts and that the number of these which match is a measure of the difficulty of the puzzle, with most 3 x 3 puzzles having 60 to 80 matches. The corner version of Sillke's puzzle has 78 matches and one solution. The edge version has 189 matches and many solutions, hence Sillke proposes various further conditions.

5.H.3. PATH FORMING PUZZLES

Here we have a set of pieces and one has to join them so that some path is formed. This is often due to a chain or a snake, etc. New section. Again, Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include this as one class.

Hoffmann. 1893. Chap. III, No. 18: The endless chain, pp. 99-100 & 131 = Hoffmann-Hordern, pp. 91-92, with photo. 18 pieces, some with parts of a chain, to make into an 8 x 8 array with the chain going through 34 of the cells. All the pieces are rectangles of width one. Photo shows The Endless Chain, by The Reason Manufacturing Co., 1880-1895. Hordern Collection, p. 62, shows the same and La Chaine sans fin, 1880-1905.

Loyd. Cyclopedia. 1914. Sam Loyd's endless chain puzzle, pp. 280 & 377. Chain through all 64 cells of a chessboard, cut into 13 pieces. The chessboard dissection is of type: 13: 02213 131.

Hummerston. Fun, Mirth & Mystery. 1924. The dissected serpent, p. 131. Same pieces as Hoffmann, and almost the same pattern.

Collins. Book of Puzzles. 1927. The dissected snake puzzle, pp. 126-127. 17 pieces forming an 8 x 8 square. All the piece are rectangular pieces of width one except for one L-hexomino -- if this were cut into straight tetromino and domino, the pieces would be identical to Hoffmann. The pattern is identical to Hummerston.

See Haubrich in 5.H.4.

5.H.4. OTHER AND GENERAL

These all have coloured edges unless specified. See S&B, p. 36, for examples.

Edwin L[ajette] Thurston. US Patent 487,797 -- Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 3pp + 3pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. 4 x 4 puzzles with 6-coloured corners or edges, but assuming no colour is repeated on a piece -- indeed he uses the 15 = BC(6,2) ways of choosing 4 out of 6 colours once only and then has a sixteenth with the same colours as another, but in different order. Also a star-shaped puzzle of six parallelograms.

Edwin L. Thurston. US Patent 487,798 -- Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. As far as I can see, this is the same as the 4 x 4 puzzle with 6-coloured edges given above, but he seems to be emphasising the 15 pieces.

Edwin L. Thurston. US Patent 490,689 -- Puzzle. Applied: 30 Sep 1890; patented: 31 Jun 1893. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. The patent is for 3 x 3 puzzles with 4-coloured corners or edges, but with pieces having no repeated colours and in a fixed orientation. He selects some 8 of these pieces for reasons not made clear and mentions moving them "after the manner of the old 13, 14, 15 puzzle." S&B, p. 36, describes the Calumet Puzzle, Calumet Baking Powder Co., Chicago, which is a 3 x 3 head to tail puzzle, claimed to be covered by this patent.

Le Berger Malin. France, c1900. 3 x 3 head to tail puzzle, but the edges are numbered and the matching edges must add to 10. ??NYS -- described by K. Takizawa, N. Takashima & N. Yoshigahara; Vess Puzzle and Its Family -- A Compendium of 3 by 3 Card Puzzles; published by the authors, Tokyo, 1983. Slocum has this in two different boxes and dates it to c1900 -- I had c1915 previously. Haubrich has one version, Produced by GB&O N.K. Atlas.

Angus K. Rankin. US Patent 1,006,878 -- Puzzle. Applied: 3 Feb 1911; patented: 24 Oct 1911. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. Described in S&B, p. 36. Grandpa's Wonder Puzzle. 3 x 3 square puzzle. Each piece has corners coloured, using four colours, and the colours meeting at a corner must differ. The patent doesn't show the advertiser's name -- Grandpa's Wonder Soap -- but is otherwise identical to S&B's photo.

Daily Mail World Record Net Sale puzzle. 1920-1921. Instructions and picture of the pieces. Letter from Whitehouse to me describing its invention. 19 6-coloured hexagons without repeated colours. Daily Mail articles as follows. There may be others that I missed and sometimes the page number is a bit unclear. Note that 5 Dec was a Sunday.

9 Nov 1920, p. 5. "Daily Mail" puzzle. To be issued on 7 Dec.

13 Nov 1920, p. 4. Hexagon mystery.

17 Nov 1920, p. 5. New mystery puzzle. Asserts the inventor does not know the solution -- i.e. the solution has been locked up in a safe.

20 Nov 1920, p. 4. What is it?

23 Nov 1920, p. 5. Fascinating puzzle. The most fascinating puzzle since "Pigs in Clover".

25 Nov 1920, p. 5. Can you do it?

29 Nov 1920, p. 5. £250 puzzle.

1 Dec 1920, p. 4. Mystery puzzle clues.

2 Dec 1920, p. 5. £250 puzzle race.

3 Dec 1920, p. 5. The puzzle.

4 Dec 1920, p. 4. The puzzle. Amplifies on the inventor not knowing the solution -- after the idea was approved, a new pattern was created by someone else and locked up.

6 Dec 1920, unnumbered back page. Photo with caption: £250 for solving this.

7 Dec 1920, p. 7. "Daily Mail" Puzzle. Released today. £100 for getting the locked up solution. £100 for the first alternative solution and £50 for the next alternative solution. "It is believed that more than one solution is possible."

8 Dec 1920, p. 5. "Daily Mail" puzzle.

9 Dec 1920, p. 5. Can you do it?

10 Dec 1920, p. 4. It can be done.

13 Dec 1920, p. 9. Most popular pastime. "More than 500,000 Daily Mail Puzzles have been sold."

15 Dec 1920, p. 4. Puzzle king & the 19 hexagons. Dudeney says he does not think it can be solved "except by trial."

16 Dec 1920, p. 4. Tantalising 19 hexagons.

16 Dec 1920, unnumbered back page. Banner at top has: "The Daily Mail" puzzle. Middle of page has a cartoon of sailors trying to solve it.

17 Dec 1920, p. 5? The Xmas game.

18 Dec 1920, p. 7. Puzzle Xmas 'card'.

20 Dec 1920, p. 7. Hexagon fun.

22 Dec 1920, p. 3. 3,000,000 fascinated. It is assumed that about 5 people try each example and so this indicates that nearly 600,000 have been sold.

23 Dec 1920, p. 3. Too many cooks.

23 Dec 1920, unnumbered back page. Cartoon: The hexagonal dawn!

28 Dec 1920, p. 3? Puzzled millions. "On Christmas Eve the sales exceeded 600,000 ...."

29 Dec 1920, p. 3? "I will do it."

30 Dec 1920, p. 8. Puzzle fun.

3 Jan 1921, p. 3. The Daily Mail Puzzle. C. Lewis, aged 21, a postal clerk solved it within two hours of purchase and submitted his solution on 7 Dec. Hundreds of identical solutions were submitted, but no alternative solutions have yet appeared. There are two pairs of identical pieces: 1 & 12, 4 & 10.

3 Jan 1921, p. 10 = unnumbered back page. Hexagon Puzzle Solved, with photo of C. Lewis and diagram of solution.

10 Jan 1921, p. 4. Hexagon puzzle. Since no alternative hexagonal solutions were received, the other £150 is awarded to those who submitted the most ingenious other solution -- this was judged to be a butterfly shape, submitted by 11 persons, who shared the £150.

Horace Hydes & Francis Reginald Beaman Whitehouse. UK Patent 173,588 -- Improvements in Dominoes. Applied: 29 Sep 1920; complete application: 29 Jun 1921; accepted: 29 Dec 1921. Reproduced in Haubrich, About ..., 1996, op. cit. below. 3pp + 1p diagrams. This is the patent for the above puzzle, corresponding to provisional patent 27599/20 on the package. The illustration shows a solved puzzle based on 'A stitch in time saves nine'.

George Henry Haswell. US Patent 1,558,165 -- Puzzle. Applied: 3 Jul 1924; patented: 11 Sep 1925. Reproduced in Haubrich, About ..., 1996, op. cit. below. 2pp + 1p diagrams. For edge-matching hexagons with further internal markings which have to be aligned. [E.g. one could draw a diagonal and require all diagonals to be vertical -- this greatly simplifies the puzzle!] If one numbers the vertices 1, 2, ..., 6, he gives an example formed by drawing the diagonals 13, 15, 42, 46 which produces six triangles along the edges and an internal rhombus.

C. Dudley Langford. Note 2829: Dominoes numbered in the corners. MG 43 (No. 344) (May 1959) 120-122. Considers triangles, squares and hexagons with numbers at the corners. There are the same number of pieces as with numbers on the edges, but corner numbering gives many more kinds of edges. E.g. with four numbers, there are 24 triangles, but these have 16 edge patterns instead of 4. The editor (R. L. Goodstein) tells Langford that he has made cubical dominoes "presumably with faces numbered". Langford suggests cubes with numbers at the corners. [I find 23 cubes with two corner numbers and 333 with three corner numbers. ??check]

Piet Hein. US Patent 4,005,868 -- Puzzle. Applied: 23 Jun 1975; patented: 1 Feb 1977. Front page + 8pp diagrams + 5pp text. Basically non-matching puzzles using marks at the corners of faces of the regular polyhedra. He devises boards so the problems can be treated as planar.

Kiyoshi Takizawa; Naoaki Takashima & Nob. Yoshigahara. Vess Puzzle and Its Family -- A Compendium of 3 by 3 Card Puzzles. Published by the authors, Tokyo, Japan, 1983. Studies 32 types (in 48 versions) of 3 x 3 'head to tail' matching puzzles and 4 related types (in 4 versions). All solutions are shown and most puzzles are illustrated with colour photographs of one solution. (Haubrich counts 51 versions -- check??)

Melford D. Clark. US Patent 4,410,180 -- Puzzle. Applied: 16 Nov 1981; patented: 18 Oct 1983. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams. Corner matching squares, but with the pieces marked 1, 2, ..., so that the pieces marked 1 form a 1 x 1 square, the pieces marked 2 allow this to be extended to a 2 x 2 square, etc. There are n2 - (n-1)2 pieces marked n.

Jacques Haubrich. Compendium of Card Matching Puzzles. Printed by the author, Aeneaslaan 21, NL-5631 LA Eindhoven, Netherlands, 1995. 2 vol., 325pp. describing over 1050 puzzles. He classifies them by the nine most common matching rules: Heads and Tails; Edge Matching (i.e. MacMahon); Path Matching; Corner Matching; Corner Dismatching; Jig-Saw-Like; Continuous Path; Edge Dismatching; Hybrid. He does not include Jig-Saw-Like puzzles here. Using the number of cards and their shape, then the matching rules, he has 136 types. 31 different numbers of cards occur: 4, 6-16, 18-21, 23-25, 28, 30, 36, 40, 45, 48, 56, 64, 70, 80, 85, 100. There is an index of 961 puzzle names. He says Hoffmann is the earliest published example. He notes that most path puzzles have a global criterion that the result have a single circuit which slightly removes them from his matching criterion and he does not treat them as thoroughly. He has developed computer programs to solve each type of puzzle and has checked them all.

Jacques Haubrich. About, Beyond and Behind Card Matching Puzzles. [= Vol. 3 of above]. Ibid, Apr 1996, 87pp. This is a general discussion of the different kinds of puzzles, how to solve them and their history, reproducing ten patents and two obituaries.

5.I. LATIN SQUARES AND EULER SQUARES

This topic ties in with certain tournament problems but I have not covered them. See also Hoffmann and Loughlin & Flood in 5.A.2 for examples of two orthogonal 3 x 3 Latin squares. The derangement problems in 5.K.2 give Latin rectangles.

Ahrens-1 & Ahrens-2. Opp. cit. in 7.N. 1917 & 1922. Ahrens-1 discusses and cites early examples of Latin squares, going back to medieval Islam (c1200), where they were used on amulets. Ahrens-2 particularly discusses work of al-Buni -- see below.

(Ahmed [the h should have an underdot] ibn ‘Alî ibn Jûsuf) el-Bûni, (Abû'l-‘Abbâs, el-Qoresî.) = Abu-l‘Abbas al-Buni. (??= Muhyi'l-Dîn Abû’l-‘Abbâs al-Bûnî -- can't relocate my source of this form.) Sams al-ma‘ârif = Shams al-ma‘ârif al-kubrâ = Šams al-ma‘ārif. c1200. ??NYS. Ahrens-1 describes this briefly and incorrectly. He expands and corrects this work in Ahrens-2. See 7.N for more details. Ahrens notes that a 4 x 4 magic square can be based on the pattern of two orthogonal Latin squares of order 4, and Al-Buni's work indicates knowledge of such a pattern, exemplified by the square

8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15 considered (mod 4). He also has Latin squares of order 4 using letters from a name of God. He goes on to show 7 Latin squares of order 7, using the same 7 letters each time -- though four are corrupted. (Throughout, the Latin squares also have 'Latin' diagonals, i.e. the diagonals contain all the values.) These are arranged so each has a different letter in the first place. It is conjectured that these are associated with the days of the week or the planets.

Tagliente. Libro de Abaco. (1515). 1541. F. 18v. 7 x 7 Latin square with entries 1, 13, 2, 14, 3, 10, 4 cyclically shifted forward -- i.e. the second row starts 13, 2, .... This is an elaborate plate which notes that the sum of each file is 47 and has a motto: Sola Virtu la Fama Volla, but I could find no text or other reason for its appearance!

Inscription on memorial to Hannibal Bassett, d. 1708, in Meneage parish church, St. Mawgan, Cornwall. I first heard of this from Chris Abbess, who reported it in some newsletter in c1993. However, [Peter Haining; The Graveyard Wit; Frank Graham, Newcastle, 1973, p. 133] cites this as being at Cunwallow, near Helstone, Cornwall. [W. H. Howe; Everybody's Book of Epitaphs Being for the Most Part What the Living Think of the Dead; Saxon & Co., London, nd [c1895] (facsimile by Pryor Publications, Whitstable, 1995); p. 173] says it is in Gunwallow Churchyard. Spelling and punctuation vary a bit. The following gives a detailed account.

Alfred Hayman Cummings. The Churches and Antiquities of Cury & Gunwalloe, in the Lizard District, including Local Traditions. E. Marlborough & Co., London & Truro, 1875, pp. 130-131. ??NX. "It has been said that there once existed ... the curious epitaph --" and gives a considerable rearrangement of the inscription below. He continues "But this is in all probability a mistake, as repeated search has been made for it, not only by the writer, but by a former Vicar of Gunwalloe, and it could nowhere be found, while there is a plate with an inscription in the church at Mawgan, the next parish, which might be very easily the one referred to." He gives the following inscription, saying it is to Hannibal Basset, d. 1708-9. Chris Weeks was kind enough to actually go to the church of St. Winwaloe, Gunwalloe, where he found nothing, and to St. Mawgan in Meneage, a few miles away. Chris Weeks sent pictures of Gunwallowe -- the church is close to the cliff edge and it looks like there could once have been more churchyard on the other side of the church where the cliff has fallen away. In the church at St. Mawgan is the brass plate with 'the Acrostic Brass Inscription', but it is not clearly associated with a grave and I wonder if it may have been moved from Gunwallowe when a grave was eroded by the sea. It is on the left of the arch by the pulpit. I reproduce Chris Weeks' copy of the text. He has sent a photograph, but it was dark and the photo is not very clear, but one can make out the Latin square part.

Hanniball Ba((et here Inter'd doth lye

Who dying lives to all Eternitye

hee departed this life the 17th of Ian

1709/8 in the 22th year of his age ~

A lover of learning

Shall wee all dye

Wee shall dye all

all dye shall wee

dye all wee shall

The (( are old style long esses. The superscript th is actually over the numeral. The 9 is over the 8 in the year and there is no stroke. This is because it was before England adopted the Gregorian calendar and so the year began on 25 Mar and was a year behind the continent between 1 Jan and 25 Mar. Correspondence of the time commonly would show 1708/9 at this time, and I have used this form for typographic convenience, but with the 9 over the 8 as on the tomb.

A word game book points out that this inscription is also palindromic!!

Richard Breen. Funny Endings. Penny Publishing, UK, 1999, p. 35. Gives the following form: Shall we all die? / We shall die all. / All die shall we? / Die all we shall and notes that it is a word palindrome and says it comes from Gunwallam [sic], near Helstone.

Joseph Sauveur. Construction générale des quarrés magiques. Mémoires de l'Académie Royale des Sciences 1710(1711) 92-138. ??NYS -- described in Cammann-4, p. 297, (see 7.N for details of Cammann) which says Sauveur invented Latin squares and describes some of his work.

Ozanam. 1725. 1725: vol. IV, prob. 29, p. 434 & fig. 35, plate 10 (12). Two 4 x 4 orthogonal squares, using A, K, Q, J of the 4 suits, but it looks like:

J(, A(, K(, Q(; Q(, K(, A(, J(; A(, J(, Q(, K(; K(, Q(, J(, A(; but the ( and ( look very similar. From later versions of the same diagram, it is clear that the first row should have its ( and ( reversed. Note the diagonals also contain all four ranks and suits. (I have a reference for this to the 1723 edition.)

Minguet. 1733. Pp. 146-148 (1864: 142-143; not noticed in other editions). Two 4 x 4 orthogonal squares, using A, K, Q, J (= As, Rey, Caballo (knight), Sota (knave)) of the 4 suits, but the Spanish suits, in descending order, are: Espadas, Bastos, Oros, Copas. The result is described but not drawn, as:

RO, AE, CC, SB; SC, CB, AO, RE; AB, RC, SE, CO; CE, SO, RB, AC;

which would translate into the more usual cards as:

K(, A(, Q(, J(; J(, Q(, A(, K(; A(, K(, J(, Q(; Q(, J(, K(, A(.

However, I'm not sure of the order of the Caballo and Sota; if they were reversed, which would interchange Q and J in the latter pattern, then both Ozanam and Minguet would have the property that each row is a cyclic shift or reversal of A, K, Q, J.

Alberti. 1747. Art. 29, p. 203 (108) & fig. 36, plate IX, opp. p. 204 (108). Two 4 x 4 orthogonal squares, figure simplified from the correct form of Ozanam, 1725.

L. Euler. Recherches sur une nouvelle espèce de Quarrés Magiques. (Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen (= Flessingue) 9 (1782) 85-239.) = Opera Omnia (1) 7 (1923) 291-392. (= Comm. Arithm. 2 (1849) 302-361.)

Manuel des Sorciers. 1825. Pp. 78-79, art. 39. ??NX Correct form of Ozanam.

The Secret Out. 1859. How to Arrange the Twelve Picture Cards and the four Aces of a Pack in four Rows, so that there will be in Neither Row two Cards of the same Value nor two of the same Suit, whether counted Horizontally or Perpendicularly, pp. 90-92. Two 4 x 4 orthogonal Latin squares, not the same as in Ozanam.

Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. XI, 1884: 200-202. Two 4 x 4 orthogonal squares.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles, No. XVI, pp. 17-18. Similar to Ozanam.

Hoffmann. 1893. Chap. X, no. 14: Another card puzzle, pp. 342 & 378-379 = Hoffmann-Hordern, pp. 234 & 236. Two orthogonal Latin squares, but the diagonals do not contain all the suits and ranks.

A(, J(, Q(, K(; J(, A(, K(, Q(; Q(, K(, A(, J(; K(, Q(, J(, A(.

G. Tarry. Le probleme de 36 officiers. Comptes Rendus de l'Association Française pour l'Avancement de Science Naturel 1 (1900) 122-123 & 2 (1901) 170-203. ??NYS

Dudeney. Problem 521. Weekly Dispatch (1 Nov, 15 Nov, 1903) both p. 10.

H. A. Thurston. Latin squares. Eureka 9 (Apr 1947) 19-21. Survey of current knowledge.

T. G. Room. Note 2569: A new type of magic square. MG 39 (No. 330) (Dec 1955) 307. Introduces 'Room Squares'. Take the 2n(2n-1)/2 combinations from 2n symbols and insert them in a 2n-1 x 2n-1 grid so that each row and column contains all 2n symbols. There are n entries and n-1 blanks in each row and column. There is an easy solution for n = 1. n = 2 and n = 3 are impossible. Gives a solution for n = 4. This is a design for a round-robin tournament with the additional constraint of 2n-1 sites such that each player plays once at each site.

Parker shows there are two orthogonal Latin squares of order 10 in 1959.

R. C. Bose & S. S. Shrikande. On the falsity of Euler's conjecture about the nonexistence of two orthogonal Latin squares of order 4t+2. Proc. Nat. Acad. Sci. (USA) 45: 5 (1959) 734-737.

Gardner. SA (Nov 1959) c= New MD, chap. 14. Describes Bose & Shrikande's work. SA cover shows a 10 x 10 counterexample in colour. Kara Lynn and David Klarner actually made a quilt of this, thereby producing a counterpane counterexample! They told me that the hardest part of the task was finding ten sufficiently contrasting colours of material.

H. Howard Frisinger. Note: The solution of a famous two-centuries-old problem: the Leonhard Euler-Latin square conjecture. HM 8 (1981) 56-60. Good survey of the history.

Jacques Bouteloup. Carrés Magiques, Carrés Latins et Eulériens. Éditions du Choix, Bréançon, 1991. Nice systematic survey of this field, analysing many classic methods. An Eulerian square is essentially two orthogonal Latin squares.

5.I.1. EIGHT QUEENS PROBLEM

See MUS I 210-284. S&B 37 shows examples. See also 5.Z. See also 6.T for examples where no three are in a row.

Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904. Pp. 1082-1084 discusses history and results for the n queens problem, with many references.

Paul J. Campbell. Gauss and the eight queens problem. HM 4:4 (Nov 1977) 397-404. Detailed history. Demonstrates that Gauss did not obtain a complete solution and traces how this misconception originated and spread.

"Schachfreund" (Max Bezzel). Berliner Schachzeitung 3 (Sep 1848) 363. ??NYS

Solutions. Ibid. 4 (Jan 1849) 40. ??NYS (Ahrens says this only gives two solutions. A. C. White says two or three. Jaenisch says a total of 5 solutions were published here and in 1854.)

Franz Nauck. Eine in das Gebiet der Mathematik fallende Aufgabe von Herrn Dr. Nauck in Schleusingen. Illustrirte Zeitung (Leipzig) 14 (No. 361) (1 Jun 1850) 352. Reposes problem. [The papers do not give a first name or initial. The only Nauck in the first six volumes of Poggendorff is Ernst Friedrich (1819-1875), a geologist. Ahrens gives no initial. Campbell gives Franz.]

Franz Nauck. Briefwechseln mit Allen für Alle. Illustrirte Zeitung (Leipzig) 15 (No. 377) (21 Sep 1850) 182. Complete solution.

Editorial comments: Briefwechsel. Illustrirte Zeitung (Leipzig) 15 (No. 378) (28 Sep 1850) 207. Thanks 6 correspondents for the complete solution and says Nauck reports that a blind person has also found all 92 solutions.

Gauss read the Illustrirte Zeitung and worked on the problem, corresponding with his friend Schumacher starting on 1 Sep 1850. Campbell discusses the content of the letters, which were published in: C. A. F. Peters, ed; Briefwechsel zwischen C. F. Gauss und H. C. Schumacher; vol. 6, Altona, 1865, ??NYS. John Brillhart writes that there is some material in Gauss' Werke, vol. XII: Varia kleine Notizen verschiednen Inhalts ... 5, pp. 19-28, ??NYS -- not cited by Campbell.

F. J. E. Lionnet. Question 251. Nouvelles Annales de Mathématiques 11 (1852) 114-115. Reposes problem and gives an abstract version.

Giusto Bellavitis. Terza rivista di alcuni articoli dei Comptes Rendus dell'Accademia delle Scienze di Francia e di alcuni questioni des Nouvelles Annales des mathématiques. Atti dell'I. R. Istituto Veneto di Scienze, Lettere ed Arti (3) 6 [= vol. 19] (1860/61) 376-392 & 411-436 (as part of Adunanza del Giorno 17 Marzo 1861 on pp. 347-436). The material of interest is: Q. 251. Disposizione sullo scacchiere di otto regine, on pp. 434-435. Gives the 12 essentially different solutions. Lucas (1895) says Bellavitis was the first to find all solutions, but see above. However this may be the first appearance of the 12 essentially different solutions.

C. F. de Jaenisch. Op. cit. in 5.F.1. 1862. Vol. 1, pp. 122-135. Gives the 12 basic solutions and shows they produce 92. Notes that in every solution, 4 queens are on white squares and 4 are on black.

A. C. Cretaine. Études sur le Problème de la Marche du Cavalier au Jeu des Échecs et Solution du Problème des Huit Dames. A. Cretaine, Paris, 1865. ??NYS -- cited by Lucas (1895). Shows it is possible to solve the eight queens problem after placing one queen arbitrarily.

G. Bellavitis. Algebra N. 72 Lionnet. Atti dell'Istituto Veneto (3) 15 (1869/70) 844-845.

Siegmund Günther. Zur mathematische Theorie des Schachbretts. Grunert's Archiv der Mathematik und Physik 56 (1874) 281-292. ??NYS. Sketches history of the problem -- see Campbell. He gives a theoretical, but not very practical, approach via determinants which he carries out for 4 x 4 and 5 x 5.

J. W. L. Glaisher. On the problem of the eight queens. Philosophical Magazine (4) 48 (1874) 457-467. Gives a sketch of Günther's history which creates several errors, in particular attributing the solution to Gauss -- see Campbell, who suggests Glaisher could not read German well. (However, in 1921 & 1923, Glaisher published two long articles involving the history of 15-16C German mathematics, showing great familiarity with the language.) Simplifies and extends Günther's approach and does 6 x 6, 7 x 7, 8 x 8 boards.

Lucas. RM2, 1883. Note V: Additions du Tome premier. Pp. 238-240. Gives the solutions on the 9 x 9 board, due to P. H. Schoute, in a series of articles titled Wiskundige Verpoozingen in Eigen Haard. Gives the solutions on the 10 x 10 board, found by M. Delannoy.

S&B, p. 37, show an 1886 puzzle version of the six queens problem.

A. Pein. Aufstellung von n Königinnen auf einem Schachbrett von n2 Feldern. Leipzig. ??NYS -- cited by Ball, MRE, 4th ed., 1905 as giving the 92 inequivalent solutions on the 10 x 10.

Ball. MRE, 1st ed., 1892. The eight queens problem, pp. 85-88. Cites Günther and Glaisher and repeats the historical errors. Sketches Günther's approach, but only cites Glaisher's extension of it. He gives the numbers of solutions and of inequivalent solutions up through 10 x 10 -- see Dudeney below for these numbers, but the two values in ( ) are not given by Dudeney. He states results for the 9 x 9 and 10 x 10, citing Lucas. Says that a 6 x 6 version "is sold in the streets of London for a penny".

Hoffmann. 1893.

Chap. VI, pp. 272-273 & 286 = Hoffmann-Hordern, pp. 187-189, with photo.

No. 24: No two in a row. Eight queens. Photo on p. 188 shows Jeu des Sentinelles, by Watilliaux, dated 1874-1895.

No. 25: The "Simple" Puzzle. Nine queens. Says a version was sold by Messrs. Feltham, with a notched board but the pieces were allowed to move over the gaps, so it was really a 9 x 9 board.

Chap. X, No. 18: The Treasure at Medinet, pp. 343-344 & 381 = Hoffmann-Hordern, pp. 237-239. This is a solution of the eight queens problem, cut into four quadrants and jumbled. The goal is to reconstruct the solution. Photo on p. 239 shows Jeu des Manifestants, with box.

Hordern Collection, p. 94, and S&B, p. 37, show a version of this with same box, but which divides the board into eight 2 x 4 rectangles.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 1: The famous Italian pin puzzle. 6 queens puzzle. No solution.

Lucas. L'Arithmétique Amusante. 1895. Note IV: Section I: Les huit dames, pp. 210-220. Asserts Bellavitis was the first to find all solutions. Discusses symmetries and shows the 12 basic solutions. Correctly describes Jaenisch as obscure. Gives an easy solution of Cretaine's problem which can be remembered as a trick. Shows there are six solutions which can be superimposed with no overlap, i.e. six solutions using disjoint sets of cells.

C. D. Locock, conductor. Chess Column. Knowledge 19 (Jan 1896) 23-24; (Feb 1896) 47-48; (May 1896) 119; (Jul 1896) 167-168. This series begins by saying most players know there is a solution, "but, possibly, some may be surprised to learn that there are ninety-two ways of performing the feat, ...." He then enumerates them. Second article studies various properties of the solutions, particularly looking for examples where one solution shifts to produce another one. Third article notes some readers' comments. Fourth article is a long communication from W. J. Ashdown about the number of distinct solutions, which he gets as 24 rather than the usual 12.

T. B. Sprague. Proc. Edinburgh Math. Soc. 17 (1898-9) 43-68. ??NYS -- cited by Ball, MRE, 4th ed., 1905, as giving the 341 inequivalent solutions on the 11 x 11.

Benson. 1904. Pins and dots puzzle, p. 253. 6 queens problem, one solution.

Ball. MRE, 4th ed., 1905. The eight queens problem, pp. 114-120. Corrects some history by citing MUS, 1st ed., 1901. Gives one instance of Glaisher's method -- going from 4 x 4 to 5 x 5 and its results going up to 8 x 8. Says the 92 inequivalent solutions on the 10 x 10 were given by Pein and the 341 inequivalent solutions on the 11 x 11 were given by Sprague. The 5th ed., pp. 113-119 calls it "One of the classical problems connected with a chess-board" and adds examples of solutions up to 21 x 21 due to Mr. Derington.

Pearson. 1907. Part III, no. 59: Stray dots, pp. 59 & 130. Same as Hoffmann's Treasure at Medinet.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The eight provinces, pp. 14-15 & 65. Same as Hoffmann's Treasure at Medinet.

A. C. White. Sam Loyd and His Chess Problems. 1913. Op. cit. in 1. P. 101 says Loyd discovered that all solutions have a piece at d1 or equivalent.

Williams. Home Entertainments. 1914. A draughtboard puzzle, p. 115. "Arrange eight men on a draughtboard in such a way that no two are upon the same line in any direction." This is not well stated!! Gives one solution: 52468317 and says "Work out other solutions for yourself."

Dudeney. AM. 1917. The guarded chessboard, pp. 95-96. Gives the number of ways of placing n queens and the number of inequivalent ways. The values in ( ) are given by Ball, but not by Dudeney.

n 4 5 6 7 8 9 10 11 12 13

ways 2 10 4 40 92 (352) (724) - - -

inequivalent ways 1 2 1 6 12 46 92 341 (1766) (1346)

Ball. MRE, 9th ed., 1920. The eight queens problem, pp. 113-119. Omits references to Pein and Sprague and adds the number of inequivalent solutions for the 12 x 12 and 13 x 13.

Blyth. Match-Stick Magic. 1921. No pairs allowed, p. 74. 6 queens problem.

Hummerston. Fun, Mirth & Mystery. 1924. No two in a line, p. 48. Chessboard. Place 'so that no two are upon the same line in any direction along straight or diagonal lines?' Gives one solution: 47531682, 'but there are hundreds of other ways'. You can let someone place the first piece.

Rohrbough. Puzzle Craft. 1932. Houdini Puzzle, p. 17. 6 x 6 case.

Rohrbough. Brain Resters and Testers. c1935. Houdini Puzzle, p. 25. 6 x 6 problem. "-- From New York World some years ago, credited to Harry Houdini." I have never seen this attribution elsewhere.

Pál Révész. Mathematik auf dem Schachbrett. In: Endre Hódi, ed. Mathematisches Mosaik. (As: Matematikai Érdekességek; Gondolat, Budapest, 1969.) Translated by Günther Eisenreich. Urania-Verlag, Leipzig, 1977. Pp. 20-27. On p. 24, he says that all solutions have 4 queens on white and 4 on black. He says that one can place at most 5 non-attacking queens on one colour.

Doubleday - 2. 1971. Too easy?, pp. 97-98. The two solutions on the 4 x 4 board are disjoint.

Dean S. Clark & Oved Shisha. Proof without words: Inductive construction of an infinite chessboard with maximal placement of nonattacking queens. MM 61:2 (1988) 98. Consider a 5 x 5 board with queens in cells (1,1), (2,4), (3,2), (4,5), (5,3). 5 such boards can be similarly placed within a 25 x 25 board viewed as a 5 x 5 array of 5 x 5 boards and this has no queens attacking. Repeating the inflationary process gives a solution on the board of edge 53, then the board of edge 54, .... They cite their paper: Invulnerable queens on an infinite chessboard; Annals of the NY Acad. of Sci.: Third Intern. Conf. on Comb. Math.; to appear. ??NYS.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Squares before your eyes, pp. 21 & 106. Asks for solutions of the eight queens problem with no piece on either main diagonal. Two of the 12 basic solutions have this, but one of these is the symmetric case, so there are 12 solutions of this problem.

Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the author. See the discussion in 6.F. He finds the following numbers of solutions for placing n queens, n = 1, 2, ..., 18.

1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 3 65596, 22 79184, 147 72512, 958 15104, 6660 90624.

5.I.2. COLOURING CHESSBOARD WITH NO REPEATS IN A LINE

New section. I know there is a general result that an n x n board can be n-coloured if n satisfies some condition like n ( 1 or 5 (mod 6), but I don't recall any other old examples of the problem.

Dudeney. Problem 50: A problem in mosaics. Tit-Bits 32 (11 Sep 1897) 439 & 33 (2 Oct 1897) 3. An 8 x 8 board with two adjacent corners omitted can be 8-coloured with no two in a row, column or diagonal. = Anon. & Dudeney; A chat with the Puzzle King; The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89.

Dudeney. AM. 1917. Prob. 302: A problem in mosaics, pp. 90 & 215-216. The solution to the previous problem is given and then it is asked to relay the tiles so that the omitted squares are the (3,3) and (3,6) cells.

Hummerston. Fun, Mirth & Mystery. 1924. Q.E.D. -- The office boy problem, Puzzle no. 30, pp. 82 & 176. Wants to mark the cells of a 4 x 4 board with no two the same in any 'straight line ..., either horizontally, vertically, or diagonally.' His answer is: ABCD,  CDEA, EABC, BCDA, which has no two the same on any short diagonal. The problem uses coins of values: A, B, C, D, E = 12, 30, 120, 24, 6 and the object is to maximize the total value of the arrangement. In fact, there are only two ways to 5-colour the board and they are mirror images. Four colours are used three times and one is used four times -- setting the value 120 on the latter cells gives the maximum value of 696.

5.J. SQUARED SQUARES, ETC.

NOTE. Perfect means no two squares are the same size. Compound means there is a squared subrectangle. Simple means not compound.

Dudeney. Puzzling Times at Solvamhall Castle: Lady Isabel's casket. London Mag. 7 (No. 42) (Jan 1902) 584 & 8 (No. 43) (Feb 1902) 56. = CP, prob. 40, pp. 67 & 191-193. Square into 12 unequal squares and a rectangle.

Max Dehn. Über die Zerlegung von Rechtecken in Rechtecke. Math. Annalen 57 (1903) 314-332. Long and technical. No examples. Shows sides must be parallel and commensurable.

Loyd. The patch quilt puzzle. Cyclopedia, 1914, pp. 39 & 344. = MPSL1, prob. 76, pp. 73 & 147-148. c= SLAHP: Building a patchquilt, pp. 30 & 92. 13 x 13 into 11 squares, not simple nor perfect. (Gardner, in 536, says this appeared in Loyd's "Our Puzzle Magazine", issue 1 (1907), ??NYS.)

Loyd. The darktown patch quilt party. Cyclopedia, 1914, pp. 65 & 347. 12 x 12 into 11 squares, not simple nor perfect, in two ways.

P. J. Federico. Squaring rectangles and squares -- A historical review with annotated bibliography. In: Graph Theory and Related Topics; ed. by J. A. Bondy & U. S. R. Murty; Academic Press, NY, 1979, pp. 173-196. Pp. 189-190 give the background to Moroń's work. Moroń later found the first example of Sprague but did not publish it.

Z. Moroń. O rozkładach prostokątów na kwadraty (In Polish) (On the dissection of a rectangle into squares). Przegląd Matematyczno-Fizyczny (Warsaw) 3 (1925) 152-153. Decomposes rectangles into 9 and 10 unequal squares. (Translation provided by A. Mąkowski, 1p. Translation also available from M. Goldberg, ??NYS.)

M. Kraitchik. La Mathématique des Jeux, 1930, op. cit. in 4.A.2, p. 272. Gives Loyd's "Patch quilt puzzle" solution and Lusin's opinion that there is no perfect solution.

A. Schoenflies. Einführung in der analytische Geometrie der Ebene und des Raumes. 2nd ed., revised and extended by M. Dehn, Springer, Berlin, 1931. Appendix VI: Ungelöste Probleme der Analytischen Geometrie, pp. 402-411. Same results as in Dehn's 1903 paper.

Michio Abe. On the problem to cover simply and without gap the inside of a square with a finite number of squares which are all different from one another (in Japanese). Proc. Phys.-Math. Soc. Japan 4 (1931) 359-366. ??NYS

Michio Abe. Same title (in English). Ibid. (3) 14 (1932) 385-387. Gives 191 x 195 rectangle into 11 squares. Shows there are squared rectangles arbitrarily close to squares.

Alfred Stöhr. Über Zerlegung von Rechtecken in inkongruente Quadrate. Schr. Math. Inst. und Inst. angew. Math. Univ. Berlin 4:5 (1939), Teubner, Leipzig, pp. 119-140. ??NYR. (This was his dissertation at the Univ. of Berlin.)

S. Chowla. The division of a rectangle into unequal squares. Math. Student 7 (1939) 69-70. Reconstructs Moroń's 9 square decomposition.

Minutes of the 203rd Meeting of the Trinity Mathematical Society (Cambridge) (13 Mar 1939). Minute Books, vol. III, pp. 244-246. Minutes of A. Stone's lecture: "Squaring the Square". Announces Brooks's example with 39 elements, side 4639, but containing a perfect subrectangle.

Minutes of the 204th Meeting of the Trinity Mathematical Society (Cambridge) (24 Apr 1939). Minute Books, vol. III, p. 248. Announcement by C. A. B. Smith that Tutte had found a perfect squared square with no perfect subrectangle.

R. Sprague. Recreation in Mathematics. Op. cit. in 4.A.1. 1963. The expanded foreword of the English edition adds comments on Dudeney's "Lady Isabel's Casket", which led to the following paper.

R. Sprague. Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate. Math. Zeitschr. 45 (1939) 607-608. First perfect squared square -- 55 elements, side 4205.

R. Sprague. Zur Abschätzung der Mindestzahl inkongruenter Quadrate, die ein gegebenes Rechteck ausfüllen. Math. Zeitschrift 46 (1940) 460-471. Tutte's 1979 commentary says this shows every rectangle with commensurable sides can be dissected into unequal squares.

A. H. Stone, proposer; M. Goldberg & W. T. Tutte, solvers. Problem E401. AMM 47:1 (Jan 1940) 48 & AMM 47:8 (Oct 1940) 570-572. Perfect squared square -- 28 elements, side 1015.

R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte. The dissection of rectangles into squares. Duke Math. J. 7 (1940) 312-340. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 10-38, with commentary by Tutte on pp. 1-9. Tutte's 1979 commentary says Smith was perplexed by the solution of Dudeney's "Lady Isabel's Casket" -- see also his 1958 article.

A. H. Stone, proposer; Michael Goldberg, solver. Problem E476. AMM 48 (1941) 405 ??NYS & 49 (1942) 198-199. An isosceles right triangle can be dissected into 6 similar figures, all of different sizes. Editorial notes say that Douglas and Starke found a different solution and that one can replace 6 by any larger number, but it is not known if 6 is the least such. Stone asks if there is any solution where the smaller triangles have no common sides.

M. Kraitchik. Mathematical Recreations, op. cit. in 4.A.2, 1943. P. 198. Shows the compound perfect squared square with 26 elements and side 608 from Brooks, et al.

C. J. Bouwkamp. On the construction of simple perfect squared squares. Konink. Neder. Akad. van Wetensch. Proc. 50 (1947) 72-78 = Indag. Math. 9 (1947) 57-63. This criticised the method of Brooks, Smith, Stone & Tutte, but was later retracted.

Brooks, Smith, Stone & Tutte. A simple perfect square. Konink. Neder. Akad. van Wetensch. Proc. 50 (1947) 1300-1301. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 99-100, with commentary by Tutte on p. 98. Bouwkamp had published several notes and was unable to make the authors' 1940 method work. Here they clarify the situation and give an example. One writer said they give details of Sprague's first example, but the example is not described as being the same as in Sprague.

W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. Proc. Camb. Phil Soc. 44 (1948) 464-482. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 106-125, with commentary by Tutte on pp. 101-105.

T. H. Willcocks, proposer and solver. Problem 7795. Fairy Chess Review 7:1 (Aug 1948) 97 & 106 (misnumberings for 5 & 14). Refers to prob. 7523 -- ??NYS. Finds compound perfect squares of orders 27, 27, 28 and 24.

T. H. Willcocks. A note on some perfect squares. Canadian J. Math. 3 (1951) 304-308. Describes the result in Fairy Chess Review prob. 7795.

T. H. Willcocks. Fairy Chess Review (Feb & Jun 1951). Prob. 8972. ??NYS -- cited and described by G. P. Jelliss; Prob. 44 -- A double squaring, G&PJ 2 (No. 17) (Oct 1999) 318-319. Squares of edges 3, 5, 9, 11, 14, 19, 20, 24, 31, 33, 36, 39, 42 can be formed into a 75 x 112 rectangle in two different ways. {These are reproduced, without attribution, as Fig. 21, p. 33 of Joseph S. Madachy; Madachy's Mathematical Recreations; Dover, 1979 (this is a corrected reprint of Mathematics on Vacation, 1966, ??NYS). The 1979 ed. has an errata slip inserted for p. 33 as the description of Fig. 21 was omitted in the text, but the erratum doesn't cite a source for the result.} The G&PJ problem then poses a new problem from Willcocks involving 21 squares to be made into a rectangle in two different ways -- it is not clear if these have to be the same shape.

M. Goldberg. The squaring of developable surfaces. SM 18 (1952) 17-24. Squares cylinder, Möbius strip, cone.

W. T. Tutte. Squaring the square. Guest column for SA (Nov 1958). c= Gardner's 2nd Book, pp. 186-209. The latter = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 244-266, with a note by Tutte on p. 244, but the references have been omitted. Historical account -- cites Dudeney as the original inspiration of Smith.

R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. G. Assemble squares of sides 2, 5, 7, 9, 16, 25, 28, 33, 36 into a rectangle. The rectangle is 69 x 61 and is not either of Moroń's examples.

W. T. Tutte. The quest of the perfect square. AMM 72:2, part II (Feb 1965) 29-35. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 432-438, with brief commentary by Tutte on p. 431. General survey, updating his 1958 survey.

Blanche Descartes [pseud. of Cedric A. B. Smith]. Division of a square into rectangles. Eureka 34 (1971) 31-35. Surveys some history and Stone's dissection of an isosceles right triangle into 6 others of different sizes (see above). Tutte has a dissection of an equilateral triangle into 15 equilateral triangles -- but some of the pieces must have the same area so we consider up and down pointing triangles as + and - areas and then all the areas are different. Author then considers dissecting a square into incongruent but equiareal rectangles. He finds it can be done in n pieces for any n ( 7.

A. J. W. Duijvestijn. Simple perfect squared square of lowest order. J. Combinatorial Thy. B 25 (1978) 240-243. Finds a perfect square of minimal order 21.

A. J. W. Duijvestin, P. J. Federico & P. Leeuw. Compound perfect squares. AMM 89 (1982) 15-32. Shows Willcocks' example has the smallest order for a compound perfect square and is the only example of its order, 24.

5.J.1. MRS PERKINS'S QUILT

This is the problem of cutting a square into smaller squares.

Loyd. Cyclopedia, 1914, pp. 248 & 372, 307 & 380. Cut 3 x 3 into 6 squares: 2 x 2 and 5  1 x 1.

Dudeney. AM. 1917. Prob. 173: Mrs Perkins's quilt, pp. 47 & 180. Same as Loyd's "Patch quilt puzzle" in 5.J.

Dudeney. PCP. 1932. Prob. 117: Square of Squares, pp. 53 & 148-149. = 536, prob. 343, pp. 120 & 324-325. c= "Mrs Perkins's quilt".

N. J. Fine & I. Niven, proposers; F. Herzog, solver. Problem E724 -- Admissible Numbers. AMM 53 (1946) 271 & 54 (1947) 41-42. Cubical version.

J. H. Conway. Mrs Perkins's quilt. Proc. Camb. Phil. Soc. 60 (1964) 363-368.

G. B. Trustrum. Mrs Perkins's quilt. Ibid. 61 (1965) 7-11.

Ripley's Puzzles and Games. 1966. Pp. 16-17, item 7. "Can you divide a square into 6 perfect squares?" Answer as in Loyd.

Nick Lord. Note 72.11: Subdividing hypercubes. MG 72 (No. 459) (Mar 1988) 47-48. Gives an upper bound for impossible numbers in d dimensions.

David Tall. To prove or not to prove. Mathematics Review 1:3 (Jan 1991) 29-32. Tall regularly uses the question as an exercise in problem solving. About ten years earlier, a 14 year old girl pointed out that the problem doesn't clearly rule out rejoining pieces. E.g. by cutting along the diagonals and rejoining, one can make two squares.

5.J.2. CUBING THE CUBE

S. Chowla. Problem 1779. Math. Student 7 (1939) 80. (Solution given in Brooks, et al., Duke Math. J., op. cit. in 5.J, section 10.4, but they give no reference to a solution in Math. Student.)

5.J.3. TILING A SQUARE OF SIDE 70 WITH SQUARES OF SIDES

1, 2, ..., 24

J. R. Bitner. Use of Macros in Backtrack Programming. M.Sc. Thesis, ref. UIUCDCS-R-74-687, Univ. of Illinois, Urbana-Champaign, 1974, ??NYS. Shows such a tiling is impossible.

5.K. DERANGEMENTS

Let D(n) = the number of derangements of n things, i.e. permutations leaving no point fixed.

Eberhard Knobloch. Euler and the history of a problem in probability theory. Gaņita-Bhāratī [NOTE: ņ denotes an n with an underdot] (Bull. Ind. Soc. Hist. Math.) 6 (1984) 1-12. Discusses the history, noting that many 19C authors were unaware of Euler's work. There is some ambiguity in his descriptions due to early confusion of n as the number of cards and n as the number of the card on which a match first occurs. Describes numerous others who worked on the problem up to about 1900: De Moivre, Waring, Lambert, Laplace, Cantor, etc.

Pierre Rémond de Montmort. Essai d'analyse sur les jeux de hazards. (1708); Seconde edition revue & augmentee de plusieurs lettres, (Quillau, Paris, 1713 (reprinted by Chelsea, NY, 1980)); 2nd issue, Jombert & Quillau, 1714. Problèmes divers sur le jeu du trieze, pp. 54-64. In the original game, one has a deck of 52 cards and counts 1, 2, ..., 13 as one turns over the cards. If a card of rank i occurs at the i-th count, then the player wins. In general, one simplifies by assuming there are n distinct cards numbered 1, ..., n and one counts 1, ..., n. One can ask for the probability of winning at some time and of winning at the k-th draw. In 1708, Montmort already gives tables of the number of permutations of n cards such that one wins on the k-th draw, for n = 1, ..., 6. He gives various recurrences and the series expression for the probability and (more or less) finds its limit. In the 2nd ed., he gives a proof of the series expression, due to Nicholas Bernoulli, and John Bernoulli says he has found it also. Nicholas' solution covers the general case with repeated cards. [See: F. N. David; Games, Gods and Gambling; Griffin, London, 1962, pp. 144-146 & 157.] (Comtet and David say it is in the 1708 ed. I have seen it on pp. 54-64 of an edition which is uncertain, but probably 1708, ??NX. Knobloch cites 1713, pp. 130-143, but adds that Montmort gave the results without proofs in the 1708 ed. and includes several letters from and to John I and Nicholas I Bernoulli in the 1713 ed., pp. 290-324, and mentions the problem in his Preface -- ??NYS.)

Abraham de Moivre. The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play. W. Pearson for the Author, London, 1718. Prob. XXV, pp. 59-63. (= 2nd ed, H. Woodfall for the Author, London, 1738. Prob. XXXIV, pp. 95-98.) States and demonstrates the formula for finding the probability of p items to be correct and q items to be incorrect out of n items. One of his examples is the probability of six items being deranged being 53/144.

L. Euler. Calcul de la probabilité dans le jeu de rencontre. Mémoires de l'Académie des Sciences de Berlin (7) (1751(1753)) 255-270. = Opera Omnia (1) 7 (1923) 11-25. Obtains the series for the probability and notes it approaches 1/e.

L. Euler. Fragmenta ex Adversariis Mathematicis Deprompta. MS of 1750-1755. Pp. 287-288: Problema de permutationibus. First published in Opera Omnia (1) 7 (1923) 542-545. Obtains alternating series for D(n).

Ozanam-Montucla. 1778. Prob. 5, 1778: 125-126; 1803: 123-124; 1814: 108-109; 1840: omitted. Describes Jeu du Treize, where a person takes a whole deck and turns up the cards, counting 1, 2, ..., 13 as he goes. He wins if a card of rank i appears at the i-th count. Montucla's description is brief and indicates there are several variations of the game. Hutton gives a lengthier description of one version. Cites Montmort for the probability of winning as .632..

L. Euler. Solutio quaestionis curiosae ex doctrina combinationum. (Mem. Acad. Sci. St. Pétersbourg 3 (1809/10(1811)) 57-64.) = Opera Omnia (1) 7 (1923) 435-440. (This was presented to the Acad. on 18 Oct 1779.) Shows D(n) = (n-1) [D(n-1) + D(n-2)] and D(n) = nD(n-1) + (-1)n.

Ball. MRE. 1st ed., 1892. Pp. 106-107: The mousetrap and Treize. In the first, one puts out n cards in a circle and counts out. If the count k occurs on the k-th card, the card is removed and one starts again. Says Cayley and Steen have studied this. It looks a bit like a derangement question.

Bill Severn. Packs of Fun. 101 Unusual Things to Do with Playing Cards and To Know about Them. David McKay, NY, 1967. P. 24: Games for One: Up and down. Using a deck of 52 cards, count through 1, 2, ..., 13 four times. You lose if a card of rank i appears when you count i, i.e. you win if the cards are a generalized derangement. Though a natural extension of the problem, I can't recall seeing it treated, perhaps because it seems to get very messy. However, a quick investigation reveals that the probability of such a generalized derangement should approach e-4.

Brian R. Stonebridge. Derangements of a multiset. Bull. Inst. Math. Appl. 28:3 (Mar 1992) 47-49. Gets a reasonable extension to multisets, i.e. sets with repeated elements.

5.K.1. DERANGED BOXES OF A, B AND A & B

Three boxes contain A or B or A & B, but they have been shifted about so each is in one of the other boxes. You can look at one item from one box to determine what is in all of them. This is just added and is certainly older than the examples below.

Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 84: Marble garble, pp. 40 & 110. Black and white marbles.

Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 26: Mexican jumping beans, pp. 40-41 & 96. Red and black beans in matchboxes. The problem continues with a Bertrand box paradox -- see 8.H.1.

Doubleday - 3. 1972. Open the box, pp. 147-148. Black and white marbles.

5.K.2. OTHER LOGIC PUZZLES BASED ON DERANGEMENTS

These typically involve a butcher, a baker and a brewer whose surnames are Butcher, Baker and Brewer, but no one has the profession of his name. I generally only state the beginning of the problem.

New section -- there must be older examples. Gardner, in an article: My ten favorite brainteasers in Games (collected in Games Big Book of Games, 1984, pp. 130-131) says this is one of his favorite problems. ??locate

I now see these lead to Latin rectangles, cf Section 5.I.

R. Turner, proposer: The sons of the dons; Eureka 2 (May 1939) 9-10. K. Tweedie, solver: On the problem of the sons of the dons. Eureka 4 (May 1940) 21-23. Six dons, in analysis, geometry, algebra, dynamics, physics and astronomy, each have a son who studies one of these subjects, but none studies the subject of his father. Several further restrictions, e.g., there are no two students who each study the subject of the other's father.

M. Adams. Puzzle Book. 1939. Prob. B.91: Easter bonnet, pp. 80 & 107. Women named Green, Black, Brown and White with 4 colours of hats and 4 colours of dresses, but name, hat and dress are always distinct.

J. B. Parker. Round the table. Eureka 5 (Jan 1941) 20-21 & 6 (May 1941) 11. Seven men, whose names are colours, with ties, socks and cars, being coloured with three of the names of other men and all colours used for each item, sitting at a table with eight places.

Anonymous. The umbrella problem. Eureka 9 (Apr 1947) 22 & 10 (Mar 1948) 25. Six men 'of negligible honesty' each go away with another's umbrella.

Jonathan Always. Puzzles to Puzzle You. Tandem, London, 1965. No. 30: Something about ties, pp. 16 & 74-75. Black, Green and Brown are wearing ties, but none has the colour of his name, remarked the green tie wearer to Mr. Black.

David Singmaster. The deranged secretary. If a secretary puts n letters all in wrong envelopes, how many envelopes must one open before one knows what is each of the unopened envelopes?

Problem proposal and solution 71.B. MG 71 (No. 455) (Mar 1987) 65 & 71 (No. 457) (Oct 1987) 238-239.

Open question. The Weekend Telegraph (11 Jun 1988) XV & (18 Jun 1988) XV.

5.K.3. CAYLEY'S MOUSETRAP

This is a solitaire (= patience) game developed by Cayley, based on Treize. Take a deck of cards, numbered 1, 2, ..., n, and shuffle them. Count through them. If a card does not match its count, put it on bottom and continue. If it matches, set it aside and start counting again from  1. One wins if all cards are set aside. In this case, pick up the deck and start a new game.

T. W. O. Richards, proposer; Richard I. Hess, solver. Prob. 1828. CM 19 (1993) 78 & 20 (1994) 77-78. Asks whether there is any arrangement which allows three or more consecutive wins. No theoretical solution. Searching finds one solution for n = 6 and n = 8 and 8 solutions for n = 9.

5.L. MÉNAGE PROBLEM

How many ways can n couples be seated, alternating sexes, with no couples adjacent?

A. Cayley. On a problem of arrangements. Proc. Roy. Soc. Edin. 9 (1878) 338-342. Problem raised by Tait. Uses inclusion/exclusion to get a closed sum.

T. Muir. On Professor Tait's problem of arrangements. Ibid., 382-387. Uses determinants to get a simple n-term recurrence.

A. Cayley. Note on Mr. Muir's solution of a problem of arrangement. Ibid., 388-391. Uses generating function to simplify to a usable form.

T. Muir. Additional note on a problem of arrangement. Ibid., 11 (1882) 187-190. Obtains Laisant's 2nd order and 4th order recurrences.

É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 123, example II, p. 215 & Note III, pp. 491-495. Lucas appears not to have known of the work of Cayley and Muir. He describes Laisant's results. The 2nd order, non-homogeneous recurrence, on pp. 494-495, is attributed to Moreau.

C. Laisant. Sur deux problèmes de permutations. Bull. Soc. Math. de France 19 (1890-91) 105-108. General approach to problems of restricted occupancy. His work yields a 2nd order non-homogeneous recurrence and homogeneous 3rd and 4th order recurrences. He cites Lucas, but says Moreau's work is unpublished.

H. M. Taylor. A problem on arrangements. Messenger of Math. 32 (1903) 60-63. Gets almost to Muir & Laisant's 4th order recurrence.

J. Touchard. Sur un problème de permutations. C. R. Acad. Sci. Paris 198 (1934) 631-633. Solution in terms of a complicated integral. States the explicit summation.

I. Kaplansky. Solution of the "problème des ménages". Bull. Amer. Math. Soc. 49 (1943) 784-785. Obtains the now usual explicit summation.

I. Kaplansky & J. Riordan. The problème de ménages. SM 12 (1946) 113-124. Gives the history and a uniform approach.

J. Touchard. Permutations discordant with two given permutations. SM 19 (1953) 109-119. Says he prepared a 65pp MS developing the results announced in 1934 and rediscovered in Kaplansky and in Kaplansky & Riordan. Proves Kaplansky's lemma on selections by finding the generating functions which involve Chebyshev polynomials. Obtains the explicit summation, as done by Kaplansky. Extends to more general problems.

M. Wyman & L. Moser. On the 'problème des ménages'. Canadian J. Math. 10 (1958) 468-480. Analytic study. Updates the history -- 26 references. Gives table of values for n = 0 (1) 65.

Jacques Dutka. On the 'Problème des ménages'. Math. Intell. 8:3 (1986) 18-25 & 33. Thorough survey & history -- 25 references.

Kenneth P. Bogart & Peter G. Doyle. Non-sexist solution of the ménage problem. AMM 93 (1986) 514-518. 14 references.

5.M. SIX PEOPLE AT A PARTY -- RAMSEY THEORY

In a group of six people, there is a triple who all know each other or there is a triple who are all strangers. I.e., the Ramsey number R(3,3) = 6. I will not go into the more complex aspects of this -- see Graham & Spencer for a survey.

P. Erdös & G. Szekeres. A combinatorial problem in geometry. Compositio Math. 2 (1935) 463-470. [= Paul Erdös; The Art of Counting; Ed. by Joel Spencer, MIT Press, 1973, pp. 5-12.] They prove that if n ( BC(a+b-2, a-1), then any two-colouring of Ka contains a monochromatic Ka or Kb.

William Lowell Putnam Examination, 1953, part I, problem 2. In: L. E. Bush; The William Lowell Putnam Mathematical Competition; AMM 60 (1953) 539-542. Reprinted in: A. M. Gleason, R. E. Greenwood & L. M. Kelly; The William Lowell Putnam Mathematical Competition Problems and Solutions -- 1938-1964; MAA, 1980; pp. 38 & 365-366. The classic six people at a party problem.

R. E. Greenwood & A. M. Gleason. Combinatorial relations and chromatic graphs. Canadian J. Math. 7 (1955) 1-7. Considers n = n(a,b,...) such that a two colouring of Kn contains a Ka of the first colour or a Kb of the second colour or .... Thus n(3,3) = 6. They find the bound and many other results of Erdös & Szekeres.

C. W. Bostwick, proposer; John Rainwater & J. D. Baum, solvers. Problem E1321 -- A gathering of six people. AMM 65 (1958) 446 & 66 (1959) 141-142.

Gamow & Stern. 1958. Diagonal strings. Pp. 93-95.

G. J. Simmons. The Game of Sim. JRM 2 (1969) 66.

M. Gardner. SA (Jan 1973) c= Knotted, chap. 9. Exposits Sim. Reports Simmons' result that it is second person (determined after his 1969 article above). The Addendum in Knotted reports that several people have shown that Sim on five points is a draw. Numerous references.

Ronald L. Graham & Joel H. Spencer. Ramsey theory. SA 263:1 (Jul 1990) 80-85. Popular survey of Ramsey theory beginning from Ramsey and Erdös & Szekeres.

5.N. JEEP OR EXPLORER'S PROBLEM

See Ball for some general discussion and notation.

Alcuin. 9C. Prob. 52: Propositio de homine patrefamilias. Wants to get 90 measures over a distance of 30 leagues. He is trying to get the most to the other side, so this is different than the 20C versions. Solution is confusing, but Folkerts rectifies a misprint and this makes it less confusing. Alcuin's camels only eat when loaded!! (Or else they perish when their carrying is done??) The camel take a load to a point 20 leagues away and leaves 10 there, then returns. This results in getting 20 to the destination.

The optimum solution is for the camel to make two return trips and a single trip to 10 leucas, so he will have consumed 30 measures and he has 60 measures to carry on. He now makes one return and a single trip of another 15 leucas, so he will have consumed another 30 measures, leaving 30 to carry on the last 5 leucas, so he reaches home with 25 measures.

Pacioli. De Viribus. c1500. Probs. 49-52. Agostini only describes Prob. 49 in some detail.

Ff. 94r - 95v. XLIX. (Capitolo) de doi aportare pome ch' piu navanza (Of two ways to transport as many apples as possible). = Peirani 134-135. One has 90 apples to transport 30 miles from Borgo [San Sepolcro] to Perosia [Perugia], but one eats one apple per mile and one can carry at most 30 apples. He carries 30 apples 20 miles and leaves 10 there and returns, without eating on the return trip! (So this = Alcuin.) Pacioli continues and gives the optimum solution!

F. 95v. L. C(apitolo). de .3. navi per .30. gabelle 90. mesure (Of three ships holding 90 measures, passing 30 customs points). Each ship has to pay one measure at each customs point. Mathematically the same as the previous.

F. 96r. LI. C(apitolo). de portar .100. perle .10. miglia lontano 10. per volta et ogni miglio lascia 1a (To carry 100 pearls 10 miles, 10 at a time, leaving one every mile). = Peirani 136-137. Takes them 2 miles in ten trips, giving 80 there. Then takes them to the destination in 8 trips, getting 16 to the destination.

Ff. 96v - 97r. LII. C(apitolo). el medesimo con piu avanzo per altro modo (The same with more carried by another method). Continues the previous problem and takes them 5 miles in ten trips, giving 50 there. Then takes them to the destination in 5 trips, getting 25 to the destination.

[This is optimal for a single stop -- if one makes the stop at distance a, then one gets a(10-a) to the destination. One can make more stops, but this is restricted by the fact that pearls cannot be divided. Assuming that the amount of pearls accumulated at each depot is a multiple of ten, one can get 28 to the destination by using depots at 2 and 7 or 5 and 7. One can get 27 to the destination with depots at 4 and 9 or 5 and 9. These are all the ways one can put in two depots with integral multiples of 10 at each depot and none of these can be extended to three such depots. If the material being transported was a continuous material like grain, then I think the optimal method is to first move 1 mile to get 90 there, then move another 10/9 to get 80 there, then another 10/8 to get 70 there, ..., continuing until we get 40 at 8.4563..., and then make four trips to the destination. This gets 33.8254 to the destination. Is this the best method??]

Cardan. Practica Arithmetice. 1539. Chap. 66, section 57, ff. EE.vi.v - EE.vii.v (pp. 152-153). Complicated problem involving carrying food and material up the Tower of Babel! Tower is assumed 36 miles high and seems to require 15625 porters.

Mittenzwey. 1880. Prob. 135, pp. 28-29; 1895?: 153, p. 32; 1917: 153, pp. 29. If eight porters can carry eight full loads from A to B in an hour, how long will it take four porters? The obvious answer is two hours, but he observes that the porters have to return from B to A and it will take three hours. [Probably a little less as they should return in less time than they go.]

Pearson. 1907. Part II, pp. 139 & 216. Two explorers who can carry food for 12 days. (No depots, i.e. form A of Ball, below.)

Loyd. A dash for the South Pole. Ladies' Home Journal (15 Dec 1910). ??NYS -- source?? -- WS??

Ball. MRE, 5th ed., 1911. Exploration problems, pp. 23-24. He distinguishes two forms of the problem, with n explorers who can carry food for d days.

A. Without depots, they can get one man nd/(n+1) days into the desert and back.

B. With depots permitted, they can get a man d/2 (1/1 + 1/2 + ... + 1/n) into the desert and back. This is the more common form.

Dudeney. Problem 744: Exploring the desert. Strand Mag. (1925). ??NX. (??= MP 49)

Dudeney. MP. 1926. Prob. 49: Exploring the desert, pp. 21 & 111 (= 536, prob. 76, pp. 22 & 240). A version of Ball's form A, with n = 9, d = 10, but replacing days by stages of length 40 miles.

Abraham. 1933. Prob. 34 -- The explorers, pp. 13 & 25 (9-10 & 112). 4 explorers, each carrying food for 5 days. Mentions general case. This is Ball's form A.

Haldeman-Julius. 1937. No. 10: The four explorers, pp. 4 & 21. Ball's form A, with n = 4, d = 5.

Olaf Helmer. Problem in logistics: The Jeep problem. Project Rand Report RA-15015 (1 Dec 1946) 7pp.

N. J. Fine. The jeep problem. AMM 54 (1947) 24-31.

C. G. Phipps. The jeep problem: a more general solution. AMM 54 (1947) 458-462.

G. G. Alway. Note 2707: Crossing the desert. MG 41 (No. 337) (1957) 209. If a jeep can carry enough fuel to get halfway across, how much fuel is needed to get across? For a desert of width 2, this leads to the series 1 + 1/3 + 1/5 + 1/7 + .... See Lehmann and Pyle below.

G. C. S[hephard, ed.] The problems drive. Eureka 11 (Jan 1949) 10-11 & 30. No. 2. Four explorers, starting from a supply base. Each can carry food for 100 miles and goes 25 miles per day. Two men do the returning to base and bringing out more supplies. the third man does ferrying to the fourth man. How far can the fourth man get into the desert and return? Answer is 100 miles. Ball's form B would give 104 1/6.

Gamow & Stern. 1958. Refueling. Pp. 114-115.

Pyle, I. C. The explorer's problem. Eureka 21 (Oct 1958) 5-7. Considers a lorry whose load of fuel takes it a distance which we assume as the unit. What is the widest desert one can cross? And how do you do it? This is similar to Alway, above. He starts at the far side and sees you have to have a load at distance 1 from the far side, then two loads at distance 1 + 1/3 from the far side, then three loads at 1 + 1/3 + 1/5, .... This diverges, so any width can be crossed. Does examples with given widths of 2, 3 and 4 units. Editor notes that he is not convinced the method is optimal.

Martin Gardner, SA (May & June 1959) c= 2nd Book, chap. 14, prob. 1. (The book gives extensive references which were not in SA.)

R. L. Goodstein. Letter: Explorer's problem. Eureka 22 (Oct 1959) 23. Says Alway shows that Pyle's method is optimal. Editor notes Gardner's article and that Eureka was cited in the solutions in Jun.

David Gale. The jeep once more or jeeper by the dozen & Correction to "The Jeep once more or jeeper by the dozen". AMM 77:5 (May 1970) 493-501 & 78:6 (Jun-Jul 1971) 644-645. Gives an elaborate approach via a formula of Banach for path lengths in one dimension. This formally proves that the various methods used are actually optimal and that a continuous string of depots cannot help, etc. Notes that the cost for a round trip is only slightly more than for a one-way trip -- but the Correction points out that this is wrong and indeed the round trip is nearly four times as expensive as a one-way trip. Considers sending several jeeps. Says he hasn't been able to do the round trip problem when there is fuel on both sides of the desert. Comments on use of dynamic programming, noting that R. E. Bellman [Dynamic Programming; Princeton Univ. Press, 1955, p. 103, ex. 54-55] gives the problem as exercises without solution and that he cannot see how to do it!

Birtwistle. Math. Puzzles & Perplexities. 1971.

The expedition, pp. 124-125, 183 & 194. Ball's form A, first with n = 5, d = 6, then in general.

Second expedition, pp. 125-126. Ball's form B, done in general.

Third expedition, pp. 126, 183-184 & 194. Three men want to cross a 180 mile wide desert. They can travel 20 miles per day and can carry food for six days, which can be stored at depots. Minimize the total distance travelled. Solution seems erroneous to me.

A. K. Austin. Jeep trips and card stacks. MTg 58 (1972) 24-25. There are n flags located at distances a1, a1 + a2, a1 + a2 + a3, .... Jeep has to begin at the origin, go to the first flag, return to the origin, go to the second flag, return, .... He can unload and load fuel at the flags. Can he do this with F fuel? Author shows this is equivalent to successfully stacking cards over a cliff with successive overhangs being a1, a2, a3, ....

Doubleday - 3. 1972. Traveller's Tale, pp. 63-64. d = 8 and we want one man to get across the desert of width 12. How many porters, who return to base, are needed? The solution implies that no depots are used. Reasoning as in Ball's case A, we see that n men can support one man crossing a desert of width 2nd/(n+1). If depots are permitted, this is essentially the jeep problem and n men can support a man getting across a desert of width d [1 + 1/3 + 1/5 + ... + 1/(2n-1)]

Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 13, pp. 27 & 129. d = 4 and we want to get a man across a desert of width 6. Similar to Doubleday - 3.

Pierre Berloquin. [Le Jardin du Sphinx. Dunod, Paris, 1981.] Translated by Charles Scribner Jr as: The Garden of the Sphinx. Scribner's, NY, 1985.

Prob. 1: Water in the desert, pp. 3 & 85.

Prob. 40: Less water in the desert, pp. 26 & 111.

Prob. 80: Beyond thirst, pp. 48 & 140.

Prob. 141: The barrier of thirst, pp. 79 & 181.

Prob. 150: No holds barred, pp. 82 & 150.

In all of these, d = 5 and we want to get a man across a desert of width 4, and sometimes back, which is slightly different than the problem of getting to the maximum distance and back.

Prob. 1 is Ball's form A, with n = 4 men, using 20 days' water.

Prob. 40 is Ball's form B, but using only whole day trips, using 14 days' water.

Prob. 80 is Ball's form B, optimized for width 4, using 11½ days' water.

Prob. 141 uses depots and bearers who don't return, as in Alcuin?? You can get one man, who is the only one to return, a distance d (1/2 + 1/3 + ... + 1/(n+1)) into the desert this way. He gives the optimum form for width 4, using 9½ days' water.

Prob. 150 is like prob. 141, except that no one returns! You can get him d (1 + 1/2 + ... + 1/n) into the desert this way. The optimum here uses 4 days' water.

D. R. Westbrook. Note 74.7: The desert fox, a variation of the jeep problem. MG 74 (No. 467) (1990) 49-50. A more complex version, posed by A. K. Dewdney in SA (Jan 1987), is solved here.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Round-trip, pp. 96-97 & 140. Plane wants to circle the earth, but can only carry fuel to go half-way. Other planes can accompany and transfer fuel, but must return to base.

Dylan Gow. Flyaway. MS 25:3 (1992/3) 84-86. Considers the standard problem without return as in Alway, Pyle and Lehmann -- but finds a non-optimal solution.

Wolfram Hinderer. Optimal crossing of a desert. MS 26:4 (1993/4) 100-102. Finds optimal solutions for Gow's problem and for the case with return -- i.e. Ball's B. Also considers use of extra jeeps that do not return, i.e. Berloquin's 141 & 150. Notes that extra jeeps that must return to base do not change the distance that one jeep can reach. [But it changes the time required.]

Harold Boas. Letter: Crossing deserts. MS 26:4 (1993/4) 122. Notes the problem has a long history and cites Fine, Phipps, Gale (and correction), Alway.

David Singmaster. Letter: Crossing deserts. MS 27:3 (1994/5) 63. Points out that the history is far older and sketches the history given above.

Günter Rote & Guochuan Zhang. Optimal Logistics for Expeditions: The Jeep Problem with Complete Refilling. Karl-Franzens-Universität Graz & Technische Universität Graz. Bericht 71 (24 Jun 1996). This deals with a variant. "We have n cans of fuel on the edge of a desert and a jeep with an empty tank whose capacity is just one can. The jeep can carry one can in addition to the fuel in its tank. Moreover, when a can is opened, the fuel must immediately be filled into the jeep's tank. The goal is to find the farthest point in the desert which the jeep can reach by consuming the n cans of fuel. Derick Wood [1984] treated this problem similarly to the classical problem and gave the first solution. Ute and Wilfried Brauer [1989] presented a new strategy and got a better solution than Wood's. They also conjectured that their solution was optimal for infinitely many values of n. We give an algorithm which produces a better solution than Brauers' for all n > 6, and we use a linear programming formulation to derive an upper bound which shows that our solution is optimal." 14 references, several not given above.

5.O. TAIT'S COUNTER PUZZLE: BBBBWWWW TO WBWBWBWB

See S&B 125.

The rules are that one can move two counters as an ordered pair, e.g. from BBBBWWWW to BBB..WWWBW, but not to BBB..WWWWB -- except in Lucas (1895) and AM prob. 237, where such reversal must be done. Also, moving to BBB..WWW.BW is sometimes explicitly prohibited, but it is not always clear just where one can move to. It is also not always specified where the blank spaces are at the beginning and end positions.

Gardner, 1961, requires that the two counters must be BW or WB.

Barbeau, 1995, notes that moving to BWBWBWBW is a different problem, requiring an extra move. I had not noticed this difference before -- indeed I previously had it the wrong way round in the heading of this section. I must check to see if this occurs earlier. See Achugbue & Chin, 1979-80, for this version.

Genjun Nakane (= Hōjiku Nakane). Kanja-otogi-soshi (Book of amusing problems for the entertainment of thinkers). 1743. ??NYS. (See: T. Hayashi; Tait's problem with counters in the Japanese mathematics; Bibl. Mathem. (3) 6 (1905) 323, for this and other Japanese references of 1844 and 1879, ??NYS.)

P. G. Tait. Listing's Topologie. Philosophical Mag. (Ser. 5) 17 (No. 103) (Jan 1884) 30-46 & plate opp. p. 80. Section 12, pp. 39-40. He says he recently saw it being played on a train.

George Hope Verney (= Lloyd-Verney). Chess eccentricities. Longmans, 1885. P. 193: The pawn puzzle. ??NX With 4 & 4.

Lucas. Amusements par les jetons. La Nature 15 (1887, 2nd sem.) 10-11. ??NYS -- cited by Ahrens, title obtained from Harkin. Probably c= the material in RM3, below.

Ball. MRE, 1st ed., 1892, pp. 48-49.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. XIV: The eight-card puzzle, pp. 14-15. Uses cards: BRBRBRBR and asks to bring the colours together, explicitly requiring the moved cards to be placed in contact with the unmoved cards.

Hoffmann. 1893. Chap. VI, pp. 270-271 & 284-285 = Hoffmann-Hordern, pp. 184-186, with photo.

No. 19: The "Four and Four" puzzle. Photo on p. 184 shows a version named Monkey Puzzle advertising Brooke's Soap to go from BBBBBWWWW.. to ..WBWBWBWB .

No. 20: The "Five and Five" puzzle.

No. 21: The "Six and Six" puzzle.

Lucas. RM3. 1893. Amusements par les jetons, pp. 145-151. He gives Delannoy's general solution for n of each colour in n moves. Remarks that one can reverse the moved pair.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 11: The Egyptian disc puzzle. 4 & 4. "Two discs adjoining each other to be moved at a time; no gaps to be left in the line." -- this seems to prevent one from making any moves at all!! No solution.

Lucas. L'Arithmétique Amusante. 1895. Pp. 84-108.

Prob. XXI - XXIV and Méthode générale, pp. 84-97. Gives solution for 4, 5, 6, 7 and the general solution for n & n in n moves due to Delannoy.

Rouges et noires, avec interversion, prob. XXV - XXVIII and Méthode générale, pp. 97-108. Interversion means that the two pieces being moved are reversed or turned over, e.g. from BBBBWWWW to BBB..WWWWB, but not to BBB..WWWBW. Gives solutions for 4, 5, 6, 7, 8 pairs and in general in n moves, but he ends with a gap, e.g. ....BB..BB and it takes an extra move to close up the gap.

Ball. MRE, 3rd ed., 1896, pp. 65-66. Cites Delannoy's solution as being in La Nature (Jun 1887) 10. ??NYS.

Ahrens. MUS I. 1910. Pp. 14-15 & 19-25. Cites Tait and gives Delannoy's general solution, from Lucas.

Ball. MRE, 5th ed., 1911, pp. 75-77. Adds a citation to Hayashi, but incorrectly gives the date as 1896.

Loyd. Cyclopedia. 1914. After dinner tricks, pp. 41 & 344. 4 & 4.

Williams. Home Entertainments. 1914. The eight counters puzzle, pp. 116-117. Standard version, but with black and white reversed, in four moves. Says the moved counters must be placed in line with and touching the others.

Dudeney. AM. 1917.

Prob. 236: The hat puzzle, pp. 67 & 196-197. BWBWBWBWBW.. to have the Bs and Ws together and two blanks at an end. Uses 5 moves to get to ..WWWWWBBBBB.

Prob. 237: Boys and girls, pp. 67-68 & 197. ..BWBWBWBW to have the Bs and Ws together with two blanks at an end, but pairs must be reversed as they are moved. Solution in 5 moves to WWWWBBBB... = Putnam, no. 2. Cf Lucas, 1895.

Blyth. Match-Stick Magic. 1921. Transferring in twos, pp. 80-81. WBWBWBWB.. to ..BBBBWWWW in four moves.

King. Best 100. 1927. No. 66, pp. 27 & 55. = Foulsham's, no. 9, pp. 9 & 13. BWBWBWBW.. to ..WWWWBBBB, specifically prescribed.

Rohrbough. Brain Resters and Testers. c1935. Alternate in Four Moves, p. 4. ..BBBBWWWW to WBWBWBWB.. , but he doesn't specify the blanks, showing all stages as closed up to 8 spaces, except the first two stages have a gap in the middle.

McKay. At Home Tonight. 1940.

Prob. 43: Arranging counters, pp. 73 & 87-88. RBRBRB.... to ....BBBRRR in three moves. Sketches general solution.

Prob. 45: Triplets, pp. 74 & 88. YRBYRBYRB.. to BBBYYYRRR.. in 5 moves.

McKay. Party Night. 1940. Heads and tails again, p. 151. RBRBR.. to ..BBRRR in three moves. RBRBRB.. to ..BBBRRR in four moves. RBRBRBRB.. to ..BBBBRRRR in four moves. Notes that the first move takes coins 2 & 3 to the end and thereafter one is always filling the spaces just vacated.

Gardner. SA (Jun & Jul 1961) = New MD, chap. 19, no. 1: Collating the coins. BWBWB to BBBWW, moving pairs of BW or WB only, but the final position may be shifted. Gardner thanks H. S. Percival for the idea. Solution in 4 moves, using gaps and with the solution shifted by six spaces to the right. Thanks to Heinrich Hemme for this reference.

Joseph S. Madachy. Mathematics on Vacation. (Scribners, NY, 1966, ??NYS); c= Madachy's Mathematical Recreations. Dover, 1979. Prob. 3: Nine-coin move, pp. 115 & 128-129 (where the solution is headed Eight-coin move). This uses three types of coin, which I will denote by B, R, W. BRWBRWBRW ( WWWRRRBBB by moving two adjacent unlike coins at a time and not placing the two coins away from the rest. Eight move solution leaves the coins in the same places, but uses two extra cells at each end. From the discussion of Bergerson's problem, see below, it is clear that the earlier book omitted the word unlike and had a nine move solution, which has been replaced by Bergerson's eight move solution.

Yeong-Wen Hwang. An interlacing transformation problem. AMM 67 (1967) 974-976. Shows the problem with 2n pieces, n > 2, can be solved in n moves and this is minimal.

Doubleday - 1. 1969. Prob. 70: Oranges and lemons, pp. 86 & 170. = Doubleday - 4, pp. 95-96. BWBWBWBWBW.. considered as a cycle. There are two solutions in five moves: to ..WWWWWBBBBB, which never uses the cycle; and to: BBWWWWWW..BBB.

Howard W. Bergerson, proposer; Editorial discussion; D. Dobrev, further solver; R. H. Jones, further solver. JRM 2:2 (Apr 1969) 97; 3:1 (Jan 1970) 47-48; 3:4 (Oct 1970) 233-234; 6:2 (Spring 1973) 158. Gives Madachy's 1966 problem and says there is a shorter solution. The editor points out that Madachy's book and Bergerson have omitted unlike. Bergerson has an eight move solution of the intended problem, using two extra cells at each end, and Leigh James gives a six move solution of the stated problem, also using two extra cells at each end. Dobrev gives solutions in six and five steps, using only two extra cells at the right. Jones notes that the problem does not state that the coins have to be adjacent and produces a four move solution of the stated problem, going from ....BRWBRWBRW.... to WWW..R..R..R..BBB.

Jan M. Gombert. Coin strings. MM 42:5 (Nov 1969) 244-247. Notes that BWBWB...... ( ......BBBWW can be done in four moves. In general, BWB...BWB, with n Ws and n+1 Bs alternating can be transformed to BB...BWW...W in n2 moves and this is minimal. This requires shifting the whole string n(n+1) to the right and a move can go to places separated from the rest of the pieces. By symmetry, ......BWBWB ( WWBBB...... in the same number of moves.

Doubleday - 2. 1971. Two by two, pp. 107-108. ..BWBWBWBW to WWWWBBBB... He doesn't specify where the extra spaces are, but says the first two must move to the end of the row, then two more into the space, and so on. The solution always has two moving into an internal space after the first move.

Wayne A. Wickelgren. How to Solve Problems. Freeman, 1974. Checker-rearrangement problem, pp. 144-146. BWBWB to BBBWW by moving two adjacent checkers, of different colours, at a time. Solves in four moves, but the pattern moves six places to the left.

Putnam. Puzzle Fun. 1978.

No. 1: Nickles [sic] & dimes, pp. 1 & 25. Usual version with 8 coins. Solution has blanks at the opposite end to where they began.

No. 2: Nickles [sic] & dimes variation, pp. 1 & 25. Same, except the order of each pair must be reversed as it moves. Solution in five moves with blanks at opposite end to where they started. = AM 237. Cf Lucas, 1895.

James O. Achugbue & Francis Y. Chin. Some new results on a shuffling problem. JRM 12:2 (1979-80) 125-129. They demonstrate that any pattern of n & n occupying 2n consecutive cells can be transformed into any other pattern in the same cells, using only two extra cells at the right, except for the case n = 3 where 10 cells are used. They then find an optimal solution for BB...BW...WW ( BWBW...BW in n+1 moves using two extra cells. They seem to leave open the question of whether the number of moves could be shortened by using more cells.

Walter Gibson. Big Book of Magic for All Ages. Kaye & Ward, Kingswood, Surrey, 1982.

Six cents at a time, p. 117. Uses pennies and nickels. .....PNPNP to NNPPP..... in four moves.

Tricky turnover, p. 137. HTHTHT to HHHTTT in two moves. This requires turning over one of the two coins on each move.

Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Pp. 117, 119 & 123-126. He asks to move BBBWWW to WBWBWB and to BWBWBW and notes that the latter takes an extra move. He sketches the general solutions.

5.P. GENERAL MOVING PIECE PUZZLES

See also under 5.A.

5.P.1. SHUNTING PUZZLES

See Hordern, op. cit. in 5.A, pp. 167-177, for a survey of these puzzles. The Chifu-Chemulpo (or Russo-Jap Railway) Puzzle of 1903 is actually not of this type since all the pieces can move by themselves -- Hordern, pp. 124-125 & plate VIII.

See S&B 124-125.

A 'spur' is a dead-end line. A 'side-line' is a line or siding joined to another at both ends.

Mittenzwey. 1880. Prob. 219-221, pp. 39-40 & 91; 1895?: 244-246, pp. 43-44 & 93; 1917: 244-246, pp. 40 & 89. First two have a canal too narrow to permit boats to pass, with a 'bight', or widening, big enough to hold one boat while another passes. First problem has two boats meeting one boat; second problem has two boats meeting two boats. The third problem has a single track railway with a side-line big enough to hold an engine and 16 wagons on the side-line or on the main line between the switches. Two trains consisting of an engine and 20 wagons meet.

Lucas. RM2, 1883, pp. 131-133. Passing with a spur and with a side-line.

Alexander Henry Reed. UK Patent 15,051 -- Improvements in Puzzles. Complete specification: 8 Dec 1885. 4pp + 1p diagrams. Reverse a train using a small turntable on the line. This has forms with one line and with two crossing lines. One object is to spell 'Humpty Dumptie'. He also has a circular line with three turntables (equivalent to the recent Top-Spin Puzzle of F. Lammertinck).

Pryse Protheroe. US Patent 332,211 -- Puzzle. Applied: 18 Sep 1885; patented: 8 Dec 1885. 3pp + 1p diagrams. Described in Hordern, p. 167. Identical to the Reed patent above! Both Reed and Protheroe are described as residents of suburban London. The Reed patent says it was communicated from abroad by an Israel J. Merritt Jr of New York and it doesn't assert that Reed is the inventor, so perhaps Reed and Merritt were agents for Protheroe.

Jeffrey & Son (Syracuse, NY). Great Railroad Puzzle. Postcard puzzle produced in 1888. ??NYS. Described in Hordern, pp. 175-176. Passing with a turntable that holds two wagons.

Arthur G. Farwell. US Patent 437,186 -- Toy or Puzzle. Applied: 20 May 1889; patented: 30 Sep 1890. 1p + 1p diagrams. Described in Hordern, pp. 167-169. Great Northern Puzzle. This requires interchanging two cars on the legs of a 'delta' switch which is too short to allow the engine through, but will let the cars through. Hordern lists 6 later patents on the same basic idea.

Ball. MRE, 1st ed., 1892, pp. 43-44. Great Northern Puzzle "which I bought some eight or nine years ago." (Hordern, p. 167, erroneously attributes this quote to Ahrens.)

Loyd. Problem 28: A railway puzzle. Tit-Bits 32 (10 Apr & 1 May 1897) 23 & 79. Engine and 3 cars need to pass 4 cars by means of a 'delta' switch whose branches and tail hold only one car. Solution with 28 reversals.

Loyd. Problem 31: The turn-table puzzle. Tit-Bits 32 (1 & 22 May 1897) 79 & 135. Reverse an engine and 9 cars with an 8 track turntable whose lines hold 3 cars. The turntable is a double curved connection which connects, e.g. track 1 to tracks 4 or 6.

E. Fourrey. Récréations Arithmétiques. Op. cit. in 4.A.1. 1899. Art. 239: Problèmes de Chemin de fer, pp. 184-189.

I. Three parallel tracks with two switched crossing tracks. Train of 21 wagons on the first track must leave wagons 9 & 12 on third track.

II. Delta shape with a turntable at the point of the delta, which can only hold the wagons and not the engine, so this is isomorphic to Farwell.

III. This is a more complex railway problem involving timetables on a circular line.

J. W. B. Shunting! c1900. ??NYS. Described in Hordern, pp. 176-177 & plate XII. Reversing a train with a turntable that holds three wagons.

Orril L. Hubbard. US Patent 753,266 -- Puzzle. Applied: 21 Apr 1902; patented: 1 Mar 1904. 3pp + 1p diagrams. Great Railroad Puzzle, described in Hordern, pp. 175-176. Improved version of the Jeffrey & Son puzzle of 1888. Engine & 2 cars to pass engine & 3 cars, using a turntable that holds two cars, preserving order of each train.

Harry Lionel Hook & George Frederick White. UK Patent 26,645 -- An Improved Puzzle or Game. Applied: 3 Dec 1902; accepted: 11 Jun 1903. 2pp + 1p diagrams. This is very cryptic, but appears to be a kind of sliding piece Puzzle using turntables.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:1 (May 1903) 50-51 & 10:2 (Jun 1903) 140-141 & 10:4 (Aug 1903) 336-337. A railway puzzle. One north-south line with a spur heading north which is holding 7 trucks, but cannot hold the engine as well, so the engine is on the main line heading south. An engine pulling seven trucks arrives from the north and wants to get past. First solution uses 17 stages; second uses 12 stages.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:5 (Sep 1903) 426-427 & 10:6 (Oct 1903) 530-531. A shunting problem. Same as Fourrey - II, hence isomorphic to Farwell. Solution in 17 stages.

Celluloid Starch Puzzle. c1905. Described in Hordern, pp. 169-170. Cars on the three parts of a 'delta' switch with an engine approaching. Reverse the engine, leaving all cars on their original places. More complexly, suppose the tail of the 'delta' only holds one car or the engine.

Livingston B. Pennell. US Patent 783,589 -- Game Apparatus. Applied: 20 Mar 1902; patented: 28 Feb 1905. 3pp + 1p diagrams. Described in Hordern, p. 173. Passing with a side line -- engine & 3 cars to pass engine & 3 cars using a siding which already contains 3 cars, without couplings, so these three can only be pushed. Also the engines can move at most three cars at a time.

William Rich & Harry Pritchard. UK Patent 7647 -- Railway Game and Puzzle. Applied: 11 Apr 1905; complete specification: 11 Oct 1905; accepted: 14 Dec 1905. 2pp + 1p diagrams. Main line with two short and two long spurs.

Ball. MRE, 4th ed., 1905, pp. 61-63, adds a problem with a side-line, "on sale in the streets in 1905“. The 5th ed., 1911, pp. 69-71 & 82, adds the name "Chifu-Chemulpo Puzzle" and that the minimum number of moves is 26, in more than one way. P. 82 gives solutions of both problems.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Great Northern Puzzle. He says the "Railway puzzle" was very popular "about twenty years ago".

Ahrens. MUS I. 1910. Pp. 3-4. Great Northern. Says it is apparently modern and cites Fourrey for other examples.

Anon. Prob. 6. Hobbies 32 (No. 814) (20 May 1911) 145 & (No. 817) (10 Jun 1911) 208. Great Northern Puzzle. Solution asks if readers know any other railway puzzles.

Loyd. The switch problem & Primitive railroading problem. Cyclopedia, 1914, pp. 167 & 361; 89 & 350 (= MPSL2, prob. 24, pp. 18-19; MPSL1, prob. 95, pp. 92 & 155). Passing with a 'delta' switch & passing with a spur. The first is like Tit-Bits Problem 28, but the engine and 3 cars have to pass 5 cars. Solution in 32 moves. See Hordern, pp. 170-171.

Hummerston. Fun, Mirth & Mystery. 1924. The Chinese railways, pp. 103 & 188. Imagine a line of positions: ABCEHGJLMN with single positions D, I, F, K attached to positions C, H, G, L. You have eight engines at ABCD and KLMN and the object is to exchange them, preserving the order. He does it in 18 moves, where a move can be of any length.

King. Best 100. 1927. No. 14, pp. 12 & 41. Side-line with a bridge over it too low for the engine. Must interchange two wagons on the side-line which are on opposite sides of the bridge.

B. M. Fairbanks. Railroad switching problems. IN: S. Loyd Jr., ed.; Tricks and Puzzles; op. cit. in 5.D.1 under Chapin; 1927. P. 85 & Answers p. 7. Three realistic problems with several spurs and sidelines.

Loyd Jr. SLAHP. 1928. Switching cars, pp. 54 & 106. Great Northern puzzle. See Hordern, pp. 168-169.

Doubleday - 2. 1971. Traffic jam, pp. 85-86. Version with cars in a narrow lane and a lay-by. Two cars going each way. Though the lay-by is three cars wide and just over a car long, he restricts its use so that it acts like it is two cars wide.

5.P.2. TAQUIN

Jacques Haubrich has kindly enlightened me that 'taquin' simply means 'teaser'. So these items should be re-categorised.

Lucas. RM3. 1893. 3ème Récréation -- Le jeu du caméléon et le jeu des jonctions de points, pp. 89-103. Pp. 91-97 -- Le taquin de neuf cases avec un seul port. I thought that taquin was the French generic term for such puzzles, but I find no other usage than that below, except in referring to the 15 Puzzle -- see references to taquin in 5.A.

Au Bon Marché (the Paris department store). Catalogue of 1907, p. 13. Reproduced in Mary Hillier; Automata and Mechanical Toys; An Illustrated History; Jupiter Books, London, 1976, p. 179. This shows Le Taquin Japonais Jeu de Patience Casse-tete. This comprises 16 hexagonal pieces, looking like a corner view of a die, so each has three rhombic parts containing a pattern of pips. They are to be placed as the corners of four interlocked hexagons with the numbers on adjacent rhombi matching.

5.Q. NUMBER OF REGIONS DETERMINED BY N LINES OR PLANES

Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 & 88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and the children get equal pieces. He makes two perpendicular diametrical cuts and then a circular cut around the middle. He seems to mean the adults get equal pieces and the children get equal pieces, not necessarily the same. But if the circular cut is at (2/2 of the radius, then the areas are all equal. Not clear where this should go -- also entered in 5.T.

Jakob Steiner. Einige Gesetze über die Theilung der Ebene und des Raumes. (J. reine u. angew. Math. 1 (1826) 349-364) = Gesam. Werke, 1881, vol. 1, pp. 77-94. Says the plane problem has been raised before, even in a Pestalozzi school book, but believes he is first to consider 3-space. Considers division by lines and circles (planes and spheres) and allows parallel families, but no three coincident.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. "3. A man marks 6 straight lines on a field in such a way as to enclose 10 spaces. How does he manage this?" Solution begins: "III. To inclose ten spaces by six ropes fastened to nine pegs." Take (0,0), (1,0), ..., (n,0), (0,n), ..., (0,1), as 2n+1 points, using n+2 ropes from (0,0) to (n,0) and to (0,n) and from (i,0) to (0,n+1-i) to enclose n(n+1)/2 areas.

Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40. Describes several ways of solving previous problem and asks for a symmetric version.

G. Chrystal. Algebra -- An Elementary Text-Book. Vol. 2, A. & C. Black, Edinburgh, 1889. [Note -- the 1889 version of vol. 1 is a 2nd ed.] Chap. 23, Exercises IV, p. 34. Several similar problems and the following.

No. 7 -- find number of interior and of exterior intersections of the diagonals of a convex n-gon.

No. 8 -- n points in general position in space, draw planes through every three and find number of lines and of points of intersection.

L. Schläfli. Theorie der vielfachen Kontinuität. Neue Denkschriften der allgemeinen schweizerischen Gesellschaft für die Naturwissenschaften 38:IV, Zürich, 1901, 239 pp. = Ges. Math. Abh., Birkhäuser, Basel, 1950-1956, vol. 1, pp. 167-392. (Pp. 388-392 are a Nachwort by J. J. Burckhardt.) Material of interest is Art. 16: Über die Zahl der Teile, ..., pp. 209-212. Obtains formula for k hyperplanes in n space.

Loyd, Dudeney, Pearson & Loyd Jr. give various puzzles based on this topic.

Howard D. Grossman. Plane- and space-dissection. SM 11 (1945) 189-190. Notes Schläfli's result and observes that the number of regions determined by k+1 hyperspheres in n space is twice the number of regions determined by k hyperplanes and gives a two to one correspondence for the case n = 2.

Leo Moser, solver. MM 26 (Mar 1953) 226. ??NYS. Given in: Charles W. Trigg; Mathematical Quickies; (McGraw-Hill, NY, 1967); corrected ed., Dover, 1985. Quickie 32: Triangles in a circle, pp. 11 & 90-91. N points on a circle with all diagonals drawn. Assume no three diagonals are concurrent. How many triangles are formed whose vertices are internal intersections?

Timothy Murphy. The dissection of a circle by chords. MG 56 (No. 396) (May 1972) 113-115 + Correction (No. 397) (Oct 1972) 235-236. N points on a circle, in a plane or on a sphere; or N lines in a plane or on a sphere, all simply done, using Euler's formula.

Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Slicing cakes, pp. 33 & 61. Cut a circular cake into 12 equal pieces with 4 cuts. [From this, we see that N full cuts can yield either 2N or 4(N-1) equal pieces. Further, if we make k circular cuts producing k+1 regions of equal area and then make N-k diametric cuts equally spaced, we get 2(k+1)(N-k) pieces of the same size.]

Looking at this problem, I see that one can obtain any number of pieces from N+1 up through the maximum.

5.Q.1. NUMBER OF INTERSECTIONS DETERMINED BY N LINES

Chrystal. Text Book of Algebra. 2nd ed., vol. 2, 1889, p. 34, ex. 7. See above.

Loyd Jr. SLAHP. 1928. When drummers meet, pp. 74 & 115. Six straight railroads can meet in 15 points.

Paul Erdös, proposer; Norbert Kaufman & R. H. Koch and Arthur Rosenthal, solvers. Problem E750. AMM 53 (1946) 591 & 54 (1947) 344. The first solution is given in Trigg, op. cit. in 5.Q, Quickie 191: Intersections of diagonals, pp. 53 & 166-167. In a convex n-gon, how many intersections of diagonals are there? This counts a triple intersection as three ordinary (i.e. double) intersections or assumes no three diagonals are concurrent. Editorial notes add some extra results and cite Chrystal.

5.R. JUMPING PIECE GAMES

See also 5.O. Some of these are puzzles, but some are games and are described in the standard works on games -- see the beginning of 4.B.

5.R.1. PEG SOLITAIRE

See MUS I 182-210.

Ahrens, MUS I 182-183, gives legend associating this with American Indians. Bergholt, below, and Beasley, below, find this legend in the 1799 Encyclopédie Méthodique: Dictionnaire des Jeux Mathématique (??*), ??NYS. Ahrens also cites some early 19C material which has not been located. Bergholt says some maintain the game comes from China.

Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a 5 x 5 board with each side having 12 men, but the description is extremely brief. It seems to have two players, but this may simply refer to the two types of piece. I'm not clear whether it's played like solitaire (with the jumped pieces being removed) or like frogs & toads. I would be grateful if someone could read the Latin carefully. The name of the puzzle is clearly Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the pages I have) -- I would be grateful to anyone who can track down and translate Arabic sources.

G. W. Leibniz. Le Jeu du Solitaire. Unpublished MS LH XXXV 3 A 10 f. 1-2, of c1678. Transcribed in: S. de Mora-Charles; Quelques jeux de hazard selon Leibniz; HM 19 (1992) 125-157. Text is on pp. 152-154. 37 hole board. Says the Germans call it 'Die Melancholy' and that it is now the mode at the French court.

Claude-Auguste Berey. Engraving: Madame la Princesse de Soubize jouant au Jeu de Solitaire. 1697(?). Beasley (below) discovered and added this while his book was in proof. It shows the 37-hole French board. Reproduced in: Pieter van Delft & Jack Botermans; Creative Puzzles of the World; op. cit. in 5.E.2.a, p. 170.

G. W. Leibniz. Jeu des Productions. Unpublished MS LH XXXV 8,30 f. 4, of 1698. Transcribed in: de Mora-Charles, loc. cit. above. Text is on pp. 154-155. 37 hole board. Considers the game in reverse.

Trouvain. Engraving: Dame de Qualité Jouant au Solitaire. 1698(?).

Claude-Auguste Berey. Engraving: Nouveau Jeu de Solitaire. Undated, but Berey was active c1690-c1730. Reproduced in: R. C. Bell; The Board Game Book; Marshall Cavendish, London, 1979, pp. 54-55 and in: Jasia Reichardt, ed.; Play Orbit [catalogue of an exhibition at the ICA, London, and elsewhere in 1969-1970]; Studio International, 1969, p. 38. Beasley's additional notes point out that this engraving is well known, but he had not realised its date until the earlier Berey engraving was discovered. This engraving includes the legend associating the game with the American indians -- "son origine vient de l'amerique ou les Peuples vont seuls à la chasse, et au retour plantent leurs flèches en des trous de leur cases, ce qui donna idée a un françois de composer ce jeu ...." Reichardt says the original is in the Bibliothèque Nationale.

The three engravings above are reproduced in: Henri d'Allemagne; Musée rétrospectif de la classe 100, Jeux, à l'exposition universelle international de 1900 à Paris, Tome II, pp. 152-158. D'Allemagne says the originals are in the Bibliothèque Nationale, Paris. He (and de Mora-Charles) also cites Rémond de Montmort, 2nd ed., 1713 -- see below.

G. W. Leibniz. Annotatio de quibusdam Ludis; inprimis de Ludo quodam Sinico, differentiaque Scachici et Latrunculorum & novo genere Ludi Navalis. Misc. Berolinensia (= Misc. Soc. Reg., Berlin) 1 (1710) 24. Last para. on p. 24 relates to solitaire. (English translation on p. xii of Beasley, below.)

Pierre Rémond de Montmort. Essai d'analyse sur les jeux de hazards. (1708); Seconde edition revue & augmentee de plusieurs lettres, (Quillau, Paris, 1713 (reprinted by Chelsea, NY, 1980)); 2nd issue, Jombert & Quillau, 1714. Avertissement (to the 2nd ed.), xli-xl. "J'ai trouvé dans le premier volume de l'Academie Royale de Berlin, ...; il propose ensuite des Problèmes sur un jeu qui a été à la mode en France il y a douze ou quinze ans, qui se nomme Le Solitaire."

Edward Hordern's collection has a wooden 37 hole board on the back of which is inscribed "Invented by Lord Derwentwater when Imprisoned in the Tower". The writing is old, at least 19C, possibly earlier. However the Encyclopedia Britannica article on Derwentwater and the DNB article on Radcliffe, James, shows that the relevant Lord was most likely to have been James Radcliffe (1689-1716), the 3rd Earl from 1705, who joined the Stuart rising in 1715, was captured at Preston, was imprisoned in the Tower and was beheaded on 24 Feb 1716, so the implied date of invention is 1715 or 1716. The third Earl became a figure of romance and many stories and books appeared about him, so the invention of solitaire could well have been attributed to him.

Though the title was attainted and hence legally extinct, it was claimed by relatives. Both James's brother Charles (1693-1746), the claimed 5th Earl from 1731, and Charles's son James Bartholomew (1725-1786), the claimed 6th Earl from 1746, spent time in prison for their Stuart sympathies. Charles escaped from Newgate Prison after the 1715 rising, but both were captured on their way to the 1745 rising and taken to the Tower where Charles was beheaded. If either of these is the Lord Derwentwater referred to, then the date must be 1745 or 1746. A guide book to Northumberland, where the family lived at Dyvelston (or Dilston) Castle, near Hexham, asserts the last Derwentwater was executed in 1745, while the [Blue Guide] says the last was executed for his part in the 1715 uprising.

In any case, the claim seems unlikely.

G. W. Leibniz. Letter to de Montmort (17 Jan 1716). In: C. J. Gerhardt, ed.; Die Philosophischen Schriften von Gottfried Wilhelm Leibniz; (Berlin, 1887) = Olms, Hildesheim, 1960; Vol. 3, pp. 667-669. Relevant passage is on pp. 668-669. (Poinsot, op. cit. in 5.E, p. 17, quotes this as letter VIII in Leibn. Opera philologica.)

J. C. Wiegleb. Unterricht in der natürlichen Magie. Nicolai, Berlin & Stettin, 1779. Anhang von dreyen Solitärspielen, pp. 413-416, ??NYS -- cited by Beasley. First known diagram of the 33-hole board.

Catel. Kunst-Cabinet. 1790. Das Grillenspiel (Solitaire), p. 50 & fig. 167 on plate VI. 33 hole board. (Das Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is a game on the 33-hole board.)

Bestelmeier. 1801. Item 511: Ein Solitair, oder Nonnenspiel. 33 hole board.

Strutt. Op. cit. in 4.B.1. The Solitary Game. (1801: Book IV, p. 238. ??NYS -- cited by Beasley -- may be actually 1791??) 1833: Book IV, chap. II, art. XV, p. 319. c= Strutt-Cox, p. 259. Beasley says this is the first attribution to a prisoner in the Bastille. The description is vague: "fifty or sixty" holes and "a certain number of pegs". Strutt-Cox adds a note that "The game of Solitaire, reimported from France, ..., came again into Fashion in England in the late" 1850s and early 1860s.

Ada Lovelace. Letter of 16 Feb 1840 to Charles Babbage. BM MSS 37191, f. 331. ??NYS -- reproduced in Teri Perl; Math Equals; Addison-Wesley, Menlo Park, California, 1978, pp. 109-110. Discusses the 37 hole board and wonders if there is a mathematical formula for it.

M. Reiss. Beiträge zur Theorie des Solitär-Spiels. J. reine angew. Math. 54 (1857) 376-379.

St. v. Kosiński & Louis Wolfsberg. German Patent 42919 -- Geduldspiel. Patented: 25 Sep 1877. 1p + 1p diagrams. 33 hole version.

The Sociable. 1858. The game of solitaire, pp. 282-284. 37 hole board. "It is supposed to have been invented in America, by a Frenchman, to beguile the wearisomeness attendant upon forest life, and for the amusement of the Indians, who pass much of their time alone at the chase, ...."

Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and Sons, London, 1883, HB. Section 135: Solitaire, p. 49. Mentions a 37 hole board but shows a 33 hole board. This material presumably goes back some time before this edition. It later shows Fox and Geese on the 33 hole board.

Hoffmann. 1893. Chap. X, no. 11: Solitaire problems, pp. 339-340 & 376-377 = Hoffmann-Hordern, pp. 232-233, with photo on p. 235. Three problems. Photo on p. 235 shows a 33-hole board in a square frame, 1820-1840, and a 37-hole board with a holding handle, 1840-1890.

Ernest Bergholt. Complete Handbook to the Game of Solitaire on the English Board of Thirty-three Holes. Routledge, London, nd [Preface dated Nov 1920] -- facsimile produced by Naoaki Takashima, 1993. This is the best general survey of the game prior to Beasley.

King. Best 100. 1927. No. 68, pp. 28 & 55. = Foulsham's no. 24, pp. 9 & 13. 3 x 3 array of men in the middle of a 5 x 5 board. Men can jump diagonally as well as orthogonally. Object is to leave one man in the centre.

Rohrbough. Puzzle Craft. 1932. Note on Solitaire & French Solitaire, pp. 14-15 (= pp. 6-7 of 1940s?). 33 hole board, despite being called French.

B. M. Stewart. Solitaire on a checkerboard. AMM 48 (1941) 228-233. This surveys the history and then considers the game on the 32 cell board comprising the squares of one colour on a chessboard. He tilts this by 45o to get a board with 7 rows, having 2, 4, 6, 8, 6, 4, 2 cells in each row. He shows that each beginning-ending problem which is permitted by the parity rules is actually solvable, but he gives examples to show this need not happen on other boards.

Gardner. SA (Jun 1962). Much amended as: Unexpected, chap. 11, citing results of Beasley, Conway, et al. Cites Leibniz and mentions Bastille story.

J. D. Beasley. Some notes on solitaire. Eureka 25 (Oct 1962) 13-18. No history of the game.

Jeanine Cabrera & René Houot. Traité Pratique du Solitaire. Librairie Saint-Germain, Paris, 1977. On p. 2, they give the story that it was invented by a prisoner in the Bastille, late 18C, and they even give the name of the reputed inventor: "Comte"(?) Pellisson. They say that a Paul Pellisson-Fontanier was in the Bastille in 1661-1666 and was a man of some note, but history records no connection between him and the game.

The Diagram Group. Baffle Puzzles -- 3: Practical Puzzles. Sphere, 1983. No. 12. On the 33-hole board place 16 markers: 1 in row 2; 3 in row 3; 5 in row 4; 7 in row 5; making a triangle centred on the mid-line. Can you remove all the men, except for one in the central square? Gives a solution in 15 jumps.

J. D. Beasley. The Ins and Outs of Peg Solitaire. OUP, 1985. History, pp. 3-7; Selected Bibliography, pp. 253-261. PLUS Additional notes, from the author, 1p, Aug 1985. 57 references and 5 patents, including everything known before 1850.

Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. This gives the legend of the nobleman in the Bastille. Then says that "it would appear that a very similar game" is mentioned by Ovid "and again, it was widely played in ancient China -- hence its still frequent alternative name, "Chinese checkers."" I have included this as an excellent example of how unreferenced statements are made in popular literature. I have never seen either of these latter statements made elsewhere. The connection with Ovid is pretty tenuous -- he mentions a game involving three in a row and otherwise is pretty cryptic and I haven't seen anyone else claiming Ovid is referring to a solitaire game -- cf 4.B.5. The connection with Chinese checkers is so far off that I wonder if there is a translation problem -- i.e. does the Italian name refer to some game other than what is known as Chinese checkers in English??

Nob Yoshigahara. Puzzlart. Tokyo, 1992. Coin solitaire, pp. 5 & 90. Four problems on a 4 x 4 board.

Marc Wellens, et al. Speelgoed Museum Vlaanderen -- Musée du Jouet Flandre -- Spielzeug Museum Flandern -- Flanders Toy Museum. Speelgoedmuseum Mechelen, Belgium, 1996, p. 90 (in English), asserts 'It was invented by the French nobleman Palissen, who had been imprisoned in the Bastille by Louis XIV' in the early 18C.

5.R.1.a. TRIANGULAR VERSION

The triangular version of the game has only recently been investigated. The triangular board is generally numbered as below.

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

Herbert M. Smith. US Patent 462,170 -- Puzzle. Filed: 13 Mar 1891; issued: 27 Oct 1891. 2pp + 1p diagrams. A board based on a triangular lattice.

Rohrbough. Puzzle Craft. 1932. Triangle Puzzle, p. 5 (= p. 6 in 1940s?). Remove peg 13 and leave last peg in hole 13.

Maxey Brooke. (Fun for the Money, Scribner's, 1963); reprinted as: Coin Games and Puzzles; Dover, 1973. All the following are on the 15 hole board.

Prob. 1: Triangular jump, pp. 10-11 & 75. Remove one man and jump to leave one man on the board. Says Wesley Edwards asserts there are just six solutions. He removes the middle man of an edge and leaves the last man there.

Prob. 2: Triangular jump, Ltd., pp. 12-13 & 75. Removes some of the possible jumps.

Prob. 3: Headless triangle, pp. 14 & 75. Remove a corner man and leave last man there.

M. Gardner. SA (Feb 1966) c= Carnival, 1975, chap. 2. Says a 15 hole version has been on sale as Ke Puzzle Game by S. S. Adams for some years. Addendum cites Brooke and Hentzel and says much unpublished work has been done.

Irvin Roy Hentzel. Triangular puzzle peg. JRM 6:4 (1973) 280-283. Gives basic theory for the triangular version. Cites Gardner.

[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY, 1975. Pennies for your thoughts, pp. 179-182. Remove a coin and solve. Hint says to remove the coin at 13 and that you should be able to have the last coin at 13. The solution has this property.

Alan G. Henney & Dagmar R. Henney. Computer oriented solutions. CM 4:8 (1978) 212-216. Considers the 'Canadian I. Q. Problem', which is the 15 hole board, but they also permit such jumps as 1 to 13, removing 5. They find solutions from each initial removal by random trial and error on a computer.

Putnam. Puzzle Fun. 1978. No. 15: Jumping coins, pp. 5 & 28. 15 hole version, remove peg 1 and leave last man there.

Benjamin L. Schwarz & Hayo Ahlburg. Triangular peg solitaire -- A new result. JRM 16:2 (1983-84) 97-101. General study of the 15 hole board showing that starting and ending with 5 is impossible.

J. D. Beasley. The Ins and Outs of Peg Solitaire. Op. cit. above, 1985. Pp. 229-232 discusses the triangular version, citing Smith, Gardner and Hentzel, saying that little has been published on it.

Irvin Roy Hentzel & Robert Roy Hentzel. Triangular puzzle peg. JRM 18:4 (1985-86) 253-256. Develops theory.

John Duncan & Donald Hayes. Triangular solitaire. JRM 23:1 (1991) 26-37. Extended analysis. Studies army advancement problem.

William A. Miller. Triangular peg solitaire on a microcomputer. JRM 23:2 (1991) 109-115 & 24:1 (1992) 11. Summarises and extends previous work. On the 10 hole triangular board, the classic problem has essentially a unique solution -- the removed man must be an edge man (e.g. 2) and the last man must be on the adjacent edge and a neighbour of the starting hole (i.e. 3 if one starts with 2). On the 15 hole board, the removed man can be anywhere and there are many solutions in each case.

Remove man from hole: 1 2 4 5

Number of solutions: 29760 14880 85258 1550

Considers the 'tree' formed by the first four rows and hole 13.

5.R.1.b. OTHER SHAPES

New section. See also King and Stewart in 5.R.1 for some forms based on a square board.

A

B C D E

F G

H

I J

Putnam. Puzzle Fun. 1978. No. 53: Checker star, pp. 10 & 34. Use the 10 points of a pentagram, as above, and leave one of the inner points empty. Reduce to one man. [Parity shows the one man must be at an outer point and any outer point can be achieved. If one leaves an outer point empty, then the last man must be on an inner point and any of these can be achieved.]

Hummerston. Fun, Mirth & Mystery. 1924.

Perplexity, pp. 22-23. Using the octagram board shown in 5.A, place 15 markers on it, leaving cell 16 empty. It is possible to remove all but one man. [I can't see how to apply parity to this board.]

Solplex, p. 25. In playing his Perplexity, specify where you will leave the last man?

Leap frog, Puzzle no. 22, pp. 64 & 175. Take a 4 x 3 board with the long edge extended by one more cell at the upper left and lower right. Put white counters on the 4 x 3 area, put a black counter in one of the extra cells and leave the other extra cell empty. Remove all but the black man. Counting multiple jumps of the same man as a single move, he does it in eight moves, getting the black man back to its starting point.

5.R.2. FROGS AND TOADS: BBB_WWW TO WWW_BBB

In the simplest version, one has n black men at the left and n white men at the right of a strip of 2n+1 cells, e.g. BBB_WWW. One can slide a piece forward (i.e. blacks go left and whites go right) into an adjacent place or one can jump forward over one man of the other colour into an empty place. The object is to reverse the colours, i.e. to get WWW_BBB. S&B 121 & 125, shows versions.

One finds that the solution never has a man moving backward nor a man jumping another man of the same colour. Some authors have considered relaxing these restrictions, particularly if one has more blank spaces, when these unusual moves permit shorter solutions. Perhaps the most general form of the one-dimensional problem would be the following. Suppose we have m men at the left of the board, n men at the right and b blank spaces in the middle. The usual case has b = 1, but when b > 1, the kinds of move permitted do change the number of moves in a minimal solution. First, considering slides, can a piece slide backward? Can a piece slide more than one space? If so, is there a maximum distance, s, that it is allowed to slide? (The usual problem has s = 1.) Of course s ( b. Second, considering jumps, can a piece jump backward? Can a piece jump over a piece (or pieces) of its own colour and/or a blank space (or spaces) and/or a mixture of these? If so, is there a maximum number of pieces, p, that it can jump over? (The usual problem has p = 1.) It is not hard to construct simple examples with s > 1 such that shorter solutions exist when unusual moves are permitted. Are there situations where one can show that backward moves are not needed?

The game is sometimes played on a 2-dimensional board, where one colour can move down or right and the other can move up or left. See: Hyde ??; Lucas (1883); Ball; Hoffmann and 5.R.3. Chinese checkers is a later variation of this same idea. On these more complex boards, one is usually allowed to make multiple jumps and the object is usually to minimize the number of moves to accomplish the interchange of pieces.

There is a trick version to convert full and empty glasses: FFFEEE to FEFEFE in one move, which is done by pouring. I've just noted this in a 1992 book and I'll look for earlier examples.

Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a 5 x 5 board with each side having 12 men, but the description is extremely brief. It seems to have two players, but this may simply refer to the two types of piece. I'm not clear whether it's played like solitaire (with the jumped pieces being removed) or like frogs & toads. I would be grateful if someone could read the Latin carefully. The name of the puzzle is clearly Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the pages I have) -- I would be grateful to anyone who can track down and translate Arabic sources.

American Agriculturist (Jun 1867). Spanish Puzzle. ??NYR -- copy sent by Will Shortz.

Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty illustrations. Routledge, London, nd. HPL gives c1850, but the material is clearly derived from Every Boy's Book, whose first edition was 1856. But the text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), nor in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously revised edition, with Edmund Routledge as editor, nor in the 13th ed. of 1878. So this material is hard to date, though in 4.A.1, I've guessed this book may be c1868.

P. 12: Frogs and toads. "A new and fascinating game of skill for two players; played on a leather board with twelve reptiles; the toads crawling, and the frogs hopping, according to certain laws laid down in the rules. The game occupies but a few minutes, but in playing it there is scarcely any limit to the skill that can be exhibited, thus forming a lasting amusement. (Published by Jaques, Hatton Garden.)" This does not sound like our puzzle, but perhaps it is related. Unfortunately Jaques' records were destroyed in WW2, so it is unlikely they can shed any light on what the game was. Does anyone know what it was?

Hanky Panky. 1872. Checker puzzle, p. 124. Three and three, with solution.

Mittenzwey. 1880. Prob. 239, pp. 44 & 94; 1895?: 267-268, pp. 48 & 96; 1917: 267-268, pp. 44 & 91-92. Problem with 3 & 3 brown and white horses in stalls. 1895? adds a version with 4 & 4.

Bazemore Bros. (Chattanooga, Tennessee). The Great "13" Puzzle! Copyright No. 1033 - O - 1883. Hammond & Jones Printers. Advertising puzzle consisting of two 3 and 3 versions arranged in an X pattern.

Lucas. RM2. 1883.

Pp. 141-143. Finds number of moves for n and n.

Pp. 144-145. Considers game on 5 x 5, 7 x 7, ..., boards and gives number of moves.

Edward Hordern's collection has an example called Sphinxes and Pyramids from the 1880s.

Sophus Tromholt. Streichholzspiele. (1889; 5th ed, 1892.) Revised from 14th ed. of 1909 by R. Thiele; Zentralantiquariat der DDR, Leipzig, 1986. Prob. 11, 41, 81 are the game for 4 & 4, 2 & 2, 3 & 3.

Ball. MRE, 1st ed., 1892, pp. 49-51. 3 & 3 case, citing Lucas, with generalization to n & n; 7 x 7 board, citing Lucas, with generalization to 2n+1 x 2n+1.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. German counter puzzle, p. 112. 3 & 3 case.

Hoffmann. 1893. Chap. VI, pp. 269-270 & 282-284 = Hoffmann-Hordern, pp. 182-185, with photo.

No. 17: The "Right and Left" puzzle. Three and three. Photo on p. 184 shows: a cartoon from Punch (18 Dec 1880): The Irish Frog Puzzle -- with a Deal of Croaking; and an example of a handsome carved board with square pieces with black and white frogs on the tops, registered 1880. Hordern Collection, p. 77, shows the latter board and two further versions: Combat Sino-Japonais (1894-1895) and Anglais & Boers (1899-1902).

No. 18. Extends to a 7 x 5 board.

Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??) 751. 3 and 3.

Lucas. L'Arithmétique Amusante. 1895. Prob. XXXV: Le bal des crapauds et des grenouilles, pp. 117-124. Does 2 and 2, 3 and 3, 4 and 4 and the general case of n and n, showing it can be done in n(n+2) moves -- n2 jumps and 2n steps. The general solution is attributed to M. Van den Berg. M. Schoute notes that each move should make as little change as possible from the previous with respect to the two aspects of changing type of piece and changing type of move.

Clark. Mental Nuts. 1904, no. 72; 1916, no. 62. A good study. 3 and 3.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. Doola's Game, pp. 42-43 & 61-62. 3 and 3.

Anon. Prob. 47: The monkey's dilemma. Hobbies 30 (No. 762) (21 May 1910) 168 & 182 & (No. 765) (11 Jun 1910) 228. Basically 3 & 3, but there are eight posts for crossing a river, with the monkeys on 1,2,3 and 6,7,8. The monkeys can jump onto the bank and we want the monkeys to all get to the bank they are headed for, so this is not the same as BBB..WWW to WWW..BBB. The solution doesn't spell out all the steps, so it's not clear what the minimum number of moves is -- could we have a monkey jumping another of the same colour?

Ahrens. MUS I. 1910. Pp. 17-19. Basically repeats some of Lucas's work from 1883 & 1895.

Williams. Home Entertainments. 1914. The cross-over puzzle, pp. 119-120. 3 and 3 with red and white counters. Doesn't say how many moves are required.

Dudeney. AM. 1917. Prob. 216: The educated frogs, pp. 59-60 & 194. _WWWBBB to BBBWWW_ with frogs able to jump either way over one or two men of either colour. Solution in 10 jumps.

Ball. MRE, 9th ed., 1920, pp. 77-79, considers the m & n case, giving the number of steps in the solution.

Blyth. Match-Stick Magic. 1921. Matchstick circle transfer, pp. 81-82. 3 and 3 in 15 moves.

Hummerston. Fun, Mirth & Mystery. 1924. The frolicsome frogs, Puzzle no. 2, pp. 17 & 172. Two 3 & 3 problems with the boards crossing at the centre cell. He notes that the easiest solution is to solve the boards one at a a time. He says: "It is not good play to jump a counter over another of the same colour."

Lynn Rohrbough, ed. Socializers. Handy Series, Kit G, Cooperative Recreation Service, Delaware, Ohio, 1925. Six Frogs, p. 5. Dudeney's 1917 problem done in 11 moves.

Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 235 describes this as The Sphinx Puzzle, "very popular around the turn of the century, particularly in the United States and France" and they show an example of the period labelled The Sphinx and Pyramid Puzzle -- An Egyptian Novelty.

Haldeman-Julius. 1937. No. 162: Checker problem, pp. 18 & 29. 3 & 3.

See Harbin in 5.R.4 for a 1963 example.

Doubleday - 1. 1969.

Prob. 77: Square dance, pp. 93 & 171. = Doubleday - 5, pp. 103-104. Start with _WWWBBB. He says they must change places, with a piece able to move into the vacant space by sliding (either way) or by jumping one or two pieces of any colour. Asks for a solution in 10 moves. His solution gets to BBBWWW_, which does not seem to be 'changing places' to me.

Prob. 79: All change, pp. 95 & 171. = Doubleday - 5, pp. 105-106. BB_WW

Start with the pattern at the right and change the whites and BB_WW

blacks in 10 moves, where a piece can slide one place into an

adjacent vacant square or jump one or two pieces into a vacant square. However, the solution simply does each row separately.

Katharina Zechlin. (Dekorative Spiele zum Selbermachen; Verlag Frech, WWWWW

Stuttgart-Botnang, 1973.) Translated as: Making Games in Wood Games BWWWW

you can build yourself. Sterling, 1975, pp. 24-27: The chess knight game. BBOWW

5 x 5 board with 12 knights of each of two colours, arranged as at the right. BBBBW

The object is to reverse them by knight's moves. Says it can be done within BBBBB

50 moves and 'is almost impossible to do it in less than 45'.

Wickelgren. How to Solve Problems. Op. cit. in 5.O. 1974. Discrimination reversal problem, pp. 78-81. _WWWBBB to BBBWWW with the extra place not specified in the goal, with pieces allowed to move into the vacant space by sliding or by hopping over one or two pieces. Gets to BBBWW_W in 9 moves. [I find it takes 10 moves to get to BBBWWW_ .]

Joe Celko. Jumping frogs and the Dutch national flag. Abacus 4:1 (Fall 1986) 71-73. Same as Wickelgren. Celko attributes this to Dudeney. Gives a solution to BBBWWW_ in 10 moves and asks for results for higher numbers.

Johnston Anderson. Seeing induction at work. MG 75 (No. 474) (Dec 1991) 406-414. Example 2: Frogs, pp. 408-411. Careful proof that BB...BB_WW...WW to WW...WW_BB...BB, with n counters of each colour, requires n2 + 2n moves.

5.R.3. FORE AND AFT -- 3 BY 3 SQUARES MEETING AT A CORNER

This is Frogs and Toads on part of the 5 x 5 board consisting of two 3 x 3 subarrays at diagonally opposite corners. They overlap in the central square. One square has 8 black men and the other has 8 white men, with the centre left vacant.

Ball. MRE, 1st ed., 1892, pp. 51-52. 51 move solution. In the third ed., 1896, pp. 69-70, he says he believes he was the first to publish the puzzle but "that it has been since widely distributed in connexion with an advertisement and probably now is well known". He gives a 48 move solution.

Hoffmann. 1893. Chap. VI, no. 26: The "English Sixteen" puzzle, pp. 273-274 & 287 = Hoffmann-Hordern, pp. 188-189, with photo. Mentions that it is produced by Messrs Heywood, as below. Solution in 52 moves, which he believes is minimal. Hordern notes that the minimum is 46. Photo on p. 188 of the Heywood version, see next entry.

John Heywood, Manchester, produced a version called 'The English Sixteen Puzzle', undated, but by 1893 as Hoffmann cites it. Photo in Hoffmann-Hordern, p. 188, dated 1880-1895.

Charles A. Emerson. US Patent 522,250 -- Puzzle. Applied: 3 Nov 1893; patented: 3 July 1894. 2pp + 1p diagrams. The Fore and Aft Puzzle. Says it can be done in 48, 49, 50, 51 or 52 moves.

Dudeney. Problem 66: The sixteen puzzle. Tit-Bits 33 (1 Jan & 5 Feb 1898) 257 & 355. "It was produced, I believe, in America, many years ago, and has since been issued over here in the form of an advertisement by a prominent commercial house." Solution in 46 moves. He says published solutions assert the minimum number of moves is 53, 52 or 50. The 46 move solution is given in Ball, MRE, 5th ed., 1911, 79-80.

Ball. MRE, 5th ed., 1911, pp. 79-80. Drops his historical claims and includes a 46 move solution due to Dudeney.

Loyd. Fore and aft puzzle. Cyclopedia, 1914, pp. 108 & 353 (solution misprinted, but claimed to be 47 moves in contrast to 52 move solutions 'in the puzzle books'.) (c= MPSL1, prob. 4, pp. 3-4 & 121 (only referring to Dudeney's 46 move solution)).

Loyd Jr. SLAHP. 1928. A joke on granddad, pp. 29 & 93. Says 'our granddaddies, who used to play this puzzle game 75 years ago, when it was universally popular. The old-time books explain how the solution is accomplished in 52 moves, "the shortest possible method."' He then asks for and gives a 46 move solution.

M. Adams. Puzzles That Everyone Can Do. 1931. Prob. 24, pp. 17 & 132: "General post". Gives a solution which takes 46 moves, but gives no discussion of it.

Rohrbough. Puzzle Craft. 1932. Migration (or Fore and Aft), p. 12 (= p. 15 of 1940s?). Says it was popular 75 years ago and it has recently been shown that it can be done in 46 moves, then gives a solution which stops at 42 moves!

M. Gardner. SA (Sep 1959) = 2nd Book, pp. 210-219. Discusses the puzzle. On pp. 218-219, he gives Dudeney's 46 move solution and says 48 different solutions and several proofs that 46 is minimal were sent to him.

Uwe Schult. Das Seemanns-Spiel: Mathematisch erledigt. Reported in Das Mathematische Kabinett column, Bild der Wissenschaft 19:11 (Nov 1982) 181-184. (A version is given in Neues aus dem Mathematischen Kabinett, ed. by Thiagar Devendran, Hugendubel, Munich, 1985, pp. 102-103.) There are 218,790 possible patterns of the pieces. Reversing black and white takes 46 moves and there are 1026 different halfway positions that can occur in a 46 move solution. There are two patterns which require 47 moves, namely, after reversing black and white, put one of the far corner pieces in the centre.

Nob Yoshigahara, postcard to me on 18 Aug 1994, announces he has found the worst solution -- in 58 moves.

5.R.4. REVERSING FROGS AND TOADS: _12...n TO _n...21 , ETC.

A piece can slide into the empty cell or jump another piece into the empty cell.

Dudeney. AM. 1917.

Prob. 214: The six frogs, pp. 59 & 193. Case of n = 6, solved in 21 moves, which he says is minimal. In general, the minimal solution takes n(n+1)/2 moves, including n steps, when n is even and (n2+3n-8)/2 moves, including 2n-4 steps, when n is odd. "This complete general solution is published here for the first time."

Prob. 215: The grasshopper puzzle, pp. 59 & 193-194. Problem for a circular arrangement. Example has n = 12. Says he invented it in 1900. Solvable in 44 moves. General solution is complex -- he says that for n > 4, it can be done in (n2+4n-16)/4 moves when n is even and in (n2+6n-31)/4 moves when n is odd.

Rohrbough. Puzzle Craft. 1932. The Reversible Frogs, p. 22 (= The Jumping Frogs, pp. 20-21 of 1940s?). n = 8, citing Dudeney, AM.

Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. Hopover, p. 89. First gives 3 and 3 Frogs and Toads, then asks for complete reversal from 123_456 to 654_321.

[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. Tempo Books (Grosset & Dunlap), NY, 1973 (& 1978?? -- both dates are given -- I'm presuming the 1978 is a 2nd ptg or a reissue under a different imprint??). Reverse the numbers, pp. 117-118. Give the problem for n = 6 and a solution in 21 moves. For n even, the method gives a solution in n(n+1)/2, it is not shown that this is optimal, nor is a general method given for odd n.

[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. 13-hour clock, pp. 43-44. Case n = 12 considered in a circle can be done in 44 moves.

Joe Celko. Jumping frogs and the Dutch national flag. Abacus 4:1 (Fall 1986) 71-73. Cites Dudeney and gives the results.

Jim Howson. The Computer Weekly Book of Puzzlers. Computer Weekly Publications, Sutton, Surrey, 1988, unpaginated. [The material comes from his column which started in 1966, so an item may go back to then.] Prob. 54 -- same as the Scotts in Master Mind Pencil Puzzles.

5.R.5. FOX AND GEESE, ETC.

There are a number of similar games on different boards -- too many to describe completely here, so I will generally just cite extensive descriptions. See any of the main books on games mentioned at the beginning of 4.B, such as Bell or Falkener. The key feature is that one side has more, but weaker, pieces. These are sometimes called hunt games. The standard Fox and Geese is played on a 33 hole Solitaire board, with diagonal moves allowed. I have recently acquired but not yet read Murray's History of Board Games other than Chess which should have lots of material.

Gretti's Saga, late 12C. Mention of Fox and Geese. Also in Edward IV's accounts. ??NYS -- cited by Botermans et al, below.

Shackerley (or Schackerley or Shakerley) Marmion. A Fine Companion (a play). 1633. IN: The Dramatic Works of Shackerley Marmion; William Paterson, Edinburgh & H. Sotheran & Co., London, 1875. II, v, pp. 140-141. "..., let him sit in the shop ..., and play at fox and geese with the foreman, ....." Earliest English occurrence of fox-and-geese. Quoted by OED and cited by Fiske, below.

Richard Lovelace. To His Honoured Friend On His Game of Chesse-Play or To Dr. F. B. on his Book of Chesse. 1656?, published in his Posthume Poems, 1659. Lines 1-4. My edition of Lovelace notes that F. B. was Francis Beale, author of 'Royall Game of Chesse Play,' 1656. Lovelace died in 1658.

Sir, now unravell'd is the Golden Fleece,

Men that could onely fool at Fox and Geese,

Are new-made Polititians by the Book,

And can both judge and conquer with a look.

Henry Brooke. Fool of Quality. [A novel.] 1766-1768. Vol. I, p. 367. ??NYS -- quoted by Fiske, below. "Can you play at no kind of game, Master Harry?" "A little at fox-and-geese, madam."

Catel. Kunst-Cabinet. 1790.

Das Fuchs- und Hühnerspiel, pp. 51-52 & fig. 168 on plate VI. 11 chickens against one fox on a 4 x 4 board with all diagonals drawn, giving 16 + 9 playing points.

Das Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is the same game on the 33-hole solitaire board with 11 sheep and one wolf, no diagonals

Bestelmeier. 1801.

Item 83: Das Schaaf- und Wolfspiel. Same diagram and game as Catel, p. 52.

Item 833: Ein Belagerungspiel. 33 hole board with a fortress on one arm, with diagonals drawn.

Strutt. Op. cit. in 4.B.1. Fox and Geese. 1833: Book IV, chap. II, art. XIV, pp. 318-319. = Strutt-Cox, p. 258 & plate opp. p. 246. Fig. 107 (= plate opp. p. 246) shows the 33 hole board with its diagonals drawn.

Gomme. Op. cit. in 4.B.1. I 141-142 refers to Strutt and Micklethwaite.

Illustrated Boy's Own Treasury. 1860. Fox and Geese, pp. 406-407. 33 hole Solitaire board with diagonals drawn.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 320, p. 152: Fuchs und Gänse. Shows 33 hole solitaire board with diagonals drawn.

Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum for 1893, pp. 489-537. Pp. 874-877 describes: the Japanese game of Juroku Musashi (Sixteen Soldiers) with 16 men versus a general; the Chinese game of Shap luk kon tséung kwan (The sixteen pursue the commander); another Chinese game of Yeung luk sz' kon tséung kwan with 27 men against a commander (described by Hyde -- ?? I didn't see this); the Malayan game of Dam Hariman (Tiger Game), identical to the Hindu game of Mogol Putt'han (= Mogul Pathan (Mogul against Pathan)), similar to a Peruvian game of Solitario and the Mexican game of Coyote; the Siamese game of Sua ghin gnua (Tiger and Oxen) and the similar Burmese game of Lay gwet kyah, with three big tigers versus 11 or 12 little tigers; the Samoan game of Moo; the Hawaiian game of Konane; a similar Madagascarian game; the Hindu game of Pulijudam (Tiger Game) with three tigers versus 15 lambs.

Fiske. Op. cit. in 4.B.1. 1905. Fox-and-Geese, pp. 146-156 & 359, discusses the history of the game, especially as to whether it is identical to the old Norse game of Hnefatafl. On p. 359, he says that John of Salisbury (c1150) used 'vulpes' as the name of a game, but there is no indication of what it was. He says "the fox-and-geese board, in comparatively modern times, has begun to be used for games more or less different in their nature, especially for one called in England solitaire and in France "English solitaire", and for another, known in Spain and Italy as asalto (assalto), in French as assaut, in Danish as belejringsspel." He then surveys the various sources that he treated under Mérelles -- see 4.B.1 and 4.B.5 for details. He is not sure that Brunet is really describing the game in the Alfonso MS (op. cit. in 4.B.5 and below). He cites an 1855 Italian usage as Jeu de Renard or Giuoco della Volpe. In Come Posso Divertirmi? (Milan, 1901, pp. 231-233), it is said that the game is usually played with 17 geese rather than 13 -- Fiske notes that this assertion is of "some historical value, if it be true." Moulidars calls it Marelle Quintuple, quotes Maison des Jeux Académiques (Paris, 1668) for a story that it was invented by the Lydians and gives the game with 13 or 17 geese. Asalto has 2 men against 24. Fiske quotes Shackley Marmion, above, for the oldest English occurrence of fox-and-geese and then Henry Brooke, above. Fiske follows with German, Swedish and Icelandic (with 13 geese) references.

H. Parker. Ancient Ceylon. Op. cit. in 4.B.1, 1909. Pp. 580-583 & 585 describe four forms of The Leopards Game, with one tiger against seven leopards, three leopards against 15 dogs, two leopards against 24 cattle and one leopard against six cattle on a 12 x 12 board. The first two are played on a triangular board.

Robert Kanigel. The Man Who Knew Infinity. A Life of the Genius Ramanujan. (Scribner's, NY, 1991); Abacus (Little, Brown & Co. (UK)), London, 1992. Pp. 18 & 377: Ramanujan and his mother used to play the game with three tigers and fifteen goats on a kind of triangular board.

The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototypic Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913. See 4.B.5 for more details of this work. See below.

Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 147 says De Cercar La Liebre (Catch the Hare) occurs in the Alfonso MS and is the earliest example of a hunt game in European literature, but undoubtedly derived from an Arabic game of the Alquerque type -- I didn't see this when I briefly looked at the facsimile -- ??NYS. They say Murray has noted that hunt games are popular in Asia, but not in Africa, leading to the conjecture that they originated in Asia. They describe it on a 5 x 5 array of points with verticals and horizontals and some diagonals drawn, with one hare against 12 hunters.

Botermans et al. continue on pp. 148-155 to describe the following.

Shap Luk Kon Tseung Kwan (Sixteen Pursue the General) played on a 5 x 5 board like Catch the Hare with an extra triangle on one side and capturing by interception.

Yeung Luk Sz'Kon Tseung Kwan, seen in Nanking by Hyde and described by him in 1694, somewhat similar to the above, but with 26 rebels against a general. (??NYS)

Fox and Geese, mentioned in Gretti's Saga of late 12C and in Edward IV's accounts. They give a version called Lupo e Pecore from a 16C Venetian book, using a Solitaire board extended by three points on each arm, giving 45 points. They give a 1664 engraving showing Le Jeu du Roi which they say is a rather complex form of fox and geese, but looks like a four-handed game on a cross-shaped board with 7 x 5 arms on a 7 x 7 central square and 4 groups of 7 x 4 men.

Leopard games, from Southeast Asia, with a kind of triangular board. Len Choa, from Thailand, has a tiger against six leopards. Hat Diviyan Keliya, from Sri Lanka, has a tiger against seven leopards.

Tiger games, also from Southeast Asia, are similar to leopard games, but use an extended Alquerque board (as in Catch the Hare). Rimau (Tiger), from Malaysia, has 24 men versus a tiger and Rimau-Rimau (Tigers) is a version with two tigers versus 22 men.

Murray. 1913.

P. 347 cites a 1901 Indian book for 2 lions against 32 goats on a chessboard.

P. 371 cites a Soyat (North Asia) example (19C?) of Bouge-Shodra (Boar's Chess) with 2 boars against 24 calves on a chessboard.

Pp. 569 & 616-617 cite the Alfonso MS of 1283 for 'De cercar la liebre', played on a 5 x 5 board with 10, 11 or 12 men against a hare.

P. 585 shows Cott. 6 (c1275) of 8 pawns against a king on a chessboard.

Pp. 587 & 590 give Cott. 11 = K6: Le Guy de Alfins with king and 4 bishops against a king on a chessboard.

Pp. 589-590 shows K4 = CB249: Le Guy de Dames and No. 5 = K5: Le Guy de Damoyselles, which have 16 pawns against a king on a chessboard.

P. 617 discusses Fox and Geese, with 13, 15 or 17 geese against a fox on the solitaire board. Edward IV, c1470, bought "two foxis and 46 hounds". Murray says more elaborate forms exist and refers to Hyde and Fiske (see 4.B.1 and 5.F.1 for more on these), ??NYS.

Pp. 675 & 692 show CB258: Partitum regis Francorum with king and four pawns against king on the chessboard. It says the first side wins.

P. 758 describes a 16C Venetian board (then) at South Kensington (V&A??) with the Solitaire board for Fox and Geese and an enlarged board for Fox and Geese.

P. 857 mentions Fox and Geese in Iceland.

Family Friend 2 (1850) 59. Fox and geese. 4 geese against 1 fox on a chess board.

The Sociable. 1858. Fox and geese, p. 281. 17 geese against a fox on the solitaire board. Four men versus a king on the draughts board, saying the first side wins even allowing the king to be placed anywhere against the men who start on one side.

Stewart Culin. Korean Games, op. cit. in 4.B.5, 1895. Pp. 76-77 describes some games of this type, in particular a Japanese game called Yasasukari Musashi with 16 soldiers versus a general on a 5 x 5 board, taken from a 1714 (or 1712) Japanese book: Wa Kan san sai dzu e "Japanese, Chinese, Three Powers picture collection", published in Osaka.

Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and Sons, London, 1883, HB. Section 2593: Fox and Geese, p. 364. 33 hole Solitaire board with 17 geese against a fox. 4 geese against a fox on the chessboard. Says the geese should win in both cases.

Slocum. Compendium. Shows Solitaire and Solitaire & Tactic Board from Gamage's 1913 catalogue. Like Bestelmeier's 833, but without diagonals.

Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Games involving unequal forces, pp. 43-52. Discusses the following.

The Maharajah and the Sepoys. 1 against 16 on a chessboard.

Fox and Geese. Cites an Icelandic work of c1300 (probably Gretti's Saga?). 1 against 13 or 17 on a Solitaire board.

Lambs and Tigers, from India. 3 against 15.

Cows and Leopards, from SE Asia. 2 against 24.

Vultures and Crows, also called Kaooa, from India. 1 against 7 on a pentagram board.

The New Military Game of German Tactics, c1870. 2 against 24 on a Solitaire Board with a fortress, as in Bestelmeier.

Yuri I. Averbakh. Board games and real events. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 17-23. Notes that Murray believes hunt games evolved from war games, but he feels the opposite is true. He describes a Nepalese game of Baghachal with four tigers versus 20 goats -- this is Murray's 5.6.22. He corrects some of Murray's assertions about Boar Chess and describes other Tuvinian hunt games: Bull's Chess and Calves' Chess, probably borrowed from the Mongols. The latter has a three-in-a-row pattern and he wonders if there is some connection with morris or noughts and crosses (which he says is "played everywhere"). He mentions Cercar la Liebre from the Alfonso MS. Fox and Geese type games are mentioned in the Icelandic sagas as 'the fox game'. He describes several forms.

5.R.6. OCTAGRAM PUZZLE

One has an octagram and seven men. One has to place a man on a vacant point and then slide him to an adjacent vacant point, then do the same with the next man, ..., so as to cover seven of the points. The diagram is just an 8-cycle and is the same as the knight's connections on the 3 x 3 board, so the octagram puzzle is equivalent to the 7 knights problem mentioned in 5.F.1. Further, the 4 knights problem of 5.F.1 has the same 8-cycle, with men at alternate points of it.

Versions with different numbers of points.

5 points: Rohrbough.

7 points: Mittenzwey; Meyer.

9 points: Dudeney.

10 points: Bell & Cornelius; Hoffmann; Cohen; Williams; Toymaker; Rohrbough; Putnam.

13 points: Berkeley & Rowland.

Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pentalpha, p. 15. Says that a pentagram board occurs at Kurna, Egypt, c-1400 and that the solitaire game of Pentalpha is played in Crete. This has 9 men to be placed on the vertices and the intersections of the pentagram. Each man must be placed on a vacant point, then slid ahead two positions along one straight line. The intermediate point may be occupied, but the ending point must be unoccupied. Unfortunately we don't know if the Egyptian board was used for this game.

Pacioli. De Viribus. c1500.

Ff. 112r - 113v. .C(apitolo). LXVIII. D(e). cita ch' a .8. porti ch' cosa convi(e)ne arepararli (Chap. 68. Of a city with 8 gates which admits of division ??). = Peirani 158-160. Octagram puzzle with a complex story about a city with 8 gates and 7 disputing factions to be placed at the gates.

F. IVv. = Peirani 8. The Index gives the above as Problem 83. Problem 82: De .8. donne ch' sonno aun ballo et de .7. giovini quali con loro sa con pagnano (Of 8 ladies who are at a ball and of 7 youths who accompany them).

Schwenter. 1636. Part 2, exercise 36, pp. 149-150. Octagram.

Witgeest. Het Natuurlyk Tover-Boek. 1686. Prob. 4, pp. 224-225. Octagram, taken from Schwenter.

Les Amusemens. 1749. P. xxxiii.

Catel. Kunst-Cabinet. 1790. Das Achteck, pp. 12-13 & fig. 36 on plate II. The rules are not clearly described.

Bestelmeier. 1801. Item 290: Das Achteckspiel. Text copies part of Catel.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. Ff. 131-133 are an analysis of the heptagram puzzle.

Rational Recreations. 1824. Feat 34, pp. 161-164. Octagram.

Endless Amusement II. 1826? Prob. 28, pp. 203-204. = New Sphinx, c1840, pp. 137-138.

Nuts to Crack IV (1835), no. 194 -- part of a long section called Tricks upon Travellers.

Family Friend (Dec 1858) 359. Practical puzzles -- 1. I don't have the answer.

The Boy's Own Magazine 3 (1857) 159 & 192. Puzzle of the points.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles No. 6, pp. 396 & 436.

The Secret Out. 1859. To Place Seven Counters upon an Eight-Pointed Star, pp. 373-374.

J. J. Cohen, New York. Star puzzle. Advertising card for Star Soap, Schultz & Co., Zanesville, Ohio, Copyright May 1887. Reproduced in: Bert Hochberg; As advertised Puzzles from the collection of Will Shortz; Games Magazine 17:1 (No. 113) 10-13, on p. 11. Identical to pentalpha - see Bell & Cornelius above.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. IX: The reversi puzzle, pp. 8-10. Version with 13 cards in a circle and one can move ahead by any number of steps. If there are x cards and one moves ahead s steps, then x and s must have no common factor.

Hoffmann. 1893.

Chap. VI, pp. 267-268 & 280-281 = Hoffmann-Hordern, pp. 180-181, with photos.

No. 13. No name. Basic octagram puzzle. Photo on p. 181 shows: The Seven Puzzles, by W. & T. Darton, dated 1806-1811; a Tunbridge ware version dated 1825-1840; and Jeu de Zig-Zag, by M. D., Paris, 1891-1900.

No. 14. The "Okto" puzzle, pp. 268 & 281. Here the counters and points are coloured. Photo on p. 181 of The "Okto" Puzzle by McGaw, Stevenson & Orr, Ltd. for John Stewart, dated 1880-1895.

Chap. X, no. 8: Crossette, pp. 337 & 374-375 = Hoffmann-Hordern, pp. 229-230, with photo. 10 counters in a circle. Start anywhere and move ahead three. Photo on p. 230 shows The Mystic Seven, a seven counter version, by the Lord Roberts Workshops, 1914-1920.

Mittenzwey. 1895? Prob. 329, pp. 58 & 106; 1917: 329, pp. 52 & 101. Heptagram.

Dudeney. Problem 58: A wreath puzzle. Tit-Bits 33 (6 & 27 Nov 1897) 99 & 153. Complex nonagram puzzle involving moves in either direction and producing the original word again.

Clark. Mental Nuts. 1897, no. 54; 1904, no. 80; 1916, no. 69. A little puzzle. Usual octagram.

Benson. 1904. The eight points puzzle, pp. 250-251. c= Hoffmann, no. 13.

Slocum. Compendium. Shows the "Octo" Star Puzzle from Gamage's 1913 catalogue.

Williams. Home Entertainments. 1914.

Crossette, pp. 115-116. Ten points, advancing three places.

Eight points puzzle, pp. 120-121. Usual octagram.

"Toymaker". Top in Hole Puzzle. Work (23 Dec 1916) 200. 10 holes and one has to move to the third position and reverse the top in that hole.

Blyth. Match-Stick Magic. 1921. Crossing the points, pp. 83-84.

Hummerston. Fun, Mirth & Mystery. 1924.

The sacred seven, Puzzle no. 5, pp. 26 & 173. Octagram puzzle on the outer points of the diagram shown in 5.A.

The four rabbits, Puzzle no. 6, pp. 26 & 173. Using the octagram shown in 5.A, put black counters on locations 1 and 2 and white counters on 7 and 8. The object is to interchange the colours. This is like the 4 knights problem except the corresponding 8-cycle has men at positions 1, 2, 5, 6. He counts a sequence of steps by the same man as a move and hence solves it in 6 moves (comprising 16 steps).

Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Turning the tails, pp. 66-69. 8 coins in a circle, tails up. Count from a tail four ahead and reverse that coin. Get 7 heads up. Counting four ahead means that if you start at 1, you count 1, 2, 3, 4 and reverse 4.

King. Best 100. 1927. No. 64, pp. 26-27 & 54.

Rohrbough. Puzzle Craft. 1932.

Count 4, p. 6. 10 points on a circle, moving ahead 3. (= Rohrbough; Brain Resters and Testers; c1935, p. 21.)

Star Puzzle, p. 8 (= p. 10 of 1940s?). Consider the pentagram with its internal vertices. First puzzle is Pentalpha. Second is to place a counter and move ahead three positions. The object is to get four counters on the points, which is the same as the pentagram puzzle, moving one position.

Jerome S. Meyer. Fun-to-do. Op. cit. in 5.C. 1948. Prob. 18: Odd man out, pp. 27 & 184. Version with 7 positions in a circle and 6 men where one must place a man and then move him three places ahead.

Putnam. Puzzle Fun. 1978. No. 63: Ten card turnover, pp. 11 & 35. Ten face down cards in a circle. Mark a card, count ahead three and turnover.

5.R.7. PASSING OVER COUNTERS

The usual version is to have 8 counters in a row which must be converted to 4 piles of two, but each move must pass a counter over two others. Martin Gardner pointed out to me that the problem for 10, 12, 14, ... counters is easily reduced to that for 8. The problem is impossible for 2, 4, 6. There are many later appearances of the problem than given here. In describing solutions, 4/1 means move the 4th piece on top of the 1st piece.

There are trick solutions where a counter moves to a vacated space or even lands between two spaces. See: Mittenzwey; Haldeman-Julius; Hemme.

Berkeley & Rowland give a problem where each move must pass a counter over two piles. This makes the problem easier and it is solvable for any even number of counters ( 6, but it gives more solutions. See: Berkeley & Rowland; Wood; Indoor Tricks & Games; Putnam; Doubleday - 1.

One could also permit passing over one pile, which is solvable for any even number ( 4.

Mittenzwey, Double Five Puzzle, Hummerston, and Singmaster & Abbott deal with the problem in a circle and with piles to be left in specific locations.

Mittenzwey, Lucas and Putnam consider making piles of three by passing over 3, etc.

Kanchusen. Wakoku Chiekurabe. 1727. Pp. 38-39. Jukkoku-futatsu-koshi (Ten stones jumping over two). Ten counters, one solution.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has "The question of the shillings passing at each time over two or a certain number 8 is the least number Any number being given and any law of transit Dr Roget" The layout suggests that Roget had posed the general version. Adjacent is a diagram with a row of 10 counters and the first move 1 to 4 shown, but with some unclear later moves.

Endless Amusement II. 1826? Prob. 10, p. 195. 10 halfpence. One solution: 4/1 7/3 5/9 2/6 8/10. = New Sphinx, c1840, pp. 135-135.

Nuts to Crack II (1833), no. 122. 10 counters, identical to Endless Amusement II.

Nuts to Crack V (1836), no. 68. Trick of the eight sovereigns. Usual form.

Young Man's Book. 1839. P. 234. Ingenious Problem. 10 halfpence. Identical to Endless Amusement II.

Family Friend 2 (1850) 178 & 209. Practical Puzzle, No. VI. Usual form with eight counters or coins. One solution.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 2, p. 176 (1868: 187). Passing over coins. Gives two symmetric solutions.

Magician's Own Book. 1857. Prob. 34: The counter puzzle, pp. 277 & 300. Identical to Book of 500 Puzzles, prob. 34.

The Sociable. 1858. Prob. 16: Problem of money, pp. 291-292 & 308. Start with 10 half-dimes, says to pass over one, but solution has passing over two. One solution. = Book of 500 Puzzles, 1859, prob. 16, pp. 9-10 & 26.

Book of 500 Puzzles. 1859.

Prob. 16: Problem of money, pp. 9-10 & 26. As in The Sociable.

Prob. 34: The counter puzzle, pp. 91 & 114. Eight counters, two solutions given. Identical to Magician's Own Book.

The Secret Out. 1859. The Crowning Puzzle, p. 386. 'Crowning' is here derived from the idea of crowning in draughts or checkers. One solution: 4/1 6/9 8/3 2/5 7/10.

Boy's Own Conjuring Book. 1860.

Prob. 33: The counter puzzle, pp. 240 & 264. Identical to Magician's Own Book, prob. 34.

The puzzling halfpence, p. 342. Almost identical to The Sociable, prob. 16, with half-dimes replaced by halfpence.

Illustrated Boy's Own Treasury. 1860. Prob. 17, pp. 398 & 438. Same as prob. 34 in Magician's Own Book but only gives one solution.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 593, part 6, pp. 299-411: Sechs Knacknüsse. 10 counters, one solution.

Hanky Panky. 1872. Counter puzzle, p. 132. Gives two solutions for 8 counters and one for 10 counters.

Kamp. Op. cit. in 5.B. 1877. No. 12, p. 325.

Mittenzwey. 1880. Prob. 235-238, pp. 43-44 & 93-94; 1895?: 262-266, pp. 47-48 & 95-96; 1917: 262-266, pp. 43-44 & 91.

235 (262). Usual problem, with 10 counters. Two solutions.

--- (263). Added in 1895? Same with 8 counters. Two solutions.

236 (264). 12 numbered counters in a circle. Pass over two to leave six piles of two on the first six positions. Solution is misprinted in all editions!

237 (265). 12 counters in a circle. Pass over three to leave six piles of two, except the last move goes over six. The solution allows landing counters on vacated locations!

238 (266). 15 counters in a row. Pass over 3 to leave five piles of three. The solution allows landing a counter on a vacated location and landing a counter between two locations!!

Lucas. RM2. 1883. Les huit pions, pp. 139-140. Solves for 8, 10, 12, ... counters. Says Delannoy has generalized to the problem of mp counters to be formed into m piles (m ( 4) of p by passing over p counters.

[More generally, using one of Berkeley & Rowland's variations (see below), one can ask when the following problem is solvable: form a line of n = kp counters into k piles of p by passing over q [counters or piles]. Does q have to be ( p?]

Double Five Puzzle. c1890. ??NYS -- described by Slocum from his example. 10 counters in a circle, but the final piles must alternate with gaps, e.g. the final piles are at the even positions. This is also solvable for 4, 8, 12, 16, ..., and I conjectured it was only solvable for 4n or 10 counters -- it is easy to see there is no solution for 2 or 6 counters and my computer gave no solutions for 14 or 18. For 4, 8, 10 or 12 counters, one can also leave the final piles in consecutive locations, but there is no such solution for 6, 14, 16 or 18 counters. See Singmaster & Abbott, 1992/93, for the resolution of these conjectures.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles.

No. VII: The halma puzzle, pp. 6-7. Arrange the first ten cards of a suit in a row so that passing over two cards leaves five piles whose cards total 11 and are in the odd places. Arrangement is 7,6,3,4,5,2,1,8,9,10. Move 2 to 9, 4 to 7, 8 to 3, 6 to 5, 10 to 1.

No. VIII: Another version, p. 7. With the cards in order and passing over two piles, leave five piles of two. But this is so easy, he adds that one wants to leave as low a total as possible on the tops of the piles. He moves 7 to 10, 6 to 3, 4 to 9, 1 to 5, 2 to 8, leaving a total of 20.

[However, this is not minimal -- there are six solutions leaving 18 exposed, e.g. 1 to 4, 3 to 6, 7 to 10, 5 to 9, 2 to 8. For 6 cards, the minimum is 6, achieved once; for 8 cards, the minimum is 11, achieved 3 times; for 12 cards, the minimum is 27, achieved 10 times. For the more usual case of passing over two cards, the minimum for 8 cards is 15, achieved twice; for 10 cards, the minimum is 22, achieved 4 times, e.g. by 7 to 10, 5 to 2, 3 to 8, 1 to 4, 6 to 9; for 12 cards, the minimum is 31, achieved 6 times; for 14 cards, the minimum is 42, achieved 8 times. For passing over one pile, the minimum for 4 cards is 3, achieved once; the minimum for 6 cards is 7, achieved twice; the minimum for 8 cards is 13, achieved 3 times; for 10 cards, the minimum is 21, achieved 4 times; the minimum for 12 cards is 31, achieved 5 times. Maxima are obtained by taking mirror images of the minimal solutions.]

Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??) 751. 8 men, passing over two men each time. Notes that it can be extended to any even number of counters.

Clark. Mental Nuts. 1897, no. 63: Toothpicks; 1904, no. 83 & 1916, no. 70: Place 8 toothpicks in a row. One solution.

Parlour Games for Everybody. John Leng, Dundee & London, nd [1903 -- BLC], p. 32: The five pairs. 10 counter version, one solution.

Wehman. New Book of 200 Puzzles. 1908. P. 15: The counter puzzle and Problem of money. 8 and 10 counter versions, the latter using pennies. Two and one solutions.

Ahrens. MUS I. 1910. Pp. 15-17. Essentially repeats Lucas.

Manson. 1911. Decimal game, pp. 253-254. Ten rings on pegs. "Children are frequently seen playing the game out of doors with pebbles or other convenient articles."

Blyth. Match-Stick Magic. 1921. Straights and crosses, pp. 85-87. 10 matchsticks, one must pass over two of them. Two solutions, both starting with 4 to 1.

Hummerston. Fun, Mirth & Mystery. 1924.

The pairing puzzle, Puzzle no. 8, pp. 27 & 173. Essential 8 counters in a circle, with four in a row being white, the other four being black. Moving only the whites, and passing over two, form four piles of two.

Pairing the pennies, Puzzle no. 39, pp. 102 & 178. Ten pennies, one solution.

Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Marrying the coins, pp. 113-115. Ten coins or eight coins, passing over 2. Gives two solutions for 10, not noting that the case of 10 is immediately reduced to 8. Says there are several solutions for 8 and gives two.

Wood. Oddities. 1927. Prob. 45: Fish in the basket, pp. 39-40. 12 fish in baskets in a circle. Move a fish over two baskets, continuing moving in the same direction, to get get two fish in each of six baskets, in the fewest number of circuits.

Rudin. 1936. No. 121, pp. 43 & 103. 10 matches. Two solutions.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Straights and crosses, pp. 105-106. As in Blyth, 1926.

Indoor Tricks and Games. Success Publishing, London, nd [1930s??].

How to pair the pennies, p. 4. 8 pennies, one solution.

The ten rings. p. 4. 10 rings, passing over two piles, one solution.

Haldeman-Julius. 1937.

No. 91: Jumping pennies, pp. 11 & 25. Six pennies to be formed into two piles of three by jumping over three pennies each time. Solution has a trick move. Jump 1 to 5, 6 to 3 and 2 to 1/5, which gives the position: 6/3 4 2/1/5. He then says: "No. 4 jumps over 5, 1 and 2 -- then jumps back over 5, 1 and 2 and lands upon 3 and 6, ...." Since the rules are not clear about where a jumping piece can land, the trick move can be viewed as a legitimate jump to the vacant 6 position, then a legitimate move over the 2/1/5 pile and the now vacant 4 position onto the 6/3 pile. If the pennies are considered as a cycle, this trick is not needed.

No. 148: Half dimes, pp. 16 & 143. 10 half dimes, passing over one dime (i.e. two counters).

Sullivan. Unusual. 1947. Prob. 39: On the line. Ten pennies.

Doubleday - 1. 1969. Prob. 75: Money moves, pp. 91 & 170. Ten pennies. Jump over two piles. Says there are several solutions and gives one, which sometimes jumps over three or four pennies.

Putnam. Puzzle Fun. 1978.

No. 26: Pile up the coins, pp. 7 & 31. 12 in a row. Make four piles of three, passing over three coins each time.

No. 27: Pile 'em up again, pp. 7 & 31. 16 in a row. Make four piles of four, passing over four or fewer each time.

No. 60: Coin assembly, pp. 11 & 35. Ten in a row, passing over two each time.

No. 61: Alternative coin assembly, pp. 11 & 35. Ten in a row, passing over two piles each time.

David Singmaster, proposer; H. L. Abbott, solver. Problem 1767. CM 18:7 (1992) 207 & 19:6 (1993). Solves the general version of the Double Five Puzzle, which the proposer had not solved. One can leave the counters on even numbered locations if and only if the number of counters is a multiple of 4 or a multiple of 10. One can leave the counters in consecutive locations if and only if the number of counters is 4, 8, 10 or 12.

Heinrich Hemme. Email of 25 Feb 1999. Points out that the rules in the usual version should say that the counter must land on a pile of a single coin. This would eliminate the trick solutions given by Mittenzwey and Haldeman-Julius. Hemme says that without this rule, the problem is easy and can be solved for 4 and 6 counters!

5.S. CHAIN CUTTING AND REJOINING

The basic problem is to minimise the cost or effort of reforming a chain from some fragments.

Loyd. Problem 25: A brace of puzzles -- No. 25: The chain puzzle. Tit-Bits 31 (27 Mar 1897) & 32 (17 Apr 1897) 41. 13 lengths: 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 12. (Not in the Cyclopedia.)

Loyd. Problem 42: The blacksmith puzzle. Tit-Bits 32 (10 & 31 Jul & 21 Aug 1897) 273, 327 & 385. Complex problem involving 10 pieces of lengths from 3 to 23 to be joined.

Clark. Mental Nuts. 1897, no. 7 & 1904, no. 14: The chain question; 1916, no. 59: The chain puzzle. 5 pieces of 3 links to make into a single length.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 9:4 (Feb 1903) 390-391 & 9:5 (Mar 1903) 490-491. The five chains. 5 pieces of 3 links to make into a single length.

Pearson. 1907. Part II, no. 67, pp. 128 & 205. 5 pieces of 3 links to make into a single length.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He attributes such puzzles to Loyd (Tit-Bits prob. 25) and gives that problem.

Cecil H. Bullivant. Home Fun. T. C. & E. C. Jack, London, 1910. Part VI, Chap IV, No. 9: The broken chain, pp. 518 & 522. 5 3-link pieces into an open chain.

Loyd. The missing link. Cyclopedia, 1914, pp. 222 (no solution) (c= MPSL2, prob. 25, pp. 19 & 129). 6 5-link pieces into a loop.

Loyd. The necklace puzzle. Cyclopedia, 1914, pp. 48 & 345 (= MPSL1, prob. 47, pp. 45-46 & 138). 12 pieces, with large and small links which must alternate.

D. E. Smith. Number Stories. 1919. Pp. 119 & 143-144. 5 pieces of 3 links to make into one length.

Hummerston. Fun, Mirth & Mystery. 1924. Q.E.D. -- The broken chain, Puzzle no. 38, pp. 99 & 178. Pieces of lengths 2, 2, 3, 3, 4, 4, 6 to make into a closed loop.

Ackermann. 1925. Pp. 85-86. Identical to the Loyd example cited by Dudeney.

Dudeney. MP. 1926. Prob. 212: A chain puzzle, pp. 96 & 181 (= 536, prob. 513, pp. 211-212 & 408). 13 pieces, with large and small links which must alternate.

King. Best 100. 1927. No. 7, pp. 9 & 40. 5 pieces of three links to make into one length.

William P. Keasby. The Big Trick and Puzzle Book. Whitman Publishing, Racine, Wisconsin, 1929. A linking problem, pp. 161 & 207. 6 pieces comprising 2, 4, 4, 5, 5, 6 links to be made into one length.

5.S.1. USING CHAIN LINKS TO PAY FOR A ROOM

The landlord agrees to accept one link per day and the owner wants to minimise the number of links he has to cut. The solution depends on whether the chain is closed in a cycle or open at the ends. Some weighing problems in 7.L.2.c and 7.L.3 are phrased in terms of making daily payment, but these are like having the chain already in pieces. See the Fibonacci in 7.L.2.c.

New section. I recall that there are older versions.

Rupert T. Gould. The Stargazer Talks. Geoffrey Bles, London, 1944. A Few Puzzles -- write up of a BBC talk on 10 Jan 1939, pp. 106-113. 63 link chain with three cuts. On p. 106, he says he believes it is quite modern -- he first heard it in 1935. On p. 113, he adds a postscript that he now believes it first appeared in John O'London's Weekly (16 Mar 1935) ??NYS.

Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 3. Man has closed chain of 182 links and wants to stay 182 days. What is the minimum number of links to be opened?

Birtwistle. Math. Puzzles & Perplexities. 1971. Pp. 13-16. Begins with seven link open-ended bracelet. Then how big a bracelet can be dealt with using only two cuts? Gets 23. Then does general case, getting n + (n+1)(2n+1 - 1).

Angela Fox Dunn. Second Book of Mathematical Bafflers. Dover, 1983. Selected from Litton's Problematical Recreations, which appeared in 1959-1971. Prob. 26, pp. 28 & 176. 23 link case.

Howson. Op. cit. in 5.R.4. 1988. Prob. 30. Says a 23 link chain need only be cut twice, giving lengths 1, 1, 3, 6, 12, which make all values up to 23. Asks for three cuts in a 63 link chain and the maximum length chain one can deal with in n cuts.

5.T. DIVIDING A CAKE FAIRLY

Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 & 88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and the children get equal pieces. He makes two perpendicular diametrical cuts and then a circular cut around the middle. He seems to mean the adults get equal pieces and the children get equal pieces, not necessarily the same. But if the circular cut is at (2/2 of the radius, then the areas are all equal. Not clear where this should go -- also entered in 5.Q.

B. Knaster. Sur le problème du partage pragmatique de H. Steinhaus. Annales de la Société Polonaise de Mathématique 19 (1946) 228-230. Says Steinhaus proposed the problem in a 1944 letter to Knaster. Outlines the Banach & Knaster method of one cutting 1/n and each being allowed to diminish it -- last diminisher takes the piece. Also shows that if the valuations are different, then everyone can get > 1/n in his measure. Gives Banach's abstract formulations.

H. Steinhaus. Remarques sur le partage pragmatique. Ibid., 230-231. Says the problem isn't solved for irrational people and that Banach & Knaster's method can form a game.

H. Steinhaus. The problem of fair division. Econometrica 16:1 (Jan 1948) 101-104. This is a report of a paper given on 17 Sep. Gives Banach & Knaster's method.

H. Steinhaus. Sur la division pragmatique. (With English summary) Econometrica 17 (Supplement) (1949) 315-319. Gives Banach & Knaster's method.

Max Black. Critical Thinking. Prentice-Hall, Englewood Cliffs, (1946, ??NYS), 2nd ed., 1952. Prob. 12, pp. 12 & 432. Raises the question but only suggests combining two persons.

5.U. PIGEONHOLE RECREATIONS

van Etten. 1624. Prob. 89, part II, pp. 131-132 (not in English editions). Two men have same number of hairs. Also: birds & feathers, fish & scales, trees & leaves, flowers or fruit, pages & words -- if there are more pages than words on any page.

E. Fourrey. Op. cit. in 4.A.1, 1899, section 213: Le nombre de cheveux, p. 165. Two Frenchmen have the same number of hairs. "Cette question fut posée et expliquée par Nicole, un des auteurs de la Logique de Port-Royal, à la duchesse de Longueville." [This would be c1660.]

The same story is given in a review by T. A. A. Broadbent in MG 25 (No. 264) (May 1941) 128. He refers to MG 11 (Dec 1922) 193, ??NYS. This might be the item reproduced as MG 32 (No. 300) (Jul 1948) 159.

The question whether two trees in a large forest have the same number of leaves is said to have been posed to Emmanuel Kant (1724-1804) when he was a boy. [W. Lietzmann; Riesen und Zwerge im Zahlbereich; 4th ed., Teubner, Leipzig, 1951, pp. 23-24.] Lietzmann says that an oak has about two million leaves and a pine has about ten million needles.

Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 9, pp. 2-3 & 53. Two people in the world have the same number of hairs on their head.

Manuel des Sorciers. 1825. Pp. 84-85. ??NX Two men have the same number of hairs, etc.

Gustave Peter Lejeune Dirichlet. Recherches sur les formes quadratiques à coefficients et à indéterminées complexes. (J. reine u. angew. Math. (24 (1842) 291-371) = Math. Werke, (1889-1897), reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle to find good rational approximations. He doesn't give it a name. In later works he called it the "Schubfach Prinzip".

Illustrated Boy's Own Treasury. 1860. Arithmetical and Geometrical Problems, No. 34, pp. 430 & 434. Hairs on head.

Pearson. 1907. Part II, no. 51, pp. 123 & 201. "If the population of Bristol exceeds by two hundred and thirty-seven the number of hairs on the head of any one of its inhabitants, how many of them at least, if none are bald, must have the same number of hairs on their heads?" Solution says 474!

Dudeney. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 670-676 (= AM, pp. 137-141). Two people have same number of hairs.

Ahrens. A&N, 1918, p. 94. Two Berliners have same number of hairs.

Abraham. 1933. Prob. 43 -- The library, pp. 16 & 25 (12 & 113). All books have different numbers of words and there are more books than words in the largest book. (My copy of the 1933 ed. is a presentation copy inscribed 'For the Athenaeum Library No 43 p 16 R M Abraham Sept 19th 1933'.)

Perelman. FMP. c1935? Socks and gloves. Pp. 277 & 283-284. = FFF, 1957: prob. 25, pp. 41 & 43; 1977, prob. 27, pp. 53-54 & 56. = MCBF, prob. 27, pp. 51 & 54. Picking socks and gloves to get pairs from 10 pairs of brown and 10 pairs of black socks and gloves.

P. Erdös & G. Szekeres. Op. cit. in 5.M. 1935. Any permutation of the first n2 - 1 integers contains an increasing or a decreasing subsequence of length > n.

P. Erdös, proposer; M. Wachsberger & E. Weiszfeld, M. Charosh, solvers. Problem 3739. AMM 42 (1935) 396 & 44 (1937) 120. n+1 integers from first 2n have one dividing another.

H. Phillips. Question Time. Dent, London, 1937. Prob. 13: Marbles, pp. 7 & 179. 12 black, 8 red & 6 white marbles -- choose enough to get three of the same colour.

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. Pp. 148-149, prob. 6. Blind maid bringing stockings from a drawer of white and black stockings.

I am surprised that the context of picking items does not occur before Perelman, Phillips and Home Book.

Sullivan. Unusual. 1943. Prob. 18: In a dark room. Picking shoes and socks to get pairs.

H. Phillips. News Chronicle "Quiz" No. 3: Natural History. News Chronicle, London, 1946. Pp. 22 & 43. 12 blue, 9 red and 6 green marbles in a bag. Choose enough to have three of one colour and two of another colour.

H. Phillips. News Chronicle "Quiz" No. 4: Current Affairs. News Chronicle, London, 1946. Pp. 17 & 40. 6 yellow, 5 blue and 2 red marbles in a bag. Choose enough to have three of the same colour.

L. Moser, proposer; D. J. Newman, solver. Problem 4300 -- The identity as a product of successive elements. AMM 55 (1948) 369 & 57 (1950) 47. n elements from a group of order n have a a subinterval with product = 1.

Doubleday - 2. 1971.

In the dark, pp. 145-146. How many socks do you have to pick from a drawer of white and black socks to get two pairs (possibly different)?

Lucky dip, pp. 147-148. How many socks do you have to pick from a drawer of with many white and black socks to get nine pairs (possibly different)? Gives the general answer 2n+1 for n pairs. [Many means that the drawer contains more than n pairs.]

Doubleday - 3. 1972. In the dark, pp. 35-36. Four sweaters and 5, 12, 4, 9 socks of the same colours as the sweaters. Lights go out. He can only find two of the sweaters. How many socks must he bring down into the light to be sure of having a pair matching one of the sweaters?

5.V. THINK-A-DOT, ETC.

I managed to acquire one of these without instructions or packaging some years ago. Michael Keller provided an example complete with instructions and packaging. I have recently seen Dockhorn's article on variations of the idea. This is related to Binary Recreations, 7.M.

The device was produced by E.S.R., Inc. The box or instructions give an address of 34 Label St., Montclair, New Jersey, 07042, USA, but the company has long been closed. In Feb 2000, Jim McArdle wrote that he believed that this became the well known Edmund Scientific Co. (101 East Gloucester Pike, Barrington, New Jersey, 08007, USA; tel: 609-547 3488; email: scientifics@; web: ). But he later wrote that investigation of the manuals of DifiComp, one of their other products, reveals that there appears to be no connection. E.S.R. = Education Science Research. The inventors of DigiComp, as listed in the patent for it, are: Irving J. Lieberman, William H, Duerig and Charles D. Hogan, all of Montclair, and they were the founders of the company. The DigiComp manuals say Think-A-Dot was later invented by John Weisbacker. There is a website devoted to DigiComp which contains this material and/or pointers to related sites and has a DigiComp emulator: . has a Yahoo club called Friends of DigiComp. There is another website with the DigiComp manual: .

E.S.R. Instructions, 8pp, nd -- but box says ©1965. No patent number anywhere but leaflet says the name Think-A-Dot is trademarked.

E.S.R., Inc. Corporation. US trademark registration no. 822,770. Filed: 8 Dec 1965; registered: 24 Jan 1967. First used 23 Aug 1965. Expired. The US Patent and Trademark Office website entry says the owner is the company and gives no information about the inventor(s). The name has been registered for a computer game on 23 Jul 2002.

Benjamin L. Schwartz. Mathematical theory of Think-A-Dot. MM 40:4 (Sep 1967) 187-193. Shows there are two classes of patterns and that one can transform any pattern into any other pattern in the same class in at most 15 drops.

Ray Hemmings. Apparatus Review: Think-a-Dot. MTg 40 (1967) 45.

Sidney Kravitz. Additional mathematical theory of Think-A-Dot. JRM 1:4 (Oct 1968) 247-250. Considers problems of making ball emerge from one side and of viewing only the back of the game.

Owen Storer. A think about Think-a-dot. MTg 45 (Winter 1968) 50-55. Gives an exercise to show that any possible transformation can be achieved in at most 9 drops.

T. H. O'Beirne. Letter: Think-a-dot. MTg 48 (Autumn 1969) 13. Proves Steiner's (Storer?? - check) assertion about 9 drops and gives an optimal algorithm.

John A. Beidler. Think-A-Dot revisited. MM 46:3 (May 1973) 128-136. Answers a question of Schwarz by use of automata theory. Characterizes all minimal sequences. Suggests some generalized versions of the puzzle.

Hans Dockhorn. Bob's binary boxes. CFF 32 (Aug 1993) 4-6. Bob Kootstra makes boxes with the same sort of T-shaped switch present in Think-A-Dot, but with just one entrance. One switch with two exits is the simplest case. Kootstra makes a box with three switches and four exits along the bottom, and the successive balls come out of the exits in cyclic sequence. Using a reset connection between switches, he also makes a two switch, three exit, box.

Boob Kootstra. Box seven. CFF 32 (Aug 1993) 7. Says he has managed to design and make boxes with 5, 6, 7, 8 exits, again with successive balls coming out the exits in cyclic order, but he cannot see any general method nor a way to obtain solutions with a minimal number of movable parts (switches and reset levers). Further his design for 7 exits is awkward and the design of an optimal box for seven is posed as a contest problem.

5.W. MAKING THREE PIECES OF TOAST

This involves an old-fashioned toaster which does one side of two pieces at a time. An alternative version is frying steaks or hamburgers on a grill which holds two objects, assuming each side has to be cooked the same length of time. The problem is probably older than these examples.

Sullivan. Unusual. 1943. Prob. 7: For the busy housewife.

J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. P. 4 (26). Mentions problem and solution.

Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 40: Minute toast, pp. 18 & 93.

D. St. P. Barnard. 50 Daily Telegraph Brain-Twisters. 1985. Op. cit. in 4.A.4. Prob. 5: Well done, pp. 16, 80, 103-104. Grilling three steaks on a grill which only holds two. He complicates the problem in two ways: a) each side takes a minute to season before cooking; b) the steaks want to be cooked 4, 3, 2 minutes per side.

Edward Sitarski. When do we eat? CM 27:2 (Mar 2001) 133-135. Hamburgers which require time T per side. After showing that three hamburgers take 3T, he asks how long it will take to cook H hamburgers. Easily shows that it can be done in HT, except for H = 1, which takes 2T. Then remarks that this is an easy version of a scheduling problem -- in reality, the hamburgers would have different numbers of sides, there would be several grills and each hamburger would have different parts requiring different grills, but in a particular order!

5.W.1. BOILING EGGS

New section. These are essentially parodies of the Cistern Problem, 7.H.

McKay. Party Night. 1940. No. 28, p. 182. "An egg takes 3½ minutes to boil. How long should 12 eggs take?"

Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 88: A boiling problem, pp. 29 & 82. "If it takes 3½ minutes to boil 2 eggs, how long will it take to boil 4 eggs?"

John King, ed. John King 1795 Arithmetical Book. Published by the editor, who is the great-great-grandson of the 1795 writer, Twickenham, 1995. P. 161, the editor mentions "If a girl on a hilltop can see two miles, how far would two girls be able to see?"

5.X. COUNTING FIGURES IN A PATTERN

New section -- there must be older examples. There are two forms of such problems depending on whether one must use the lattice lines or just the lattice points.

For counting several shapes, see: Young World (c1960); Gooding (1994) in 5.X.1.

5.X.1. COUNTING TRIANGLES

Counting triangles in a pattern is always fraught with difficulties, so I have written a program to do this, but I haven't checked all the examples here.

Pearson. 1907. Part II.

No. 74: A triangle of triangles, p. 74. Triangular array with four on a side, but with all the altitudes also drawn. Gets 653 triangles of various shapes.

No. 75: Pharaoh's seal, pp. 75 & 174. Isosceles right triangles in a square pattern with some diagonals.

Anon. Prob. 76. Hobbies 31 (No. 791) (10 Dec 1910) 256 & (No. 794) (31 Dec 1910) 318. Make as many triangles as possible with six matches. From the solution, it seems that the tetrahedron was expected with four triangles, but many submitted the figure of a triangle with its altitudes drawn, but only one solver noted that this figure contains 16 triangles! However, if the altitudes are displaced to give an interior triangle, I find 17 triangles!!

Loyd. Cyclopedia. 1914. King Solomon's seal, pp. 284 & 378. = MPSL2, No. 142, pp. 100 & 165 c= SLAHP: Various triangles, pp. 25 & 91. How many triangles in the triangular pattern with 4 on a side? Loyd Sr. has this embedded in a larger triangle.

Collins. Book of Puzzles. 1927. The swarm of triangles, pp. 97-98. Same as Pearson No. 74. He says there are 653 triangles and that starting with 5 on a side gives 1196 and 10,000 on a side gives 6,992,965,420,382. When I gave August's problem in the Weekend Telegraph, F. R. Gill wrote that this puzzle with 5 on a side was given out as a competition problem by a furniture shop in north Lancashire in the late 1930s, with a three piece suite as a prize for the first correct solution.

Evelyn August. The Black-Out Book. Harrap, London, 1939. The eternal triangle, pp. 64 & 213. Take a triangle, ABC, with midpoints a, b, c, opposite A, B, C. Take a point d between a and B. Draw Aa, ab, bc, ca, bd, cd. How many triangles? Answer is given as 24, but I (and my program) find 27 and others have confirmed this.

Anon. Test your eyes. Mathematical Pie 7 (Oct 1952) 51. Reproduced in: Bernard Atkin, ed.; Slices of Mathematical Pie; Math. Assoc., Leicester, 1991, pp. 15 & 71 (not paginated - I count the TP as p. 1). Triangular pattern with 2 triangles on a side, with the three altitudes drawn. Answer is 47 'obtained by systematic counting'. This is correct. Cf Hancox, 1978.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. How many triangles, pp. 43 & 130. Take a pentagon and draw the pentagram inside it. In the interior pentagon, draw another pentagram. How many triangles are there? Answer is 85.

Young World. c1960. P. 57: One for Pythagoras. Consider a L-tromino. Draw all the midlines to form 12 unit squares. Or take a 4 x 4 square array and remove a 2 x 2 array from a corner. Now draw the two main diagonals of the 4 x 4 square - except half of one diagonal would be outside our figure. How many triangles and how many squares are present? Gives correct answers of 26 & 17.

J. Halsall. An interesting series. MG 46 (No. 355) (Feb 1962) 55-56. Larsen (below) says he seems to be the first to count the triangles in the triangular pattern with n on a side, but he does not give any proof.

Although there are few references before this point, the puzzle idea was pretty well known and occurs regularly. E.g. in the children's puzzle books of Norman Pulsford which start c1965, he gives various irregular patterns and asks for the number of triangles or squares.

J. E. Brider. A mathematical adventure. MTg (1966) 17-21. Correct derivation for the number of triangles in a triangle. This seems to be the first paper after Halsall but is not in Larsen.

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/12, pp. 23 & 75. Consider an isosceles right triangle with legs along the axes from (0,0) to (4,0) and (0,4). Draw the horizontals and verticals through the integer lattice points, except that the lines through (1,1) only go from the legs to this point and stop. Draw the diagonals through even-integral lattice points, e.g. from (2,0) to (0,2). How many triangles. Says he found 27, but his secretary then found 29. I find 29.

Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting triangles.

Item 3. Consider a Star of David with the diameters of its inner hexagon drawn. How many triangles are in it? Answer: 20, which I agree with.

Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. How many triangles are there? Answer: 150, however, there are only 124.

Item 5. Consider five squares, with their midlines and diagonals drawn, formed into a Greek cross. How many triangles are there? Answer: 104, but there are 120.

Doubleday - 2. 1971. Count down, pp. 127-128. How many triangles in the pentagram (i.e. a pentagon with all its diagonals)? He says 35.

Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Triangular, pp. 27 & 63. Count triangles in an irregular pattern.

[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. An unusual star, pp. 49-50. Consider a pentagram and draw lines from each star point through the centre to the opposite crossing point. How many triangles? They say 110.

[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. 1973. Op. cit. in 5.R.4.

Diamonds are forever, pp. 35-36. Hexagon with Star of David inside and another Star of David in the centre of that one. How many triangles? Answer is 76.

Count the triangles, pp. 55-56. Ordinary Greek cross of five squares, with all the diagonals and midlines of the five squares drawn. How many triangles> Answer is 104.

C. P. Chalmers. Note 3353: More triangles. MG 58 (No. 403) (Mar 1974) 52-54. How many triangles are determined by N points lying on M lines? (Not in Larsen.)

Nicola Davies. The 2nd Target Book of Fun and Games. Target (Universal-Tandem), London, 1974. Squares and triangles, pp. 18 & 119. Consider a chessboard of 4 x 4 cells. Draw all the diagonals, except the two main ones. How many squares and how many triangles?

Shakuntala Devi. Puzzles to Puzzle You. Op. cit. in 5.D.1. 1976. Prob. 136: The triangles, pp. 85 & 133. How many triangles in a Star of David made of 12 equilateral triangles?

Michael Holt. Figure It Out -- Book Two. Granada, London, 1978. Prob. 67, unpaginated. How many triangles in a Star of David made of 12 equilateral triangles?

Putnam. Puzzle Fun. 1978. No. 91: Counting triangles, pp. 12 & 37. Same as Doubleday - 2.

D. J. Hancox, D. J. Number Puzzles For all The Family. Stanley Thornes, London, 1978.

Puzzle 8, pp. 2 & 47. Draw a line with five points on it, say A, B, C, D, E, making four segments. Connect all these points to a point F on one side of the line and to a point G on the other side of the line, with FCG collinear. How many triangles are there? Answer is 24, which is correct.

Puzzle 53, pp. 24 & 54. Same as Anon.; Test Your Eyes, 1952. Answer is 36, but there are 47.

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984.

Problem 40, with Solution at the back of the book. Same as Doubleday - 2.

Problem 116, with Solution at the back of the book. Count the triangles in a 'butterfly' pattern.

Sue Macy. Mad Math. The Best of DynaMath Puzzles. Scholastic, 1987. (Taken from Scholastic's DynaMath magazine.) Shape Up, pp. 5 & 56.

Take a triangle, trisect one edge and join the points of trisection to the opposite vertex. How many triangles? [More generally, if one has n points on a line and joins them all to a vertex, there are 1 + 2 + ... + n-1 = n(n-1)/2 triangles.]

Take a triangle, join up the midpoints of the edges, giving four smaller triangles, and draw one altitude of the original triangle. How many triangles?

1980 Celebration of Chinese New Year Contest Problem No. 5; solution by Leroy F. Meyers. CM 17 (1991) 2 & 18 (1992) 272-273. n x n array of squares with all diagonals drawn. Find the number of isosceles right triangles. [Has this also been done in half the diagram? That is, how many isosceles right triangles are in the isosceles right triangle with legs going from (0,0) to (n,0) and (0,n) with all verticals, horizontals and diagonals through integral points drawn?]

Mogens Esrom Larsen. The eternal triangle -- a history of a counting problem. Preprint, 1988. Surveys the history from Halsall on. The problem was proposed at least five times from 1962 and solved at least ten times. I have sent him the earlier references.

Marjorie Newman. The Christmas Puzzle Book. Hippo (Scholastic Publications), London, 1990. Star time, pp. 26 & 117. Consider a Star of David formed from 12 triangles, but each of the six inner triangles is subdivided into 4 triangles. How many triangles in this pattern? Answer is 'at least 50'. I find 58.

Erick Gooding. Polygon counting. Mathematical Pie No. 131 (Spring 1994) 1038 & Notes, pp. 1-2. Consider the pentagram, i.e. the pentagon with its diagonals drawn. How many triangles, quadrilaterals and pentagons are there? Gets 35, 25, 92, with some uncertainty whether the last number is correct.

When F. R. Gill (See Pearson and Collins above) mentioned the problem of counting the triangles in the figure with all the altitudes drawn, I decided to try to count them myself for the figure with N intervals on each side. The theoretical counting soon gets really messy and I adapted my program for counting triangles in a figure (developed to verify the number found for August's problem). However, the number of points involved soon got larger than my simple Basic could handle and I rewrote the program for this special case, getting the answers of 653 and 1196 and continuing to N = 22. I expected the answers to be like those for the simpler triangle counting problem so that there would be separate polynomials for the odd and even cases, or perhaps for different cases (mod 3 or 4 or 6 or 12 or ??). However, no such pattern appeared for moduli 2, 3, 4 and I did not get enough data to check modulus 6 or higher. I communicated this to Torsten Sillke and Mogens Esrom Larsen. Sillke has replied with a detailed answer showing that the relevant modulus is 60! I haven't checked through his work yet to see if this is an empirical result or he has done the theoretical counting.

Heather Dickson, Heather, ed. Mind-Bending Challenging Optical Puzzles. Lagoon Books, London, 1999, pp. 40 & 91. Gives the version m = n = 4 of the following. I have seen other versions of this elsewhere, but I found the general solution on 4 Jul 2001 and am submitting it as a problem to AMM.

Consider a triangle ABC. Subdivide the side AB into m parts by inserting m-1 additional points. Connect these points to C. Subdivide the side AC into n parts by inserting n-1 additional points and connect them to B. How many triangles are in this pattern? The number is [m2n + mn2]/2. When m = n, we get n3, but I cannot see any simple geometric interpretation for this.

5.X.2. COUNTING RECTANGLES OR SQUARES

I have just seen M. Adams. There are probably earlier examples of these types of problems.

Anon. Prob. 63. Hobbies 30 (No. 778) (10 Sep 1910) 488 & 31 (No. 781) (1 Oct 1910) 2. How many rectangles on a 4 x 4 chessboard? Solution says 100, which is correct, but then says they are of 17 different types -- I can only get 16 types.

Blyth. Match-Stick Magic. 1921. Counting the squares, p. 47. Count the squares on a 4 x 4 chessboard made of matches with an extra unit square around the central point. The extra unit square gives 5 additional squares beyond the usual 1 + 4 + 9 + 16.

King. Best 100. 1927. No. 9, pp. 10 & 40. = Foulsham's, no. 5, pp. 6 & 10. 4 x 4 board with some diagonals yielding one extra square.

Loyd Jr. SLAHP. 1928. How many rectangles?, pp. 80 & 117. Asks for the number of squares and rectangles on a 4 x 4 board (i.e. a 5 x 5 lattice of points). Says answers are 1 + 4 + 9 + 16 and (1 + 2 + 3 + 4)2 and that these generalise to any size of board.

M. Adams. Puzzles That Everyone Can Do. 1931. o o

Prob. 37, pp. 22 & 134: 20 counter problem. Given the pattern of o o

20 counters at the tight, 'how many perfect squares are o o o o o o

contained in the figure.' This means having their vertices o o o o o o

at counters. There are surprisingly more than I expected. o o

Taking the basic spacing as one, one can have squares of o o

edge 1, (2, (5, (8, (13, giving 21 squares in all.

He then asks how many counters need to be removed in order to destroy all the squares? He gives a solution deleting six counters.

Prob. 217, pp. 83 & 162: Match squares. He gives 10 matches making a row of three equal squares and asks you to add 14 matches to form 14 squares. The answer is to make a 3 x 3 array of squares and count all of the squares in it.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Counting the squares, pp. 84-85. As in Blyth.

Indoor Tricks and Games. Success Publishing, London, nd [1930s??]. Square puzzle, p. 62. Start with a square and draw its diagonals and midlines. Join the midpoints of the sides to form a second level square inscribed in the first level original square. Repeat this until the 9th level. How many squares are there? Given answer is 16, but in my copy someone has crossed this out and written 45, which seems correct to me.

Meyer. Big Fun Book. 1940. No. 9, pp. 162 & 752. Draw four equidistant horizontal lines and then four equidistant verticals. How many squares are formed? This gives a 3 x 3 array of squares, but he counts all sizes of squares, getting 9 + 4 + 1 = 14. (Also in 7.AU.)

Foulsham's New Party Book. Foulsham, London, nd [1950s?]. P. 103: How many squares? 4 x 4 board with some extra diagonals giving one extra square.

Although there are few references before this point, the puzzle idea was pretty well known and occurs regularly in the children's puzzle books of Norman Pulsford which start c1965. He gives various irregular patterns and asks for the number of triangles or squares.

Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 140: A surprising answer, pp. 43 & 90. 4 x 4 chessboard with four corner cells deleted. How many rectangles are there?

Anon. Puzzle page: Strictly for squares. MTg 30 (1965) 48 & 31 (1965) 39 & 32 (1965) 39. How many squares on a chessboard? First solution gets S(8) = 1 + 4 + 9 + ... + 64 = 204. Second solution observes that there are skew squares if one thinks of the board as a lattice of points and this gives S(1) + S(2) + ... + S(8) = 540 squares.

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966.

Prob. 2/11, pp. 22 & 74. 4 x 4 array of squares bordered on two sides by bricks 1 x 2, 1 x 3, 2 x 1, 2 x 1. Count the squares and the rectangles. Gets 35 and 90.

Prob. 2/14, pp. 23 & 75. Pattern of squares making the shape of a person -- how many squares in it?

Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting squares.

Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. The solution says this has 30 squares. I get 31, but perhaps they weren't counting the whole figure. I have computed the total number of squares for an n x n array and get (2n3 + n2)/2 squares for n even and (2n3 + n2 -1)/2 squares for n odd.

Unnumbered item at lower right of p. 73. 4 x 4 array of squares with their diagonals drawn, except that the four corner squares have only one diagonal -- the one not pointing to an opposite corner -- and this reduces the number of squares by eight, agreeing with the given answer of 64.

Doubleday - 2. 1971. Bed of nails, pp. 129-130. 20 points in the form of a Greek cross with double-length arms (so that the axes are five times the width of the central square, or the shape is a 9-omino). How many squares can be located on these points? He finds 21.

W. Antony Broomhead. Note 3315: Two unsolved problems. MG 55 (No. 394) (Dec 1971) 438. Find the number of squares on an n x n array of dots, i.e. the second problem in MTg (1965) above, and another problem.

W. Antony Broomhead. Note 3328: Squares in a square lattice. MG 56 (No. 396) (May 1972) 129. Finds there are n2(n2 - 1)/12 squares and gives a proof due to John Dawes. Editorial note says the problem appears in: M. T. L. Bizley; Probability: An Intermediate Textbook; CUP, 1957, ??NYS. A. J. Finch asks the question for cubes.

Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Squares, pp. 26 & 63. Same as Briggs.

Nicola Davies. The 2nd Target Book of Fun and Games. 1974. See entry in 5.X.1.

Putnam. Puzzle Fun. 1978.

No. 107: Square the coins, pp. 17 & 40. 20 points in the form of a Greek cross made from five 2 x 2 arrays of points. How many squares -- including skew ones? Gets 21.

No. 108: Unsquaring the coins, pp. 17 & 40. How many points must be removed from the previous pattern in order to leave no squares? Gets 6.

5.X.3. COUNTING HEXAGONS

M. Adams. Puzzle Book. 1939. Prob. C.157: Making hexagons, pp. 163 & 190. The hexagon on the triangular lattice which is two units along each edge contains 8 hexagons. [It is known that the hexagon of side n contains n3 hexagons. I recently discovered this but have found that it is known, though I don't know who discovered it first.]

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 32, with Solution at the back of the book. Count the hexagons in the hexagon of side three on the triangular lattice. They get 27.

5.X.4. COUNTING CIRCLES

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/13, pp. 23 & 75. Pattern with hexagonal symmetry and lots of overlapping circles, some incomplete.

5.Y. NUMBER OF ROUTES IN A LATTICE

The common earlier form was to have the route spell a word or phrase from the centre to the boundaries of a diamond. I will call this a word diamond. Sometimes the phrase is a palindrome and one reads to the centre and then back to the edge. See Dudeney, CP, for analysis of the most common cases. I have seen such problems on the surface of a 3 x 3 x 3 cube. The problems of counting Euler or Hamiltonian paths are related questions, but dealt with under 5.E and 5.F.

New section -- in view of the complexity of the examples below, there must be older, easier, versions, but I have only found the few listed below. The first entry gives some ancient lattices, but there is no indication that the number of paths was sought in ancient times.

Roger Millington. The Strange World of the Crossword. M. & J. Hobbs, Walton-on-Thames, UK, 1974. (This seems to have been retitled: Crossword Puzzles: Their History and Cult for a US ed from Nelson, NY.)

On pp. 38-39 & 162, he gives the cabalistic triangle shown below and says it is thought to have been constructed from the opening letters of the Hebrew words Ab (Father), Ben (Son), Ruach Acadash (Holy Spirit). He then asks how many ways one can read ABRACADABRA in it, though there is no indication that the ancients did this. His answer is 1024 which is correct.

A B R A C A D A B R A

A B R A C A D A B R

A B R A C A D A B

A B R A C A D A

A B R A C A D

A B R A C A

A B R A C

A B R A

A B R

A B

A

On pp. 39-40 he describes and illustrates an inscription on the Stele of Moschion from Egypt, c300. This is a 39 x 39 square with a Greek text from the middle to the corner, e.g. like the example in the following entry. The text reads: ΟΥIΡIΔIΜΟΥΧIΩΝΥΓIΑΥΘΕIΥΤΟΝΠΟΔΑIΑΤΠΕIΑIΥ which means: Moschion to Osiris, for the treatment which cured his foot. Millington does not ask for the number of ways to read the inscription, which is 4 BC(38,19) = 14 13810 55200.

Curiosities for the Ingenious selected from The most authentic Treasures of E D C D E

Nature, Science and Art, Biography, History, and General Literature. D C B C D

(1821); 2nd ed., Thomas Boys, London, 1822. Remarkable epitaph, C B A B C

p. 97. Word diamond extended to a square, based on 'Silo Princeps Fecit', D C B C D

with the ts at the corners. An example based on 'ABCDEF' is shown E D C D E

at the right. Says this occurs on the tomb of a prince named Silo at the

entrance of the church of San Salvador in Oviedo, Spain. Says the epitaph can be read in 270 ways. I find there are 4 BC(16, 8) = 51490 ways.

In the churchyard of St. Mary's, Monmouth, is the gravestone of John Rennie, died 31 May 1832, aged 33 years. This has the inscription shown below. Further down the stone it gives his son's name as James Rennie. Apparently an N has been dropped to get a message with an odd number of letters. I have good photos. Nothing asks for the number of ways of reading the inscription. I get 4 BC(16,9) = 45760 ways.

eineRnhoJsJohnRenie

ineRnhoJsesJohnReni

neRnhoJseiesJohnRen

eRnhoJseiliesJohnRe

RnhoJseileliesJohnR

nhoJseilereliesJohn

hoJseilerereliesJoh

oJseilereHereliesJo

hoJseilerereliesJoh

nhoJseilereliesJohn

RnhoJseileliesJohnR

eRnhoJseiliesJohnRe

neRnhoJseiesJohnRen

ineRnhoJsesJohnReni

eineRnhoJsJohnRenie

Nuts to Crack I (1832), no. 200. The example from Curiosities for the Ingenious with 'SiloPrincepsFecit', but no indication of what is wanted -- perhaps it is just an amusing picture.

W. Staniforth. Letter. Knowledge 16 (Apr 1893) 74-75. Considers 1 2 3 4 5 6

"figure squares" as at the right. "In how many different ways may 2 3 4 5 6 7

the figures in the square be read from 1 to 11 consecutively?" 3 4 5 6 7 8

He computes the answers for the n x n case for the first few 4 5 6 7 8 9

cases and finds a recurrence. "Has such a series of numbers any 5 6 7 8 9 10

mathematical designation?" The editor notes that he doesn't 6 7 8 9 10 11

know.

J. J. Alexander. Letter. Knowledge 16 (May 1893) 89. Says Staniforth's numbers are the sums of the squares of the binomial coefficients BC(n, k), the formula for which is BC(2n, n). Editor say he has received more than one note pointing this out and cites a paper on such figure squares by T. B. Sprague in the Transactions of the Royal Society of Edinburgh -- ??NYS, no more details provided.

Loyd. Problem 12: The temperance puzzle. Tit-Bits 31 (2 & 23 Jan 1897) 251 & 307. Red rum & murder. = Cyclopedia, 1914, The little brown jug, pp. 122 & 355. c= MPSL2, no. 61, pp. 44 & 141. Word diamond based on 'red rum & murder', i.e. the central line is redrum&murder. He allows a diagonal move from an E back to an inner R and this gives 372 paths from centre to edge, making 3722 = 138,384 in total.

Dudeney. Problem 57: The commercial traveller's puzzle. Tit-Bits 33 (30 Oct & 20 Nov 1897) 82 & 140. Number of routes down and right on a 10 x 12 board. Gives a general solution for any board.

Dudeney. A batch of puzzles. The Royal Magazine 1:3 (Jan 1899) 269-274 & 1:4 (Feb 1899) 368-372. A "Reviver" puzzle. Complicated pattern based on 'reviver'. 544 solutions.

Dudeney. Puzzling times at Solvamhall Castle. London Magazine 7 (No. 42) (Jan 1902) 580-584 & 8 (No. 43) (Feb 1902) 53-56. The amulet. 'Abracadabra' in a triangle with A at top, two B's below, three R's below that, etc. Answer: 1024. = CP, 1907, No. 38, pp. 64-65 & 190. CF Millington at beginning of this section.

Dudeney. CP. 1907.

Prob. 30: The puzzle of the canon's yeoman, pp. 55-56 & 181-182. Word diamond based on 'was it a rat I saw'. Answer is 63504 ways. Solution observes that for a diamond of side n+1, with no diagonal moves, the number of routes from the centre to an edge is 4(2n-1) and the number of ways to spell the phrase is this number squared. Analyses four types with the following central lines: A - 'yoboy'; B - 'level'; C - 'noonoon'; D - 'levelevel'.

In A, one wants to spell 'boy', so there are 4(2n-1) solutions.

In  B, one wants to spell 'level' and there are [4(2n-1)]2 solutions.

In C, one wants to spell 'noon' and there are 8(2n-1) solutions.

In D, one wants to spell 'level' and there are complications as one can start and finish at the edge. He obtains a general formula for the number of ways. Cf Loyd, 1914.

Prob. 38: The amulet, pp. 64-65 & 190. See: Dudeney, 1902.

Pearson. 1907. Part II: A magic cocoon, p. 147. Word diamond based on 'cocoon', so the central line is noocococoon. Because one can start at the non-central Cs, and can go in as well as out, I get 948 paths. He says 756.

Loyd. Cyclopedia. 1914. Alice in Wonderland, pp. 164 & 360. = MPSL1, no. 109, pp. 107 & 161-162. Word diamond based on 'was it a cat I saw'. Cf Dudeney, 1907.

Dudeney. AM. 1917.

Prob. 256: The diamond puzzle, pp. 74 & 202. Word diamond based on 'dnomaidiamond'. This is type A of his discussion in CP and he states the general formula. 252 solutions.

Prob. 257: The deified puzzle, pp. 74-75 & 202. Word diamond based on 'deifiedeified'. This is type D in CP and has 1992 solutions. He says 'madamadam' gives 400 and 'nunun' gives 64, while 'noonoon' gives 56.

Prob. 258: The voter's puzzle, pp. 75 & 202. Word diamond built on 'rise to vote sir'. Cites CP, no. 30, for the result, 63504, and the general formula.

Prob. 259: Hannah's puzzle, pp. 75 & 202. 6 x 6 word square based on 'Hannah' with Hs on the outside, As adjacent to the Hs and four Ns in the middle. Diagonal moves allowed. 3468 ways.

Wood. Oddities. 1927. Prob. 44: The amulet problem, p. 39. Like the original ABRACADABRA triangle, but with the letters in reverse order.

Collins. Book of Puzzles. 1927. The magic cocoon puzzle, pp. 169-170. As in Pearson.

Loyd Jr. SLAHP. 1928. A strolling pedagogue, pp. 38 & 97. Number of routes to opposite corner of a 5 x 5 array of points.

D. F. Lawden. On the solution of linear difference equations. MG 36 (No. 317) (Sep 1952) 193-196. Develops use of integral transforms and applies it to find that the number of king's paths going down or right or down-right from (0, 0) to (n, n) is Pn(3) where Pn(x) is the Legendre polynomial.

Leo Moser. King paths on a chessboard. MG 39 (No. 327) (Feb 1955) 54. Cites Lawden and gives a simpler proof of his result Pn(3).

Anon. Puzzle Page: Check this. MTg 36 (1964) 61 & 27 (1964) 65. Find the number of king's routes from corner to corner when he can only move right, down or right-down. Gets 48,639 routes on 8 x 8 board.

Ripley's Puzzles and Games. 1966. P. 32. Word diamond laid out differently so A A A

one has to read from one side to the opposite side. Rotating by 45o, one gets B B 

the pattern at the right for edge three. One wants the number of ways to C C C

read ABCDEF. In general, when the first line of As has n positions, D D 

the total number of ways to reach the first row is n. For each successive E E E

row, the total number is alternately twice the number for the previous row less F F 

twice the end term of that row or just twice the the number for the previous

row. In our example with n = 3, the number of ways to reach the second row is 4 = 2x3 - 2x1. The number of ways to reach the third row is 8 = 2x4. The number of ways to reach the fourth row is 12 = 2x8 - 2x2, then we get 24 = 2 x 12; 36 = 2x24 - 2x6. It happens that the first n end terms are the central binomial coefficients BC(2k,k), so this is easy to calculate. I find the total number of routes, for n = 2, 3, ..., 7, is 4, 18, 232, 1300, 6744, 33320, the last being the desired and given answer for the given problem.

Pál Révész. Op. cit. in 5.I.1. 1969. On p. 27, he gives the number of routes for a king moving forward on a chessboard and a man moving forward on a draughtsboard.

Putnam. Puzzle Fun. 1978. No. 8: Level - level, pp. 3 & 26. Form a wheel of 16 points labelled LEVELEVELEVELEVE. Place 4 Es inside, joined to two consecutive Vs and the intervening L. Then place a V in the middle, joined to these four Es. How many ways to spell LEVEL? He gets 80, which seems right.

5.Z. CHESSBOARD PLACING PROBLEMS

See MUS I 285-318, some parts of the previous chapter and the Appendix in II 351-360. See also 5.I.1, 6.T.

There are three kinds of domination problems.

In strong domination, a piece dominates the square it is on.

In weak domination, it does not, hence more pieces may be needed to dominate the board.

Non-attacking domination is strong domination with no piece attacking another. Graph theorists say the pieces are independent. This also may require more pieces than strong domination, but it may require more or fewer pieces than weak domination.

The words 'guarded' or 'protected' are used for weak domination, but 'unguarded' or 'unprotected' may mean either strong or non-attacking domination.

Though these results seem like they must be old, the ideas seem to have originated with the eight queens problem, c1850, (cf 5.I.1) and to have been first really been attacked in the late 19C. There are many variations on these problems, e.g. see Ball, and I will not attempt to be complete on the later variations. In recent years, this has become a popular subject in graph theory, where the domination number, γ(G), is the size of the smallest strongly dominating set on the graph G and the independent domination number, i(G), is the size of the smallest non-attacking (= independent) dominating set.

Mario Velucchi has a web site devoted to the non-dominating queens problem and related sites for similar problems. See: and .

Ball. MRE, 3rd ed., 1896, pp. 109-110: Other problems with queens. Says: "Captain Turton has called my attention to two other problems of a somewhat analogous character, neither of which, as far as I know, has been hitherto published, ...." These ask for ways to place queens so as to attack as few or as many cells as possible -- see 5.Z.2.

Ball. MRE, 4th ed., 1905, pp. 119-120: Other problems with queens; Extension to other chess pieces. Repeats above quote, but replaces 'hitherto published' by 'published elsewhere', extends the previous text and adds the new section.

Ball. MRE, 5th ed., 1911. Maximum pieces problem; Minimum pieces problem, pp. 119-122. [6th ed., 1914 adds that Dudeney has written on these problems in The Weekly Dispatch, but this is dropped in the 11th ed. of 1939.] Considerably generalizes the problems. On the 8 x 8 board, the maximum number of non-attacking kings is 16, queens is 8, bishops is 14 [6th ed., 1914, adds there are 256 solutions], knights is 32 with 2 solutions and rooks is 8 with 88 solutions [sic, but changed to 8! in the 6th ed.]. The minimum number of pieces to strongly dominate the board is 9 kings, 5 queens with 91 inequivalent solutions [the 91 is omitted in the 6th ed., since it is stated later], 8 bishops, 12 knights, 8 rooks. The minimum number of pieces to weakly dominate the board is 5 queens, 10 bishops, 14 knights, 8 rooks.

Dudeney. AM. 1917. The guarded chessboard, pp. 95-96. Discusses different ways pieces can weakly or non-attackingly dominate n x n boards.

G. P. Jelliss. Multiple unguard arrangements. Chessics 13 (Jan/Jun 1982) 8-9. One can have 16 kings, 8 queens, 14 bishops, 32 knights or 8 rooks non-attackingly placed on a 8 x 8 board. He considers mixtures of pieces -- e.g. one can have 10 kings and 4 queens non-attacking. He tries to maximize the product of the numbers of each type in a mixture -- e.g. scoring 40 for the example.

5.Z.1. KINGS

Ball. MRE, 4th ed., 1905. Other problems with queens; Extension to other chess pieces, pp. 119-120. Says problems have been proposed for k kings on an n x n, citing L'Inter. des math. 8 (1901) 140, ??NYS.

Gilbert Obermair. Denkspiele auf dem Schachbrett. Hugendubel, Munich, 1984. Prob. 27, pp. 29 & 58. 9 kings strongly, and 12 kings weakly, dominate an 8 x 8 board.

5.Z.2. QUEENS

Here the graph is denoted Qn, but I will denote γ(Qn) by γ(n) and i(Qn) by i(n).

Murray. Pp. 674 & 691. CB249 (c1475) shows 16 queens weakly dominating an 8 x 8 board, but the context is unclear to me.

de Jaenisch. Op. cit. in 5.F.1. Vol. 3, 1863. Appendice, pp. 244-271. Most of this is due to "un de nos anciens amis, Mr de R***". Finds and describes the 91 ways of placing 5 queens so as to non-attackingly dominate the 8 x 8 board. Then considers the n x n board for n = 2, ..., 7 with strong and non-attacking domination. Up through 5, he gives the number of pieces being attacked in each solution which allows one to determine the weak solutions. For n < 6, he gets the answers in the table below, but for n = 6, he gets 21 non-attacking solutions instead of 17?.

Ball. MRE, 3rd ed., 1896. Other problems with queens, pp. 109-110. "Captain [W. H.] Turton has called my attention to two other problems of a somewhat analogous character, neither of which, as far as I know, has been hitherto published, or solved otherwise than empirically." The first is to place 8 queens so as to strongly dominate the fewest squares. The minimum he can find is 53. (Cf Gardner, 1999.) The second is to place m queens, m ( 5, so as to strongly dominate as many cells as possible. With 4 queens, the most he can find is 62.

Dudeney. Problem 54: The hat-peg puzzle. Tit-Bits 33 (9 & 30 Oct 1897) 21 & 82. Problem involves several examples of strong domination by 5 queens on an 8 x 8 board leading to a non-attacking domination. He says there are just 728 such. This = 8 x 91. = Anon. & Dudeney; A chat with the Puzzle King; The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. = AM; 1917; pp. 93-94 & 221.

Ball. MRE, 4th ed., 1905, loc. cit. in 5.Z.1. Extends 3rd ed. by asking for the minimum number of queens to strongly dominate a whole n x n board. Says there seem to be 91 ways of having 5 non-attacking queens on the 8 x 8, citing L'Inter. des Math. 8 (1901) 88, ??NYS.

Ball. MRE, 5th ed., 1911, loc. cit. in 5.Z. On pp. 120-122, he considers queens and states the minimum numbers of queens required to strongly dominate the board and the numbers of inequivalent solutions for 2 x 2, 3 x 3, ..., 7 x 7, citing the article cited in the 4th ed. and Jaenisch, 1862, without a volume number. For n = 7, he gives the same unique solution for strongly dominating as for non-attacking dominating. [In the 6th ed., this is corrected and he says it is a solution.] He says Jaenisch also posed the question of the minimum number of non-attacking queens to dominate the board and gives the numbers and the number of inequivalent ways for the 4 x 4, .., 8 x 8, except that he follows Jaenisch in stating that there are 21 solutions on the 6 x 6. [This is changed to 17 in the 6th ed.]

Dudeney. AM. 1917. Loc. cit. in 5.Z. He uses 'protected' for 'weakly', but he seems to copy the values for 'strongly' from Jaenisch or Ball. His 'not protected' seems to mean 'non-attacking'. However, some values are different and I consequently am very uncertain as to the correct values??

Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 24-25, he shows 5 queens are sufficient to strongly dominate the board and says this is minimal.

Below, min. denotes the minimum number of queens to dominate and no. is the number of inequivalent ways to do so.

STRONG WEAK NON-ATTACKING

n min. no. min. no. min. no.

1 1 1 0 0 1 1

2 1 1 2 2 1 1

3 1 1 2 5 1 1

4 2 3 2 3 3 2

5 3 37 3 15 3 2

6 3 1 4 (2 4 17?

7 4 4 5? 4 1

8 5 (150 5 (41 5 91

Rodolfo Marcelo Kurchan, proposer; Henry Ibstedt & proposer, solver. Prob. 1738 -- Queens in space. JRM 21:3 (1989) 220 & 22:3 (1989) 237. How many queens are needed to strongly dominate an n x n x n cubical board? For n = 3, 4, ..., 9, the best known numbers are: 1, 4, 6, 8, 14, 20, 24. The solution is not clear if these are minimal, but it seems to imply this.

Martin Gardner. Chess queens and maximum unattacked cells. Math Horizons (Nov 1999). Reprinted in Workout, chap. 34. Considers the problem of Turton described in Ball, 3rd ed, above: place 8 queens on an 8 x 8 board so as to strongly dominate the fewest squares. That is, leave the maximum number of unattacked squares. More generally, place k queens on an n x n board to leave the maximum number of unattacked squares. He describes a simple problem by Dudeney (AM, prob. 316) and recent work on the general problem. He cites Velucchi, cf below, who provides the following table of maximum numbers of unattacked cells and number of solutions for the maximum. I'm not sure if some of these are still only conjectured.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Max 0 0 0 1 3 5 7 11 18 22 30 36 47 56 72 82 97

Sols 0 0 0 25 1 3 38 7 1 1 2 7 1 4 3 1

Mario Velucchi has a web site devoted to the non-dominating queens problem and related sites for similar problems. See: and .

A. P. Burger & C. M. Mynhardt. Symmetry and domination in queens graphs. Bull. Inst. Combinatorics Appl. 29 (May 2000) 11-24. Extends results to n = 30, 45, 69, 77. Summarizes the field, with 14 references, several being earlier surveys. The table below gives all known values. It will be seen that the case n = 4k + 1 seems easiest to deal with. The values separated by strokes, /, indicate cases where the value is one of the two given values, but it is not known which.

n 4 5 6 7 8 9 10 11 12 13 14 15 16 17

γ(n) 2 3 3 4 5 5 5 5 6 7 7/8 8/9 8/9 9

i(n) 3 3 4 4 5 5 5 5 7 7 8 9 9 9

n 18 19 21 25 29 30 31 33 37 41 45 49 53 57

γ(n) 9 10 11 13 15 15 16 17 19 21 23 25 27 29

i(n) 11 13 17 23

n 61 69 77

γ(n) 31 35 39

5.Z.3. BISHOPS

Dudeney. AM. 1917. Prob. 299: Bishops in convocation, pp. 89 & 215. There are 2n ways to place 2n-2 bishops non-attackingly on an n x n board. At loc. cit. in 5.Z, he says that for n = 2, ..., 8, there are 1, 2, 3, 6, 10, 20, 36 inequivalent placings.

Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 25-26, he shows the maximum number of non-attacking bishops on one colour is 7 and there are 16 ways to place them.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 17, pp. 23 & 50. 8 bishops strongly, and 10 bishops weakly, dominate the 8 x 8 board.

5.Z.4. KNIGHTS

Ball. MRE, 4th ed., 1905. Loc. cit. in 5.Z.1. Says questions as to the maximum number of non-attacking knights and minimum number to strongly dominate have been considered, citing L'Inter. des math. 3 (1896) 58, 4 (1897) 15-17 & 254, 5 (1898) 87 [5th ed. adds 230-231], ??NYS.

Dudeney. AM. 1917. Loc. cit. in 5.Z. Notes that if n is odd, one can have (n2+1)/2 non-attacking knights in one way, while if n is even, one can have n2/2 in two equivalent ways.

Irving Newman, proposer; Robert Patenaude, Ralph Greenberg and Irving Newman, solvers. Problem E1585 -- Nonattacking knights on a chessboard. AMM 70 (1963) 438 & 71 (1964) 210-211. Three easy proofs that the maximum number of non-attacking knights is 32. Editorial note cites Dudeney, AM, and Ball, MRE, 1926, p. 171 -- but the material is on p. 171 only in the 11th ed., 1939.

Gardner. SA (Oct 1967, Nov 1967 & Jan 1968) c= Magic Show, chap. 14. Gives Dudeney's results for the 8 x 8. Golomb has noted that Greenberg's solution of E1585 via a knight's tour proves that there are only two solutions. For the k x k board, k = 3, 4, ..., 10, the minimal number of knights to strongly dominate is: 4, 4, 5, 8, 10, 12, 14, 16. He says the table may continue: 21, 24, 28, 32, 37. Gives numerous examples.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 16, pp. 21 & 47. 14 knights are necessary for weak domination of the 8x8 board.

E. O. Hare & S. T. Hedetniemi. A linear algorithm for computing the knight's domination number of a k x n chessboard. Technical report 87-May-1, Dept. of Computer Science, Clemson University. 1987?? Pp. 1-2 gives the history from 1896 and Table 2 on p. 13 gives their optimal results for strong domination on k x n boards, 4 ( k ( 9, k ( n ( 12 and also for k = n = 10. For the k x k board, k = 3, ..., 10, they confirm the results in Gardner.

Anderson H. Jackson & Roy P. Pargas. Solutions to the N x N knight's cover problem. JRM 23:4 (1991) 255-267. Finds number of knights to strongly dominate by a heuristic method, which finds all solutions up through N = 10. Improves the value given by Gardner for N = 15 to 36 and finds solutions for N = 16, ..., 20 with 42, 48, 54, 60, 65 knights.

5.Z.5. ROOKS

É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 128, pp. 220-223. Determines the number of inequivalent placings of n nonattacking rooks on an n x n board in general and gives values for n ( 12. For n = 1, ..., 8, there are 1, 1, 2, 7, 23, 115, 694, 5282 inequivalent ways.

Dudeney. AM. 1917. Loc. cit. at 5.Z. Notes there are n! ways to place n non-attacking rooks and asks how many of these are inequivalent. Gives values for n = 1, ..., 5. AM prob. 296, pp. 88 & 214, is the case n = 4.

D. F. Holt. Rooks inviolate. MG 58 (No. 404) (Jun 1974) 131-134. Uses Burnside's lemma to determine the number of inequivalent solutions in general, getting Lucas' result in a more modern form.

5.Z.6. MIXTURES

Ball. MRE, 5th ed., 1911. Loc. cit. in 5.Z. P. 122: "There are endless similar questions in which combinations of pieces are involved." 4 queens and king or queen or bishop or knight or rook or pawn can strongly dominate 8 x 8.

King. Best 100. 1927.

No. 77, pp. 30 & 57. 4 queens and a rook strongly dominate 8 x 8.

No. 78, pp. 30 & 57. 4 queens and a bishop strongly dominate 8 x 8.

5.AA. CARD SHUFFLING

New section. I have been meaning to add this sometime, but I have just come across an expository article, so I am now starting. The mathematics of this gets quite formidable. See 5.AD for a somewhat related topic.

A faro, weave, dovetail or perfect (riffle) shuffle starts by cutting the deck in half and then interleaving the two halves. When the deck has an even number of cards, there are two ways this can happen -- the original top card can remain on top (an out shuffle) or it can become the second card of the shuffled deck (an in shuffle). E.g. if our deck is 123456, then the out shuffle yields 142536 and the in shuffle yields 415263. Note that removing the first and last cards converts an out shuffle on 2n cards to an in shuffle on 2n-2 cards. When the deck has an odd number of cards, say 2n+1, we cut above or below the middle card and shuffle so the top of the larger pile is on top, i.e. the larger pile straddles the smaller. If the cut is below the middle card, we have piles of n+1 and n and the top card remains on top, while cutting above the middle card leaves the bottom card on bottom. Removing the top or bottom card leaves an in shuffle on 2n cards.

Monge's shuffle takes the first card and then alternates the next cards over and under the resulting pile, so 12345678 becomes 86421357.

At G4G2, 1996, Max Maven gave a talk on some magic tricks based on card shuffling and gave a short outline of the history. The following is an attempt to summarise his material. The faro shuffle, done by inserting part of the deck endwise into the other part, but not done perfectly, began to be used in the early 18C and a case of cheating using this is recorded in 1726. The riffle shuffle, which is the common American shuffle, depends on mass produced cards of good quality and began to be used in the mid 19C. However, magicians did not become aware of the possibilities of the perfect shuffle until the mid 20C, despite the early work of Stanyans C. O. Williams and Charles T. Jordan in the 1910s.

Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, pp. 78-85: Of the combinations of the cards. This describes a shuffle, where one takes the top two cards, then puts the next two cards on top, then the next three cards underneath, then the next two on top, then the next three underneath. For ten cards 1234567890, it produces 8934125670, a permutation of order 7. Tables of the first few repetitions are given for 10, 24, 27 and 32 cards, having orders 7, 30, 30, 156.

The Secret Out. 1859. Permutation table, pp. 394-395 (UK: 128-129). Describes Hooper's shuffle for ten cards.

Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. XV, 1884: 214-222. Discusses Monge's shuffle and its period.

John Nevil Maskelyne. Sharps and Flats. 1894. ??NYS -- cited by Gardner in the Addendum of Carnival. "One of the earliest mentions". Called the "faro dealer's shuffle".

Ahrens. MUS I. 1910. Ein Kartenkunststück Monges, pp. 152-145. Expresses the general form of Monge's shuffle and finds its order for n = 1, 2, ..., 10. Mentions the general question of finding the order of a shuffle.

Charles T. Jordan. Thirty Card Mysteries. The author, Penngrove, California, 1919 (??NYS), 2nd ed., 1920 (?? I have copy of part of this). Cited by Gardner in the Addendum to Carnival. First magician to apply the shuffle, but it was not until late 1950s that magicians began to seriously use and study it. The part I have (pp. 7-10) just describes the idea, without showing how to perform it. The text clearly continues to some applications of the idea. This material was reprinted in The Bat (1948-1949).

Frederick Charles Boon. Shuffling a pack of cards and the theory of numbers. MG 15 (1930) 17-20. Considers the Out shuffle and sees that it relates to the order of 2 (mod 2n+1) and gives some number theoretic observations on this. Also considers odd decks.

J. V. Uspensky & M. A. Heaslet. Elementary Number Theory. McGraw-Hill, NY, 1939. Chap. VIII: Appendix: On card shuffling, pp. 244-248. Shows that an In shuffle of a deck of 2n cards takes the card in position i to position 2i (mod 2n+1), so the order of the permutation is the exponent or order of 2 (mod 2n+1), which is 52 when n = 26. [Though not discussed, this shows that the order of the Out shuffle is the order of 2 (mod 2n-1), which is only 8 for n = 26. And the order of a shuffle of 2n+1 cards is the order of 2 (mod 2n+1).] Monge's shuffle is more complex, but leads to congruences (mod 4n+1) and has order equal to the smallest exponent e such that 2e ( ±1 (mod 4n+1), which is 12 for n = 26.

T. H. R. Skyrme. A shuffling problem. Eureka 7 (Mar 1942) 17-18. Describes Monge's shuffle with the second card going under or over the first. Observes that in the under shuffle for an even number of cards, the last card remains fixed, while the over shuffle for an odd number of cards also leaves the last card fixed. By appropriate choice, one always has the n-th card becoming the first. Finds the order of the shuffle essentially as in Uspensky & Heaslet. Makes some further observations.

N. S. Mendelsohn, proposer and solver. Problem E792 -- Shuffling cards. AMM 54 (1947) 545 ??NYS & 55 (1948) 430-431. Shows the period of the out shuffle is at most 2n-2. Editorial notes cite Uspensky & Heaslet and MG 15 (1930) 17-20 ??NYS.

Charles T. Jordan. Trailing the dovetail shuffle to its lair. The Bat (Nov, Dec 1948; Jan, Feb, Mar, 1949). ??NYS -- cited by Gardner. I have No. 59 (Nov 1948) cover & 431-432, which reprints some of the material from his book.

Paul B. Johnson. Congruences and card shuffling. AMM 63 (1956) 718-719. ??NYS -- cited by Gardner.

Alexander Elmsley. Work in Progress. Ibidem 11 (Sep 1957) 222. He had previously coined the terms 'in' and 'out' and represented them by I and O. He discovers and shows that to put the top card into the k-th position, one writes k-1 in binary and reads off the sequence of 1s and 0s, from the most significant bit, as I and O shuffles. He asks but does not solve the question of how to move the k-th card to the top -- see Bonfeld and Morris.

Alexander Elmsley. The mathematics of the weave shuffle, The Pentagram 11:9 (Jun 1957) 70-71; 11:10 (Jul 1957) 77-79; 11:11 (Aug 1957) 85; 12 (May 1958) 62. ??NYR -- cited by Gardner in the bibliography of Carnival, but he doesn't give the Ibidem reference in the bibliography, so there may be some confusion here?? Morris only cites Pentagram.

Solomon W. Golomb. Permutations by cutting and shuffling. SIAM Review 3 (1961) 293-297. ??NYS -- cited by Gardner. Shows that cuts and the two shuffles generate all permutations of an even deck. However, for an odd deck of n cards, the two kinds of shuffles can be intermixed and this only changes the cyclic order of the result. Since cutting also only changes the cyclic order, the number of possible permutations is n times the order of the shuffle.

Gardner. SA (Oct 1966) = Carnival, chap. 10. Defines the in and out shuffles as above and gives the relation to the order of 2. Notes that it is easier to do the inverse operations, which consist of extracting every other card. Describes Elmsley's method. Addendum says no easy method is known to determine shuffles to bring the k-th card to the top.

Murray Bonfeld. A solution to Elmsley's problem. Genii 37 (May 1973) 195-196. Solves Elmsley's 1957 problem by use of an asymmetric in-shuffle where the top part of the deck has 25 cards, so the first top card becomes second and the last two cards remain in place. (If one ignores the bottom two cards this is an in-shuffle of a 50 card deck.)

S. Brent Morris. The basic mathematics of the faro shuffle. Pi Mu Epsilon Journal 6 (1975) 86-92. Obtains basic results, getting up to Elmsley's work. His reference to Gardner gives the wrong year.

Israel N. Herstein & Irving Kaplansky. Matters Mathematical. 1974; slightly revised 2nd ed., Chelsea, NY, 1978. Chap. 3, section 4: The interlacing shuffle, pp. 118-121. Studies the permutation of the in shuffle, getting same results as Uspensky & Heaslet.

S. Brent Morris. Faro shuffling and card placement. JRM 8:1 (1975) 1-7. Shows how to do the faro shuffle. Gives Elmsley's and Bonfeld's results.

Persi Diaconis, Ronald L. Graham & William M. Kantor. The mathematics of perfect shuffles. Adv. Appl. Math. 4 (1983) 175-196. ??NYS.

Steve Medvedoff & Kent Morrison. Groups of perfect shuffles. MM 60:1 (1987) 3-14. Several further references to check.

Walter Scott. Mathematics of card sharping. M500 125 (Dec 1991) 1-7. Sketches Elmsley's results. States a peculiar method for computing the order of 2 (mod 2n+1) based on adding translates of the binary expansion of 2n+1 until one obtains a binary number of all 1s. The number of ones is the order a and the method is thus producing the smallest a such that 2a-1 is a multiple of 2n+1.

John H. Conway & Richard K. Guy. The Book of Numbers. Copernicus (Springer-Verlag), NY, 1996. Pp. 163-165 gives a brief discussion of perfect shuffles and Monge's shuffle.

5.AB. FOLDING A STRIP OF STAMPS

É. Lucas. Théorie des Nombres. Gauthier-Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. P. 120.

Exemple II -- La bande de timbres-poste. -- De combien de manières peut-on replier, sur un seul, une bande de p timbres-poste?

Exemple III -- La feuille de timbres-poste. -- De combien de manières peut-on replir, sur un seul, une feuille rectangulaire de pq timbres-poste?

"Nous ne connaissons aucune solution de ces deux problèmes difficiles proposés par M. Em. Lemoine."

M. A. Sainte-Laguë. Les Réseaux (ou Graphes). Mémorial des Sciences Mathématiques, fasc. XVIII. Gauthier-Villars, Paris, 1926. Section 62: Problème des timbres-poste, pp. 39-41. Gets some basic results and finds the numbers for a strip of n, n = 1, 2, ..., 10 as: 1, 2, 6, 16, 50, 144, 448, 7472, 17676, 41600.

Jacques Devisme. Contribution a l'étude du problème des timbres-poste. Comptes-Rendus du Deuxième Congrès International de Récréation Mathématique, Paris, 1937. Librairie du "Sphinx", Bruxelles, 1937, pp. 55-56. Cites Lucas (but in the wrong book!) and Sainte-Laguë. Studies the number of different forms of the result, getting numbers: 1, 2, 3, 8, 18, 44, 115, 294, 783.

5.AC. PROPERTIES OF THE SEVEN BAR DIGITAL DISPLAY

┌─┐ 2

The seven bar display, in the form of a figure 8, as at the right, is │ │ 1 3

now the standard form for displaying digits on calculators, clocks, etc. ├─┤ 4

This lends itself to numerous problems of a combinatorial/numerical │ │ 5 7

New Section. └─┘ 6

For reference, we number the seven bars in the reverse-S pattern

shown. We can then refer to a pattern by its binary 7-tuple or its decimal equivalent. E.g. the number one is displayed by having bars 3 and 7 on, which gives a binary pattern 1000100 corresponding to decimal 68. NOTE that there is some ambiguity with the 6 / 9. Most versions use the upper / lower bar for these, i.e. 1101111 / 1111011, but the bar is sometimes omitted, giving 1001111 / 1111001. I will assume the first case unless specified.

I have been interested in these for some time for several reasons. First, my wife has such a clock on her side of the bed and she often has a glass of water in front of it, causing patterns to be reversed. At other times the clock has been on the floor upside down, causing a different reversal of patterns. Second, segments often fail or get stuck on and I have tried to analyse which would be the worst segment to fail or get stuck. As an example, the clock in my previous car went from 16:59 to 15:00. Third, I have analysed which segment(s) in a clock are used most/least often.

Birtwistle. Calculator Puzzle Book. 1978. Prob. 35: New numbers, pp. 26-27 & 83. Asks for the number of new digits one can make, subject to their being connected and full height. Says it is difficult to determine when these are distinct -- e.g. calculators differ as to the form of their 6s and 9s -- so he is not sure how to count, but he gives 22 examples. I find there are 55 connected, full-height patterns.

Gordon Alabaster, proposer & Robert Hill, solver. Problem 134.3 -- Clock watching. M500 134 (Aug 1993) 17 & 135 (Oct 1993) 14-15. Proposer notes that one segment of the units digit of the seconds on his station clock was stuck on, but that the sequence of symbols produced were all proper digits. Which segment was stuck? Asks if there are answers for 2, ..., 6 segments stuck on. Solver gives systematic tables and discusses problems of how to determine which segment(s) are stuck and whether one can deduce the correct time when the stuck segments are known.

Martin Watson. Email to NOBNET, 17 Apr 2000 08:17:32 PDT [NOBNET 2334]. Observes that the 10 digits have a total of 49 segments and asks if they can be placed on a 4 x 5 square grid. He calls these forms 'digigrams'. He had been unable to find a solution but Leonard Campbell has found 5 distinct solutions, though they do no differ greatly. He has the pieces and some discussion on his website: . Dario Uri [22 Apr 2000 14:44:35 +0200] found two extra solutions, but Rick Eason [22 Apr 2000 09:37: -0400] also found these, but points out that these have an error due to misreading the lattice which gives the two bars of the 1 being parallel instead of end to end. Eason's program also found the 5 solutions.

5.AD. STACKING A DECK TO PRODUCE A SPECIAL EFFECT

New section. This refers to the process of arranging a deck of cards or a stack of coins so that dealing it by some rule produces a special effect. In many cases, this is just inverting the permutation given by the rule and the Josephus problem (7.B) is a special case. Other cases involve spelling out the names of cards, etc.

Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Alternate heads, pp. 61-63. Stack of eight coins. Place one on the table and the next on the bottom of the stack. The sequence of placed coins is to alternate heads and tails. How do you arrange the stack? Answer is HHTHHTTT. This is the same process as counting out by 2s -- see 7.B.

Doubleday - 2. 1971. Heads and tails, pp. 105-106. Same as Blyth, but with six coins and solution HTTTHH.

5.AE. REVERSING CUPS

New section. There are several versions of this and they usually involve parity. The basic move is to reverse two of the cups. The classic problem seems to be to start with UDU and produce DDD in three moves. A trick version is to demonstrate this several times to someone and then leave him to start from DUD. Another easy problem is to leave three cups as they were after three moves. This is equivalent to a 3 x 3 array with an even number in each row and column -- see 6.AO.2. These problems must be much older than I have, but the following are the only examples I have yet noted.

Anonymous. Social Entertainer and Tricks (thus on spine, but running title inside is New Book of Tricks). Apparently a compilation with advertisements for Johnson Smith (Detroit, Michigan) products, c1890?. P. 38a: Bottoms up. Given UDU, produce DDD in three moves.

Young World. c1960. P. 39: Water switch. Full and empty glasses: FFFEEE. Make them alternately full and empty in one move.

Putnam. Puzzle Fun. 1978.

No. 3: Tea for three, pp. 1 & 25. Cups given as UDU. Produce DDD in three moves.

No. 16: Glass alignment, pp. 5 & 28. Six cups arranged UUUDDD. Produce an alternating row. He gets UDUDUD in three moves. I can get DUDUDU in four moves.

5.AF. SPOTTING DICE

New section. In the early 1980s, I asked Richard Guy what was the 'standard' configuration for a die and later asked Ray Bathke if he used a standard pattern. Assuming opposite sides add to seven there are two handednesses. But also the spot pattern of the two, three and six has two orientations, giving 16 different patterns of die. Ray said that when he furnished dice with games, some customers had sent them back because they weren't the same. Within about three years, I had obtained examples of all sixteen patterns! Indeed, I often found several patterns in a single batch from one manufacturer. Ray Bathke also pointed out that the small dice that come from the oriental games have the two arranged either horizontally or vertically rather than diagonally, giving another 16 patterns. I have only obtained five of these, but with both handednesses included. I used this idea in one of my Brain Twisters, cf below.

Since the 2, 3 and 6 faces all meet at a corner, one has just to describe this corner. The 2, 3, 6 can be clockwise around the corner or anti-clockwise. Note that 236 is clockwise if and only if 132 is clockwise. The position of the 2 and 3 can be described by saying whether the pattern points toward or away from the corner. If we place the 2 upward, then 6 will be a vertical face and we can describe it by saying whether the lines of three spots are vertical or horizontal. Guy told me a system for describing a die, but it's not in Winning Ways and I've forgotten it, so I'll invent my own.

We write the sequence 236 if 236 is arranged clockwise at the 236 corner and we write 263 otherwise. When looked at cornerwise, with the 2 on top, the pattern of the 2 may appear vertical or horizontal. We write 2 when it is vertical and 2 when it is horizontal. (For oriental dice, the 2 will appear on a diagonal and can be indicated by 2 or 2. If we now rotate the cube to bring the 3 on top, its pattern will appear either vertical or horizontal and we write 3 or 3. Putting the 2 back on top, the 6 face will be upright and the lines of three spots will be either vertical or horizontal, which we denote by 6 or 6.

David Singmaster. Dicing around. Weekend Telegraph (16 Dec 1989). = Games & Puzzles No. 15 (Jun 1995) 22-23 & 16 (Jul 1995) 43-44. How many dice are there? Describes the normal 16 and mentions the other 16.

Ian Stewart. The lore and lure of dice. SA (Nov 1997) ??. He asserts that the standard pattern has 132 going clockwise at a corner, except that the Japanese use the mirror-image version in playing mah-jongg. His picture has both 2 and 3 toward the 236 corner and the 6 being vertical, i.e. in pattern 236. He discusses crooked dice of various sorts and that the only way to make all values from 1 to 12 equally likely is to have 123456 on one die and 000666 on the other.

Ricky Jay. The story of dice. The New Yorker (11 Dec 2000) 90-95.

5.AG. RUBIK'S CUBE AND SIMILAR PUZZLES

I have previously avoided this as being too recent to be covered in a historical work, but it is now old enough that it needs to be covered, and there are some older references. Much of the history is given in my Notes on Rubik's Cube and my Cubic Circular. Jaap Scherphuis has sent me a file of puzzle patents and several dozen of them could be entered here, but I will only enter older or novel items. Scherphuis's file has about a dozen patents for the 4 x 4 x 4 and 5 x 5 x 5 cubes! See Section 5.A for predecessors of the idea. However, this Section will mostly deal with puzzles where pieces are permuted without having any empty places, so these are generally permutation puzzle.

5.AG.1. RUBIK'S CUBE

New section. Much to be added.

Richard E. Korf. Finding optimal solutions to Rubik's Cube using pattern databases. Proc. Nat. Conf. on Artificial Intelligence (AAAI-97), Providence, Rhode Island, Jul 1997, pp. 700-705. Studies heuristic methods of finding optimal solutions of the Cube. Claims to be the first to find optimal solutions for random positions of the Cube -- but I think others such as Kociemba and Reid were doing it up to a decade earlier. For ten random examples, he found optimal solutions took 16 moves in one case, 17 moves in three cases, 18 moves in six cases, from which he asserts the median optimal solution length seems to be 18. He uses the idea of axial moves and obtains the lower bound of 18 for God's Algorithm, as done in my Notes in 1980. Cites various earlier work in the field, but only one reference to the Cube literature.

Richard E. Korf & Ariel Felner. Disjoint pattern database heuristics. Artificial Intelligence 134 (2002) 9-22. Discusses heuristic methods of solving the Fifteen Puzzle, Rubik's Cube, etc. Asserts the median optimal solution length for the Cube is only 18. Seems to say one of the problems in the earlier paper took a couple of weeks running time, but improved methods of Kociemba and Reid can find optimal solutions in about an hour.

5.AG.2. HUNGARIAN RINGS, ETC.

New section. Much to be added.

William Churchill. US Patent 507,215 -- Puzzle. Applied: 28 May 1891; patented: 24 Oct 1893. 1p + 1p diagrams. Two rings of 22 balls, intersecting six spaces apart.

Hiester Azarus Bowers. US Patent 636,109 -- Puzzle. Filed: 16 Aug 1899; patented 31 Oct 1899. 2pp + 1p diagrams. 4 rotating discs which overlap in simple lenses.

Ivan Moscovich. US Patent 4,509,756 -- Puzzle with Elements Transferable Between Closed-loop Paths. Filed: 18 Dec 1981; patented: 9 Apr 1985. Cover page + 3pp + 2pp diagrams. Two rings of 18 balls, each stretched to have two straight sections with semicircular ends. The rings cross in four places, at the ends of the straight sections, so adjacent crossing points are separated by two balls. I'm not sure this was ever produced. Mentions three circular rings version, but there each pair of rings only overlaps in two places so this is a direct generalization of the Hungarian Rings.

David Singmaster. Hungarian Rings groups. Bull. Inst. Math. Appl. 20:9/10 (Sep/Oct 1984) 137-139. [The results were stated in Cubic Circular 5 & 6 (Autumn & Winter 1982) 9-10.] An article by Philippe Paclet [Des anneaux et des groupes; Jeux et Stratégie 16 (Aug/Sep 1982) 30-32] claimed that all puzzles of two rings have groups either the symmetric or the alternating group on the number of balls. This article shows this is false and determines the group in all cases. If we have rings of size m, n and the intersections are distances a, b apart on the two rings. Then the group, G(m, n, a, b) is the symmetric group on m+n-2 if mn is even and is the alternating group if mn is odd; except that G(4, 4, 1, 1) is the exceptional group described in R. M. Wilson's 1974 paper: Graph puzzles, homotopy and the alternating group -- cited in Section 5.A under The Fifteen Puzzle -- and is also the group generated by two adjacent faces on the Rubik Cube acting on the six corners on those faces; and except that G(2a, 2b, a, b) keeps antipodal pairs at antipodes and hence is a subgroup of the wreath product Z2 wr Sa+b-1, with three cases depending on the parities of a and b.

Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of Computer Science and Statistics, University of Rhode Island, Jan 1994. Top-Spin has a cycle of 20 pieces and a small turntable which permits inverting a section of four pieces. After developing the group theory and doing the Fifteen Puzzle and the Missing Link, he shows the state space of Top-Spin is S20.

6. GEOMETRIC RECREATIONS

6.A. PI

This is too big a topic to cover completely. The first items should be consulted for older material and the general history. Then I include material of particular interest. See also 6.BL which has some formulae which are used to compute π. I have compiled a separate file on the history of π.

Augustus De Morgan. A Budget of Paradoxes. (1872); 2nd ed., edited by D. E. Smith, (1915), Books for Libraries Press, Freeport, NY, 1967.

J. W. Wrench Jr. The evolution of extended decimal approximations to π. MTr 53 (Dec 1960) 644-650. Good survey with 55 references, including original sources.

Petr Beckmann. A History of π. The Golem Press, Boulder, Colorado, (1970); 2nd ed., 1971.

Lam Lay-Yong & Ang Tian-Se. Circle measurements in ancient China. HM 13 (1986) 325-340. Good survey of the calculation of π in China.

Dario Castellanos. The ubiquitous π. MM 61 (1988) 67-98 & 148-163. Good survey of methods of computing π.

Joel Chan. As easy as pi. Math Horizons 1 (Winter 1993) 18-19. Outlines some recent work on calculating π and gives several of the formulae used.

David Singmaster. A history of π. M500 168 (Jun 1999) 1-16. A chronology. (Thanks to Tony Forbes and Eddie Kent for carefully proofreading and amending my file.)

Aristophanes. The Birds. -414. Lines 1001-1005. In: SIHGM I 308-309. Refers to 'circle-squarers', possibly referring to the geometer/astronomer Meton.

E. J. Goodwin. Quadrature of the circle. AMM 1 (1894) 246-247.

House Bill No. 246, Indiana Legislature, 1897. "A bill for an act introducing a new mathematical truth ..." In Edington's paper (below), p. 207, and in several of the newspaper reports.

(Indianapolis) Journal (19 Jan 1897) 3. Mentions the Bill in the list of bills introduced.

Die Quadratur des Zirkels. Täglicher Telegraph (Indianapolis) (20 Jan 1897) ??. Surveys attempts since -2000 and notes that Lindemann and Weierstrass have shown that the problem is impossible, like perpetual motion.

A man of 'genius'. (Indianapolis) Sun (6 Feb 1897) ??. An interview with Goodwin, who says: "The astronomers have all been wrong. There's about 40,000,000 square miles on the surface of this earth that isn't here." He says his results are revelations and gives several rules for the circle and the sphere.

Mathematical Bill passed. (Indianapolis) Journal (6 Feb 1897) 5. "This is the strangest bill that has ever passed an Indiana Assembly." Gives whole text of the Bill.

Dr. Goodwin's theaorem (sic) Resolution adopted by the House of Representatives. (Indianapolis) News (6 Feb 1897) 4. Gives whole text of the Bill.

The Mathematical Bill Fun-making in the Senate yesterday afternoon -- other action. (Indianapolis) News (13 Feb 1897) 11. "The Senators made bad puns about it, ...." The Bill was indefinitely postponed.

House Bills in the Senate. (Indianapolis) Sentinel (13 Feb 1897) 2. Reports the Bill was killed.

(No heading??) (Indianapolis) Journal (13 Feb 1897) 3, col. 4. "... indefinitely postponed, as not being a subject fit for legislation."

Squaring the circle. (Indianapolis) Sunday Journal (21 Feb 1897) 9. Says Goodwin has solved all three classical impossible problems. Says π = 3.2, using the fact that (2 = 10/7, giving diagrams and a number of rules.

My thanks to Underwood Dudley for locating and copying the above newspaper items.

C. A. Waldo. What might have been. Proc. Indiana Acad. Science 26 (1916) 445-446.

W. E. Edington. House Bill No. 246, Indiana State Legislature, 1897. Ibid. 45 (1935) 206-210.

A. T. Hallerberg. House Bill No. 246 revisited. Ibid. 84 (1975) 374-399.

Manuel H. Greenblatt. The 'legal' value of pi, and some related mathematical anomalies. American Scientist 53 (Dec 1965) 427A-434A. On p. 427A he tries to interpret the bill and obtains three different values for π.

David Singmaster. The legal values of pi. Math. Intell. 7:2 (1985) 69-72. Analyses Goodwin's article, Bill and other assertions to find 23 interpretable statements giving 9 different values of π !

Underwood Dudley. Mathematical Cranks. MAA, 1992. Legislating pi, pp. 192-197.

C. T. Heisel. The Circle Squared Beyond Refutation. Published by the author, 657 Bolivar Rd., Cleveland, Ohio, 1st ed., 1931, printed by S. J. Monck, Cleveland; 2nd ed., 1934, printed by Lezius-Hiles Co., Cleveland, ??NX + Supplement: "Fundamental Truth", 1936, ??NX, distributed by the author from 2142 Euclid Ave., Cleveland. This is probably the most ambitious publication of a circle-squarer -- Heisel distributed copies all around the world.

Underwood Dudley. πt: 1832-1879. MM 35 (1962) 153-154. He plots 45 values of π as a function of time over the period 1832-1879 and finds the least-squares straight line which fits the data, finding that πt = 3.14281 + .0000056060 t, for t measured in years AD. Deduces that the Biblical value of 3 was a good approximation for the time and that Creation must have occurred when πt = 0, which was in -560,615.

Underwood Dudley. πt. JRM 9 (1976-77) 178 & 180. Extends his previous work to 50 values of π over 1826-1885, obtaining πt = 4.59183 - .000773 t. The fact that πt is decreasing is worrying -- when πt = 1, all circles will collapse into straight lines and this will certainly be the end of the world, which is expected in 4646 on 9 Aug at 20:55:33 -- though this is only the expected time and there is considerable variation in this prediction. [Actually, I get that this should be on 11 Aug. However, it seems to me that circles will collapse once πt = 2, as then the circumference corresponds to going back and forth along the diameter. This will occur when t = 3352.949547, i.e. in 3352, on 13 Dec at 14:01:54 -- much earlier than Dudley's prediction, so start getting ready now!]

6.B. STRAIGHT LINE LINKAGES

See Yates for a good survey of the field.

James Watt. UK Patent 1432 -- Certain New Improvements upon Fire and Steam Engines, and upon Machines worked or moved by the same. Granted: 28 Apr 1784; complete specification: 24 Aug 1784. 14pp + 1 plate. Pp. 4-6 & Figures 7-12 describe Watt's parallel motion. Yates, below, p. 170 quotes one of Watt's letters: "... though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other invention I have ever made."

P. F. Sarrus. Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires; et reciproquement. C. R. Acad. Sci. Paris 36 (1853) 1036-1038. 6 plate linkage. The name should be Sarrus, but it is printed Sarrut on this and the following paper.

Poncelet. Rapport sur une transformation nouvelle des mouvements rectilignes alternatifs en mouvements circulaires et reciproquement, par Sarrut. Ibid., 36 (1853) 1125-1127.

A. Peaucellier. Lettre au rédacteur. Nouvelles Annales de Math. (2) 3 (1864) 414-415. Poses the problem.

A. Mannheim. Proces-Verbaux des sceances des 20 et 27 Juillet 1867. Bull. Soc. Philomathique de Paris (1867) 124-126. ??NYS. Reports Peaucellier's invention.

Lippman Lipkin. Fortschritte der Physik (1871) 40-?? ??NYS

L. Lipkin. Über eine genaue Gelenk-Geradführung. Bull. Acad. St. Pétersbourg [=? Akad. Nauk, St. Petersburg, Bull.] 16 (1871) 57-60. ??NYS

L. Lipkin. Dispositif articulé pour la transformation rigoureuse du mouvement circulaire en mouvement rectiligne. Revue Univers. des Mines et de la Métallurgie de Liége 30:4 (1871) 149-150. ??NYS. (Now spelled Liège.)

A. Peaucellier. Note sur un balancier articulé a mouvement rectiligne. Journal de Physique 2 (1873) 388-390. (Partial English translation in Smith, Source Book, vol. 2, pp. 324-325.) Says he communicated it to Soc. Philomath. in 1867 and that Lipkin has since also found it. There is also an article in Nouv. Annales de Math. (2) 12 (1873) 71-78 (or 73?), ??NYS.

E. Lemoine. Note sur le losange articulé du Commandant du Génie Peaucellier, destiné a remplacer le parallélogramme de Watt. J. de Physique 2 (1873) 130-134. Confirms that Mannheim presented Peaucellier's cell to Soc. Philomath. on 20 Jul 1867. Develops the inversive geometry of the cell.

[J. J. Sylvester.] Report of the Annual General Meeting of the London Math. Soc. on 13 Nov 1873. Proc. London Math. Soc. 5 (1873) 4 & 141. On p. 4 is: "Mr. Sylvester then gave a description of a new instrument for converting circular into general rectilinear motion, and into motion in conics and higher plane curves, and was warmly applauded at the close of his address." On p. 141 is an appendix saying that Sylvester spoke "On recent discoveries in mechanical conversion of motion" to a Friday Evening's Discourse at the Royal Institution on 23 Jan 1874. It refers to a paper 20 pages long but is not clear if or where it was published.

H. Hart. On certain conversions of motion. Cambridge Messenger of Mathematics 4 (1874) 82-88 and 116-120 & Plate I. Hart's 5 bar linkage. Obtains some higher curves.

A. B. Kempe. On some new linkages. Messenger of Mathematics 4 (1875) 121-124 & Plate I. Kempe's linkages for reciprocating linear motion.

H. Hart. On two models of parallel motions. Proc. Camb. Phil. Soc. 3 (1876-1880) 315-318. Hart's parallelogram (a 5 bar linkage) and a 6 bar one.

V. Liguine. Liste des travaux sur les systèms articulés. Bull. d. Sci. Math. 18 (or (2) 7) (1883) 145-160. ??NYS - cited by Kanayama. Archibald; Outline of the History of Mathematics, p. 99, says Linguine is entirely included in Kanayama.

Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of Construction. A second volume to accompany his previous Mechanical Movements, Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910. This is filled with many types of mechanisms. Pp. 245-247 show five straight-line linkages and some related mechanisms.

(R. Kanayama). (Bibliography on linkages. Text in Japanese, but references in roman type.) Tôhoku Math. J. 37 (1933) 294-319.

R. C. Archibald. Bibliography of the theory of linkages. SM 2 (1933-34) 293-294. Supplement to Kanayama.

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 82-101 & 168-191. Gets up to outlining Kempe's proof that any algebraic curve can be drawn by a linkage.

R. H. Macmillan. The freedom of linkages. MG 34 (No. 307) (Feb 1960) 26-37. Good survey of the general theory of linkages.

Michael Goldberg. Classroom Note 312: A six-plate linkage in three dimensions. MG 58 (No. 406) (Dec 1974) 287-289.

6.C. CURVES OF CONSTANT WIDTH

Such curves play an essential role in some ways to drill a square hole, etc.

L. Euler. Introductio in Analysin Infinitorum. Bousquet, Lausanne, 1748. Vol. 2, chap. XV, esp. § 355, p. 190 & Tab. XVII, fig. 71. = Introduction to the Analysis of the Infinite; trans. by John D. Blanton; Springer, NY, 1988-1990; Book II, chap. XV: Concerning curves with one or several diameters, pp. 212-225, esp. § 355, p. 221 & fig. 71, p. 481. This doesn't refer to constant width, but fig. 71 looks very like a Reuleaux triangle.

L. Euler. De curvis triangularibus. (Acta Acad. Petropol. 2 (1778(1781)) 3-30) = Opera Omnia (1) 28 (1955) 298-321. Discusses triangular versions.

M. E. Barbier. Note sur le problème de l'aiguille et jeu du joint couvert. J. Math. pures appl. (2) 5 (1860) 273-286. Mentions that perimeter = π * width.

F. Reuleaux. Theoretische Kinematik; Vieweg, Braunschweig, 1875. Translated: The Kinematics of Machinery. Macmillan, 1876; Dover, 1964. Pp. 129-147.

Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of Construction. A second volume to accompany his previous Mechanical Movements, Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910.

Item 642: Turning a square by circular motion, p. 247. Plain face, with four pins forming a centred square, is turned by the lathe. A triangular follower is against the face, so it is moved in and out as a pin moves against it. This motion is conveyed by levers to the tool which moves in and out against the work which is driven by the same lathe.

Item 681: Geometrical boring and routing chuck, pp. 257-258. Shows it can make rectangles, triangles, stars, etc. No explanation of how it works.

Item 903A: Auger for boring square holes, pp. 353-354. Uses two parallel rotating cutting wheels.

H. J. Watts. US Patents 1,241,175-7 -- Floating tool-chuck; Drill or boring member; Floating tool-chuck. Applied: 30 Nov 1915; 1 Nov 1916; 22 Nov 1916; all patented: 25 Sep 1917. 2 + 1, 2 + 1, 4 + 1 pp + pp diagrams. Devices for drilling square holes based on the Reuleaux triangle.

T. Bonnesen & W. Fenchel. Theorie der konvexen Körper. Berlin, 1934; reprinted by Chelsea, 1971. Chap. 15: Körper konstanter Breite, pp. 127-141. Surveys such curves with references to the source material.

G. D. Chakerian & H. Groemer. Convex bodies of constant width. In: Convexity and Its Applications; ed. by Peter M. Gruber & Jörg M. Wills; Birkhäuser, Boston, 1983. Pp. 49-96. (??NYS -- cited in MM 60:3 (1987) 139.) Bibliography of some 250 items since 1930.

6.D. FLEXAGONS

These were discovered by Arthur H. Stone, an English graduate student at Princeton in 1939. American paper was a bit wider than English and would not fit into his notebooks, so he trimmed the edge off and had a pile of long paper strips which he played with and discovered the basic flexagon. Fellow graduate students Richard P. Feynman, Bryant Tuckerman and John W. Tukey joined in the investigation and developed a considerable theory. One of their fathers was a patent attorney and they planned to patent the idea and began to draw up an application, but the exigencies of the 1940s led to its being put aside, though knowledge of it spread as mathematical folklore. E.g. Tuckerman's father, Louis B. Tuckerman, lectured on it at the Westinghouse Science Talent Search in the mid 1950s.

S&B, pp. 148-149, show several versions. Most square versions (tetraflexagons or magic books) don't fold very far and are really just extended versions of the Jacob's Ladder -- see 11.L

Martin Gardner. Cherchez la Femme [magic trick]. Montandon Magic Co., Tulsa, Okla., 1946. Reproduced in: Martin Gardner Presents; Richard Kaufman and Alan Greenberg, 1993, pp. 361-363. [In: Martin Gardner Presents, p. 404, this is attributed to Gardner, but Gardner told me that Roger Montandon had the copyright -- ?? I have learned a little more about Gardner's early life -- he supported himself by inventing and selling magic tricks about this time, so it may be that Gardner devised the idea and sold it to Montandon.]. A hexatetraflexagon.

"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon, 1947. A trick book, pp. 42-43. Same hexatetraflexagon.

Sidney Melmore. A single-sided doubly collapsible tessellation. MG 31 (No. 294) (1947) 106. Forms a Möbius strip of three triangles and three rhombi. He sees it has two distinct forms, but doesn't see the flexing property!!

Margaret Joseph. Hexahexaflexagrams. MTr 44 (Apr 1951) 247-248. No history.

William R. Ransom. A six-sided hexagon. SSM 52 (1952) 94. Shows how to number the 6 faces. No history.

F. G. Maunsell. Note 2449: The flexagon and the hexahexaflexagram. MG 38 (No. 325) (Sep 1954) 213-214. States that Joseph is first article in the field and that this is first description of the flexagon. Gives inventors' names, but with Tulsey for Tukey.

R. E. Rogers & Leonard L. D'Andrea. US Patent 2,883,195 -- Changeable Amusement Devices and the Like. Applied: 11 Feb 1955; patented: 21 Apr 1959. 2pp + 1p correction + 2pp diagrams. Clearly shows the 9 and 18 triangle cases and notes that one can trim the triangles into hexagons so the resulting object looks like six small hexagons in a ring.

M. Gardner. Hexa-hexa-flexagon and Cherchez la femme. Hugard's MAGIC Monthly 13:9 (Feb 1956) 391. Reproduced in his: Encyclopedia of Impromptu Magic; Magic Inc., Chicago, 1978, pp. 439-442. Describes hexahexa and the hexatetra of Gardner/Montandon & Willane.

M. Gardner. SA (Dec 1956) = 1st Book, chap. 1. His first article in SA!!

Joan Crampin. Note 2672: On note 2449. MG 41 (No. 335) (Feb 1957) 55-56. Extends to a general case having 9n triangles of 3n colours.

C. O. Oakley & R. J. Wisner. Flexagons. AMM 64:3 (Mar 1957) 143-154.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, section 61, pp. 24-25: Hexaflexagons. Describes the simplest case, citing Joseph.

Roger F. Wheeler. The flexagon family. MG 42 (No. 339) (Feb 1958) 1-6. Improved methods of folding and colouring.

M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Tetraflexagons and flexatube.

P. B. Chapman. Square flexagons. MG 45 (1961) 192-194. Tetraflexagons.

Anthony S. Conrad & Daniel K. Hartline. Flexagons. TR 62-11, RIAS, (7212 Bellona Avenue, Baltimore 12, Maryland,) 1962, 376pp. This began as a Science Fair project in 1956 and was then expanded into a long report. The authors were students of Harold V. McIntosh who kindly sent me one of the remaining copies in 1996. They discover how to make any chain of polygons into a flexagon, provided certain relations among angles are satisfied. The bibliography includes almost all the preceding items and adds the references to the Rogers & D'Andrea patent, some other patents (??NYS) and a number of ephemeral items: Conrad produced an earlier RIAS report, TR 60-24, in 1960; Allan Phillips wrote a mimeographed paper on hexaflexagons; McIntosh wrote an unpublished paper on flexagons; Mike Schlesinger wrote an unpublished paper on Tuckerman tree theory.

Sidney H. Scott. How to construct hexaflexagons. RMM 12 (Dec 1962) 43-49.

William R. Ransom. Protean shapes with flexagons. RMM 13 (Feb 1963) 35-37. Describes 3-D shapes that can be formed. c= Madachy, below.

Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. The magic book, pp. 124-125. As in Gardner's Cherchez la Femme and Willane.

Pamela Liebeck. The construction of flexagons. MG 48 (No. 366) (Dec 1964) 397-402.

Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Other flexagon diversions, pp. 76-81. Describes 3-D shapes that one can form. Based on Ransom, RMM 13.

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 66-75. Describes various tetra- and hexa-flexagons.

Douglas A. Engel. Hexaflexagon + HFG = slipagon! JRM 25:3 (1993) 161-166. Describes his slipagons, which are linked flexagons.

Robert E. Neale (154 Prospect Parkway, Burlington, Vermont, 05401, USA). Self-designing tetraflexagons. 12pp document received in 1996 describing several ways of making tetraflexagons without having to tape or paste. He starts with a creased square sheet, then makes some internal tears or cuts and then folds things through to miraculously obtain a flexagon! A slightly rearranged version appeared in: Elwyn R. Berlekamp & Tom Rodgers, eds.; The Mathemagician and Pied Puzzler A Collection in Tribute to Martin Gardner; A. K. Peters, Natick, Massachusetts, 1999, pp. 117-126.

Jose R. Matos. US Patent 5,735,520 -- Fold-Through Picture Puzzle. Applied: 7 Feb 1997; patented: 7 Apr 1998. Front page + 6pp diagrams + 13pp text. Robert Byrnes sent an example of the puzzle. This is a square in thin plastic, 100mm on an edge. Imagine a 2 x 2 array of squares with their diagonals drawn. Fold along all the diagonals and between the squares. This gives an array of 16 isosceles right triangles. Now cut from the centres of the four squares to the centre of the whole array. This produces an X cut in the middle. This object can now be folded through itself in various ways to produce a double thickness square of half the area with various logos. The example is 100mm along the edge of the large square and has four logos advertising Beanoland (at Chessington, 3 versions) and Strip Cheese. The patent is assigned to Lulirama International, but Byrnes says it has not been a commercial success as it is too complicated. The patent cites 19 earlier patents, back to 1881, and discusses the history of such puzzles. It also says the puzzle can form three dimensional objects.

6.E. FLEXATUBE

This is the square cylindrical tube that can be inverted by folding. It was also invented by Arthur H. Stone, c1939, cf 6.D.

J. Leech. A deformation puzzle. MG 39 (No. 330) (Dec 1955) 307. Doesn't know source. Says there are three solutions.

M. Gardner. Flexa-tube puzzle. Ibidem 7 (Sep 1956) 129. Cites the inventors of the flexagons and the articles of Maunsell and Leech (but he doesn't have its details). (I have a note that this came with attached sample, but the copy I have doesn't indicate such.)

T. S. Ransom. Flexa-tube solution. Ibidem 9 (Mar 1957) 174.

M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Says Stone invented it and shows Ransom's solution.

H. Steinhaus. Mathematical Snapshots. Not in the Stechert, NY, 1938, ed. nor the OUP, NY, 1950 ed. OUP, NY: 1960: pp. 189-193 & 326; 1969 (1983): pp. 177-181 & 303. Erroneous attribution to the Dowkers. Shows a different solution than Ransom's.

John Fisher. John Fisher's Magic Book. Muller, London, 1968. Homage to Houdini, pp. 152-155. Detailed diagrams of the solution, but no history.

Highland Games (2 Harpers Court, Dingwall, Ross-Shire, IV15 9HT) makes a version called Table Teaser, made in a strip with end pieces magnetic. Pieces are coloured so to produce several folding and inverting problems other than the usual one. Bought in 1995.

6.F. POLYOMINOES, ETC.

See S&B, pp. 15-18. See 6.F.1, 6.F.3, 6.F.4 & 6.F.5 for early occurrences of polyominoes. See Lammertink, 1996 & 1997 for many examples in two and three dimensions.

NOTATION. Each of the types of puzzle considered has a basic unit and pieces are formed from a number of these units joined edge to edge. The notation N: n1, n2, .... denotes a puzzle with N pieces, of which ni pieces consist of i basic units. If ni are single digit numbers, the intervening commas and spaces will be omitted, but the digits will be grouped by fives, e.g. 15: 00382 11.

Polyiamonds: Scrutchin; John Bull; Daily Sketch; Daily Mirror; B. T.s Zig-Zag; Daily Mail; Miller (1960); Guy (1960); Reeve & Tyrrell; O'Beirne (2 & 9 Nov 1961); Gardner (Dec 1964 & Jul 1965); Torbijn; Meeus; Gardner (Aug 1975); Guy (1996, 1999); Knuth,

Polycubes: Rawlings (1939); Editor (1948); Niemann (1948); French (1948); Editor (1948); Niemann (1948); Gardner (1958); Besley (1962); Gardner (1972)

Solid Pentominoes: Nixon (1948); Niemann (1948); Gardner (1958); Miller (1960); Bouwkamp (1967, 1969, 1978); Nelson (2002);

Cylindrical Pentominoes: Yoshigahara (1992);

Polyaboloes: Hooper (1774); Book of 500 Puzzles (1859); J. M. Lester (1919); O'Beirne (21 Dec 61 &  18 Jan 62)

Polyhexes: Gardner (1967); Te Riele & Winter

Polysticks: Benjamin; Barwell; General Symmetrics; Wiezorke & Haubrich; Knuth; Jelliss;

Polyrhombs or Rhombiominoes: Lancaster (1918); Jones (1992).

Polylambdas: Roothart.

Polyspheres -- see Section 6.AZ.

GENERAL REFERENCES

G. P. Jelliss. Special Issue on Chessboard Dissections. Chessics 28 (Winter 1986) 137-152. Discusses many problems and early work in Fairy Chess Review.

Branko Grünbaum & Geoffrey C. Shephard. Tilings and Patterns. Freeman, 1987. Section 9.4: Polyiamonds, polyominoes and polyhexes, pp. 497-511. Good outline of the field with a number of references otherwise unknown.

Michael Keller. A polyform timeline. World Game Review 9 (Dec 1989) 4-5. This outlines the history of polyominoes and other polyshapes. Keller and others refer to polyaboloes as polytans.

Rodolfo Marcelo Kurchan (Parana 960 5 "A", 1017 Buenos Aires, Argentina). Puzzle Fun, starting with No. 1 (Oct 1994). This is a magazine entirely devoted to polyomino and other polyform puzzles. Many of the classic problems are extended in many ways here. In No. 6 (Aug 1995) he presents a labelling of the 12 hexiamonds by the letters A, C, H, I, J, M, O, P, S, V, X, Y, which he obtained from Anton Hanegraaf. I have never seen this before.

Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, recreation 23, pp. 64-66. Considers figures formed of isosceles right triangles. He has eight of these, coloured with eight colours, and uses some of them to form "chequers or regular four-sided figures, different either in form or colour".

Book of 500 Puzzles. 1859. Triangular problem, pp. 74-75. Identical to Hooper, dropping the last sentence.

Dudeney. CP. 1907. Prob. 74: The broken chessboard, pp. 119-121 & 220-221. The 12 pentominoes and a 2 x 2.

A. Aubry. Prob. 3224. Interméd. Math 14 (1907) 122-124. ??NYS -- cited by Grünbaum & Shephard who say Aubry has something of the idea or the term polyominoes.

G. Quijano. Prob. 3430. Interméd. Math 15 (1908) 195. ??NYS -- cited by Grünbaum & Shephard, who say he first asked for the number of n-ominoes.

Thomas Scrutchin. US Patent 895,114 -- Puzzle. Applied: 20 Feb 1908; patented: 4 Aug 1908. 2pp + 1p diagrams. Mentioned in S&B, p. 18. A polyiamond puzzle -- triangle of side 8, hence with 64 triangles, apparently cut into 10 pieces (my copy is rather faint -- replace??).

Thomas W. Lancaster. US Patent 1,264,944 -- Puzzle. Filed: 7 May 1917; patented: 7 May 1918. 2pp + 1p diagrams. For a general polyrhomb puzzle making a rhombus. His diagram shows an 11 x 11 rhombus filled with 19 pieces formed from 4 to 10 rhombuses.

John Milner Lester. US Patent 1,290,761 -- Game Apparatus. Filed: 6 Feb 1918; patented: 7 Jan 1919. 2pp + 3pp diagrams. Fairly general assembly puzzle claims. He specifically illustrates a polyomino puzzle and a polyabolo puzzle. The first has a Greek cross of edge 3 (hence containing 45 unit cells) to be filled with polyominoes -- 11: 01154. The second has an 8-pointed star formed by superimposing two 4 x 4 squares. This has area 20 and hence contains 40 isosceles right triangles of edge 1, which is the basic unit of this type of puzzle. There are 11: 0128 pieces.

Blyth. Match-Stick Magic. 1921. Spots and squares, pp. 68-73. He uses matchsticks broken in thirds, so it is easier to describe with units of one-third. 6 units, 4 doubles and 2 triples. Some of the pieces have black bands or spots. Object is to form polyomino shapes without pieces crossing, but every intersection must have a black spot. 19 polyomino shapes are given to construct, including 7 of the pentominoes, though some of the shapes are only connected at corners.

"John Bull" Star of Fortune Prize Puzzle. 1922. This is a puzzle with 20 pieces, coloured red on one side, containing 6 through 13 triangles to be assembled into a star of David with 4 triangles along each edge (hence 12 x 16 = 192 triangles). Made by Chad Valley. Prize of £250 for a red star matching the key solution deposited at a bank; £150 for solution closest to the key; £100 for a solution with 10 red and 10 grey pieces, or as nearly as possible. Closing date of competition is 27 Dec 1922. Puzzle made by Chad Valley Co. as a promotional item for John Bull magazine, published by Odhams Press. A copy is in the toy shop of the Buckleys Shop Museum, Battle, East Sussex, to whom I am indebted for the chance to examine the puzzle and a photocopy of the puzzle, box and solution.

Daily Sketch Jig-Saw Puzzle. By Chad Valley. Card polyiamonds. 39: 0,0,1,5,6, 12,9,6, with a path printed on one side, to assemble into a shape of 16 rows of 15 with four corners removed and so the printed sides form a continuous circuit. In box with shaped bottom. Instructions on inside cover and loose sheet to submit solution. No dates given, but appears to be 1920s, though it is somewhat similar to the Daily Mail Crown Puzzle of 1953 -- cf below -- so it might be much later.

Daily Mirror Zig-Zag £500 Prize Puzzle. By Chad Valley. Card polyiamonds. 29: 0,0,1,1,4,  5,6,3,1,4, 1,2,1. One-sided pieces to fit into frame in card box. Three pieces are duplicated and one is triplicated. Solution and claim instructions appeared in Daily Mirror (17 Jan 1930) 1-2. See: Tom Tyler & Felicity Whiteley; Chad Valley Promotional Jig-Saw Puzzles; Magic Fairy Publishing, Petersfield, Hampshire, 1990, p. 55.

B. T.s Zig-Zag. B.T. is a Copenhagen newspaper. Polyiamond puzzle. 33: 0,0,1,2,5,  6,7,2,2,4,  1,1,1, Some repetitions, so I only see 20 different shapes. To be fit into an irregular frame. Solution given on 23 Nov 1931, pp. 1-2. (I have a photocopy of the form to fill in; an undated set of rules, apparently from the paper, saying the solutions must be received by 21 Nov; and the pages giving the solution; provided by Jan de Geus.)

Herbert D. Benjamin. Problem 1597: A big cutting-out design -- and a prize offer. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:9 (Dec 1934) 92. Finds the 35 hexominoes and asks if they form a 14 x 15 rectangle. Cites Dudeney (Tribune (20 Dec 1906)); Loyd (OPM (Apr-Jul 1908)) (see 6.F.1); Dudeney (CP, no. 74) (see above) and some other chessboard dissections. Jelliss says this is the first dissection problem in this journal.

F. Kadner. Solution 1597. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:10 (Feb 1935) 104-105. Shows the 35 hexominoes cannot tile a rectangle by two arguments, both essentially based on two colouring. Gives some other results and some problems are given as 1679-1681 -- ??NYS.

William E. Lester. Correction to 1597. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:11 (Apr 1935) 121. Corrects an error in Kadner. Finds a number of near-solutions. Editor says Kadner insists the editor should take credit for the two-colouring form of the previous proof.

Frans Hansson, proposer & solver?. Problem 1844. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:12 (Jun 1935) 128 & 2:13 (Aug 1935) 135. Finds both 3 x 20 pentomino rectangles.

W. E. Lester & B. Zastrow, proposers. Problem 1923. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:13 (Aug 1935) 138. Take an 8 x 8 board and remove its corners. Fill this with the 12 pentominoes.

H. D. Benjamin, proposer. Problem 1924. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:13 (Aug 1935) 138. Dissect an 8 x 8 into the 12 pentominoes and the I-tetromino. Need solution -- ??NYS.

Thomas Rayner Dawson & William E. Lester. A notation for dissection problems. Fairy Chess Review 3:5 (Apr 1937) 46-47. Gives all n-ominoes up to n = 6. Describes the row at a time notation. Shows the pentominoes and a 2 x 2 cover the chessboard with the 2 x 2 in any position. Asserts there are 108 7-ominoes and 368 8-ominoes -- citing F. Douglas & W. E. L[ester] for the hexominoes and J. Niemann for the heptominoes.

H. D. Benjamin, proposer. Problem 3228. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect a 5 x 5 into the five tetrominoes and a pentomino so that the pentomino touches all the tetrominoes along an edge. Asserts the solution is unique. Refers to problems 3026-3030 -- ??NYS.

H. D. Benjamin, proposer. Problem 3229. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect an 8 x 8 into the 12 pentominoes and a tetromino so that all pieces touch the edge of the board. Asserts only one tetromino works.

T. R. D[awson], proposer. Problems 3230-1. Fairy Chess Review 3:12 (Jun 1938) 129. Extends prob. 3229 to ask for solutions with 12 pieces on the edge, using two other tetrominoes. Thinks it cannot be done with the remaining two tetrominoes.

Editorial note: The colossal count. Fairy Chess Review 3:12 (Jun 1938) 131. Describes progress on enumerating 8-ominoes (four people get 368 but Niemann gets 369) and 9-ominoes (numbers vary from 1237 to 1285). All workers are classifying them by the size of the smallest containing rectangle.

W. H. Rawlings, proposer. Problem 3930. Fairy Chess Review 4:3 (Nov 1939) 28. How many pentacubes are there? Ibid. 4:4 (Feb 1940) 75, reports that both 25 or 26 are claimed, but the editor has only seen 24. Ibid. 4:5 (Apr 1940) 85, reports that R. J. F[rench] has clearly shown there are 23 -- but this considers reflections as equal -- cf the 1948 editorial note.

R. J. French, proposer and solver. Problem 4149. Fairy Chess Review 4:3 (Nov 1939) 43 & 4:6 (Jun 1940) 93. Asks for arrangement of the pentominoes with the largest hole and gives one with 127 squares in the hole. (See: G. P. Jelliss; Comment on Problem 1277; JRM 22:1 (1990) 69. This reviews various earlier solutions and comments on Problem 1277.)

J. Niemann. Item 4154: "The colossal count". Fairy Chess Review 4:3 (Nov 1939) 44-45. Announces that there are 369 8-ominoes, 1285 9-ominoes and 4654 10-ominoes, but Keller and Jelliss note that he missed a 10-omino which was not corrected until 1966.

H. D. Benjamin. Unpublished notes. ??NYS -- cited and briefly described in G. P. Jelliss; Prob. 48 -- Aztec tetrasticks; G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss says Benjamin studied polysticks, which he called 'lattice dissections' around 1946-1948 and that some results by him and T. R. Dawson were entered in W. Stead's notebooks but nothing is known to have been published. For orders 1, 2, 3, 4, there are 1, 2, 5, 16 polysticks. Benjamin formed these into a 6 x 6 lattice square. Jelliss then mentions Barwell's rediscovery of them and goes on to a new problem -- see Knuth, 1999.

D. Nixon, proposer and solver. Problem 7560. Fairy Chess Review 6:16 (Feb 1948) 12 & 6:17 (Apr 1948) 131. Constructs 3 x 4 x 5 from solid pentominoes.

Editorial discussion: Space dissection. Fairy Chess Review 6:18 (Jun 1948) 141-142. Says that several people have verified the 23 pentacubes but that 6 of them have mirror images, making 29 if these are considered distinct. Says F. Hansson has found 77 6-cubes (these exclude mirror images and the 35 solid 6-ominoes). Gives many problems using n-cubes and/or solid polyominoes, which he calls flat n-cubes -- some are corrected in 7:2 (Oct 1948) 16 (erroneously printed as 108).

J. Niemann. The dissection count. Item 7803. Fairy Chess Review 7:1 (Aug 1948) 8 (erroneously printed as 100). Reports on counting n-cubes. Gets the following.

n = 4 5 6 7

flat pieces 5 12 35 108

non-flats 2 11 77 499

TOTAL 7 23 112 607

mirror images 1 6 55 416

GRAND TOTAL 8 29 167 1023

R. J. French. Space dissections. Fairy Chess Review 7:2 (Oct 1948) 16 (erroneously printed as 108). French writes that he and A. W. Baillie have corrected the number of 6-cubes to 35 + 77 + 54 = 166. Baillie notes that every 6-cube lies in two layers -- i.e. has some width ( 2 -- and asks for the result for n-cubes as prob. 7879. [I suspect the answer is that n ( 3k implies that an n-cube has some width ( k.] Editor adds some corrections to the discussion in 6:18.

Editorial note. Fairy Chess Review 7:3 (Dec 1948) 23. Niemann and Hansson confirm the number 166 given in 7:2.

Daily Mail Crown Puzzle. Made by Chad Valley Co. 1953. 26 pieces, coloured on one side, to be fit into a crown shape. 11 are border pieces and easily placed. The other 15 are polyiamonds: 15: 00112 24012 11. Prize of £100 for solution plus best slogan, entries due on 8 Jun 1953.

S. W. Golomb. Checkerboards and polyominoes. AMM 61 (1954) 675-682. Mostly concerned with covering the 8 x 8 board with copies of polyominoes. Shows one covering with the 12 pentominoes and the square tetromino. Mentions that the idea can be extended to hexagons. S&B, p. 18, and Gardner (Dec 1964) say he mentions triangles, but he doesn't.

Walter S. Stead. Dissection. Fairy Chess Review 9:1 (Dec 1954) 2-4. Gives many pentomino and hexomino patterns -- e.g. one of each pattern of 8 x 8 with a 2 x 2 square deleted. "The possibilities of the 12 fives are not infinite but they will provide years of amusement." Includes 3 x 20, 4 x 15, 5 x 12 and 6 x 10 rectangles. No reference to Golomb. In 1955, Stead uses the 108 heptominoes to make a 28 x 28 square with a symmetric hole of size 28 in the centre -- first printed as cover of Chessics 28 (1986).

Jules Pestieau. US Patent 2,900,190 -- Scientific Puzzle. Filed: 2 Jul 1956; patented: 18 Aug 1959. 2pp + 1p diagrams. For the 12 pentominoes! Diagram shows the 6 x 10 solution with two 5 x 6 rectangles and shows the two-piece non-symmetric equivalence of the N and F pieces. Pieces have markings on one side which may be used -- i.e. pieces may not be turned over. Mentions possibility of using n-ominoes.

Gardner. SA (Dec 1957) = 1st Book, chap. 13. Exposits Golomb and Stead. Gives number of n-ominoes for n = 1, ..., 7. 1st Book describes Scott's work. Says a pentomino set called 'Hexed' was marketed in 1957. (John Brillhart gave me and my housemates an example in 1960 -- it took us two weeks to find our first solution.)

Dana Scott. Programming a Combinatorial Puzzle. Technical Report No. 1, Dept. of Elec. Eng., Princeton Univ., 1958, 20pp. Uses MANIAC to find 65 solutions for pentominoes on an 8 x 8 board with square 2 x 2 in the centre. Notes that the 3 x 20 pentomino rectangle has just two solutions. In 1999, Knuth notes that the total number of solutions with the 2 x 2 being anywhere does not seem to have ever been published and he finds 16146.

M. Gardner. SA (Sep 1958) c= 2nd Book, chap. 6. First general mention of solid pentominoes, pentacubes, tetracubes. In the Addendum in 2nd Book, he says Theodore Katsanis of Seattle suggested the eight tetracubes and the 29 pentacubes in a letter to Gardner on 23 Sep 1957. He also says that Julia Robinson and Charles W. Stephenson both suggested the solid pentominoes.

C. Dudley Langford. Note 2793: A conundrum for form VI. MG 42 (No. 342) (Dec 1958) 287. 4 each of the L, N, and T (= Y) tetrominoes make a 7 x 7 square with the centre missing. Also nine pieces make a 6 x 6 square but this requires an even number of Ts.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 8. Use the pentominoes to make two 5 x 5 squares at the same time. Solution just says there are several ways to do so.

J. C. P. Miller. Pentominoes. Eureka 23 (Oct 1960) 13-16. Gives the Haselgroves' number of 2339 solutions for the 6 x 10 and says there are 2 solutions for the 3 x 20. Says Lehmer suggests assembling 12 solid pentominoes into a 3 x 4 x 5 and van der Poel suggests assembling the 12 hexiamonds into a rhombus.

C. B. & Jenifer Haselgrove. A computer program for pentominoes. Ibid., 16-18. Outlines program which found the 2339 solutions for the 6 x 10. It is usually said that they also found all solutions of the 3 x 20, 4 x 15 and 5 x 12, but I don't see it mentioned here and in JRM 7:3 (1974) 257, it is reported that Jenifer (Haselgrove) Leech stated that only the 6 x 10 and 3 x 20 were done in 1960, but that she did the 5 x 12 and 4 x 15 with a new program in c1966. See Fairbairn, c1962, and Meeus, 1973.

Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Considers handed polyominoes, i.e. polyominoes when reflections are not considered equivalent. Notes that neither the 5 plain nor the 7 handed tetrominoes can form a rectangle. The 10 chequered handed tetrominoes form 4 x 10 and 5 x 8 rectangles and he has several solutions of each. There is no 2 x 20 rectangle. Discusses MacMahon pieces -- cf 5.H.2 -- and polyiamonds. He uses the word 'hexiamond', but not 'polyiamond' -- in an email of 8 Apt 2000, Guy says that O'Beirne invented all the terms. He considers making a 'hexagon' from the 19 hexiamonds. Part II considers solid problems and uses the term 'solid pentominoes'.

Solomon W. Golomb. The general theory of polyominoes: part 2 -- Patterns and polyominoes. RMM 5 (Oct 1961) 3-14. ??NYR.

J. E. Reeve & J. A. Tyrrell. Maestro puzzles. MG 45 (No. 353) (Oct 1961) 97-99. Discusses hexiamond puzzles, using the 12 reversible pieces. [The puzzle was marketed under the name 'Maestro' in the UK.]

T. H. O'Beirne. Pell's equation in two popular problems. New Scientist 12 (No. 258) (26 Oct 1961) 260-261.

T. H. O'Beirne. Pentominoes and hexiamonds. New Scientist 12 (No. 259) (2 Nov 1961) 316-317. This is the first use of the word 'polyiamond'. He considers the 19 one-sided pieces. He says he devised the pieces and R. K. Guy has already published many solutions in Nabla. He asks for the number of ways the 18 one-sided pentominoes can fill a 9 x 10. In 1999, Knuth found this would take several months.

T. H. O'Beirne. Some hexiamond solutions: and an introduction to a set of 25 remarkable points. New Scientist 12 (No. 260) (9 Nov 1961) 378-379.

Maurice J. Povah. Letter. MG 45 (No. 354) (Dec 1961) 342. States Scott's result of 65 and the Haselgroves' result of 2339 (computed at Manchester). Says he has over 7000 solutions for the 8 x 8 board using a 2 x 2.

T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751-752.

T. H. O'Beirne. Some tetrabolic difficulties. New Scientist 13 (No. 270) (18 Jan 1962) 158-159. These two columns are the first mention of tetraboloes, so named by S. J. Collins.

R. A. Fairbairn. Pentomino Problems: The 6 x 10, 5 x 12, 4 x 15, and 3 x 20 Rectangles -- The Complete Drawings. Unpublished MS, undated, but c1962, based on the Haselgroves' work of 1960. ??NYS -- cited by various authors, e.g. Madachy (1969), Torbijn (1969), Meeus (1973). Madachy says Fairbairn is from Willowdale, Ontario, and takes some examples from his drawings. However, the dating is at variance with Jenifer Haselgrove's 1973 statement - cf Haselgrove, 1960. Perhaps this MS is somewhat later?? Does anyone know where this MS is now? Cf Meeus, 1973.

Serena Sutton Besley. US Patent 3,065,970 -- Three Dimensional Puzzle. Filed: 6 Jul 1960; issued: 27 Nov 1962. 2pp + 4pp diagrams. For the 29 pentacubes, with one piece duplicated giving a set of 30. Klarner had already considered omitting the 1 x 1 x 5 and found that he could make two separate 2 x 5 x 7s. Besley says the following can be made: 5 x 5 x 6, 3 x 5 x 10, 2 x 5 x 15, 2 x 3 x 25; 3 x 5 x 6, 3 x 3 x 10, 2 x 5 x 9, 2 x 3 x 15; 3 x 4 x 5, 2 x 5 x 6, 2 x 3 x 10 (where the latter three are made with the 12 solid pentominoes and the previous four are made with the 18 non-planar pentacubes) but detailed solutions are only given for the 5 x 5 x 6, 3 x 5 x 6, 3 x 4 x 5. Mentions possibility of n-cubes.

M. Gardner. Polyiamonds. SA (Dec 1964) = 6th Book, chap. 18. Exposits basic ideas and results for the 12 double sided hexiamonds. Poses several problems which are answered by readers. The six-pointed star using 8 pieces has a unique solution. John G. Fletcher and Jenifer (Haselgrove) Leech both showed the 3 x 12 rhombus is impossible. Fletcher found the 3 x 11 rhombus has 24 solutions, all omitting the 'bat'. Leech found 155 solutions for the 6 x 6 rhombus and 74 solutions for the 4 x 9. Mentions there are 160 9-iamonds, one with a hole.

John G. Fletcher. A program to solve the pentomino problem by the recursive use of macros. Comm. ACM 8 (1965) 621-623. ??NYS -- described by Knuth in 1999 who says that Fletcher found the 2339 solutions for the 6 x 10 in 10 minutes on an IBM 7094 and that the program remains the fastest known method for problems of placing the 12 pentominoes.

M. Gardner. Op art. SA (Jul 1965) = 6th Book, chap. 24. Shows the 24 heptiamonds and discusses which will tile the plane.

Solomon W. Golomb. Tiling with polyominoes. J. Combinatorial Theory 1 (1966) 280-296. ??NYS. Extended by his 1970 paper.

T. R. Parkin. 1966. ??NYS -- cited by Keller. Finds 4655 10-ominoes.

M. Gardner. SA (Jun 1967) = Magic Show, chap. 11. First mention of polyhexes.

C. J. Bouwkamp. Catalogue of Solutions of the Rectangular 3 x 4 x 5 Solid Pentomino Problem. Dept. of Math., Technische Hogeschool Eindhoven, July 1967, reprinted 1981, 310pp.

C. J. Bouwkamp. Packing a rectangular box with the twelve solid pentominoes. J. Combinatorial Thy. 7 (1969) 278-280. He gives the numbers of solutions for rectangles as 'known'.

2 x 3 x 10 can be packed in 12 ways, which are given.

2 x 5 x 6 can be packed in 264 ways.

3 x 4 x 5 can be packed in 3940 ways. (See his 1967 report.)

T. R. Parkin, L. J. Lander & D. R. Parkin. Polyomino enumeration results. Paper presented at the SIAM Fall Meeting, Santa Barbara, 1 Dec 1967. ??NYS -- described by Madachy, 1969. Gives numbers of n-ominoes, with and without holes, up to n = 15, done two independent ways.

Joseph S. Madachy. Pentominoes -- Some solved and unsolved problems. JRM 2:3 (Jul 1969) 181-188. Gives the numbers of Parkin, Lander & Parkin. Shows various examples where a rectangle splits into two congruent halves. Discusses various other problems, including Bouwkamp's 3 x 4 x 5 solid pentomino problem. Bouwkamp reports that the final total of 3940 was completed on 16 Mar 1967 after about three years work using three different computers, but that a colleague's program would now do the whole search in about three hours.

P. J. Torbijn. Polyiamonds. JRM 2:4 (Oct 1969) 216-227. Uses the double sided hexiamonds and heptiamonds. A few years before, he found, by hand, that there are 156 ways to cover the 6 x 6 rhombus with the 12 hexiamonds and 74 ways for the 4 x 9, but could find no way to cover the 3 x 12. The previous year, John G. Fletcher confirmed these results with a computer and he displays all of these -- but this contradicts Gardner (Dec 64) -- ?? He gives several other problems and results, including using the 24 heptiamonds to form 7 x 12, 6 x 14, 4 x 21 and 3 x 28 rhombuses.

Solomon W. Golomb. Tlling with sets of polyominoes. J. Combinatorial Theory 9 (1970) 60-71. ??NYS. Extends his 1966 paper. Asks which heptominoes tile rectangles and says there are two undecided cases -- cf Marlow, 1985. Gardner (Aug 75) says Golomb shows that the problem of determining whether a given finite set of polyominoes will tile the plane is undecidable.

C. J. Boukamp & D. A. Klarner. Packing a box with Y-pentacubes. JRM 3:1 (1970) 10-26. Substantial discussion of packings with Y-pentominoes and Y-pentacubes. Smallest boxes are 5 x 10 and 2 x 5 x 6 and 3 x 4 x 5.

Fred Lunnon. Counting polyominoes. IN: Computers in Number Theory, ed. by A. O. L. Atkin & B. J. Birch; Academic Press, 1971, pp. 347-372. He gets up through 18-ominoes, but the larger ones can have included holes. The numbers for n = 1, 2, ..., are as follows: 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107911, 192622052. These values have been quoted numerous times.

Fred Lunnon. Counting hexagonal and triangular polyominoes. IN: Graph Theory and Computing, ed. by R. C. Read; Academic Press, 1972, pp. 87-100. ??NYS -- cited by Grünbaum & Shephard.

M. Gardner. SA (Sep 1972). c= Knotted, chap. 3. Says the 8 tetracubes were made by E. S. Lowe Co. in Hong Kong and marketed as "Wit's End". Says an MIT group found 1390 solutions for the 2 x 4 x 4 box packed with tetracubes. He reports that several people found that there are 1023 heptacubes -- but see Niemann, 1948, above. Klarner reports that the heptacubes fill a 2 x 6 x 83.

Jean Meeus. Some polyomino and polyamond [sic] problems. JRM 6:3 (1973) 215-220. (Corrections in 7:3 (1974) 257.) Considers ways to pack a 5 x n rectangle with some n pentominoes. A. Mank found the number of ways for n = 2, 3, ..., 11 as follows, and the number for n = 12 was already known:

0,   7,  50, 107, 541, 1387, 3377, 5865,   6814,   4103, 1010.

Says he drew out all the solutions for the area 60 rectangles in 1972 (cf Fairbairn, c1962). Finds that 520 of the 6 x 10 rectangles can be divided into two congruent halves, sometimes in two different ways. For 5 x 12, there are 380; for 4 x 15, there are 94. Gives some hexomino rectangles by either deleting a piece or duplicating one, and an 'almost 11 x 19'. Says there are 46 solutions to the 3 x 30 with the 18 one-sided pentominoes and attributes this to Mrs (Haselgrove) Leech, but the correction indicates this was found by A. Mank.

Jenifer Haselgrove. Packing a square with Y-pentominoes. JRM 7:3 (1974) 229. She finds and shows a way to pack 45 Y-pentominoes into a 15 x 15, but is unsure if there are more solutions. In 1999, Knuth found 212 solutions. She also reports the impossibility of using the Y-pentominoes to fill various other rectangles.

S. W. Golomb. Trademark for 'PENTOMINOES'. US trademark 1,008,964 issued 15 Apr 1975; published 21 Jan 1975 as SN 435,448. (First use: November 1953.) [These appear in the Official Gazette of the United States Patent Office (later Patent and Trademark Office) in the Trademarks section.]

M. Gardner. Tiling with polyominoes, polyiamonds and polyhexes. SA (Aug 75) (with slightly different title) = Time Travel, chap. 14. Gives a tiling criterion of Conway. Describes Golomb's 1966 & 1970 results.

C. J. Bouwkamp. Catalogue of solutions of the rectangular 2 x 5 x 6 solid pentomino problem. Proc. Koninklijke Nederlandse Akad. van Wetenschappen A81:2 (1978) 177-186. Presents the 264 solutions which were first found in Sep 1967.

H. Redelmeier. Discrete Math. 36 (1981) 191-203. ??NYS -- described by Jelliss. Obtains number of n-ominoes for n ( 24.

Karl Scherer. Problem 1045: Heptomino tessellations. JRM 14:1 (1981-82) 64. XX 

Says he has found that the heptomino at the right fills a 26 x 42 rectangle. XXXXX

See Dahlke below.

David Ellard. Poly-iamond enumeration. MG 66 (No. 438) (Dec 1982) 310-314. For n = 1, ..., 12, he gets 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3342 n-iamonds. One of the 8-iamonds has a hole and there are many later cases with holes.

Anon. 31: Polyominoes. QARCH 1:8 (June 1984) 11-13. [This is an occasional publication of The Archimedeans, the student maths society at Cambridge.] Good survey of counting and asymptotics for the numbers of polyominoes, up to n = 24, polycubes, etc. 10 references.

T. W. Marlow. Grid dissections. Chessics 23 (Autumn, 1985) 78-79.

X XX

Shows XXXXX fills a 23 x 24 and XXXXX fills a 19 x 28.

Herman J. J. te Riele & D. T. Winter. The tetrahexes puzzle. CWI Newsletter [Amsterdam] 10 (Mar 1986) 33-39. Says there are: 7 tetrahexes, 22 pentahexes, 82 hexahexes, 333  heptahexes, 1448 octahexes. Studies patterns of 28 hexagons. Shows the triangle cannot be constructed from the 7 tetrahexes and gives 48 symmetric patterns that can be made.

Karl A. Dahlke. Science News 132:20 (14 Nov 1987) 310. (??NYS -- cited in JRM 21:3 and

XX

22:1 and by Marlow below.) Shows XXXXX fills a 21 x 26 rectangle.

The results of Scherer and Dahlke are printed in JRM 21:3 (1989) 221-223 and Dahlke's solution is given by Marlow below.

Karl A. Dahlke. J. Combinatorial Theory A51 (1989) 127-128. ??NYS -- cited in JRM 22:1. Announces a 19 x 28 solution for the above heptomino problem, but the earlier 21 x 26 solution is printed by error. The 19 x 28 solution is printed in JRM 22:1 (1990) 68-69.

Tom Marlow. Grid dissections. G&PJ 12 (Sep/Dec 1989) 185. Prints Dahlke's result.

Brian R. Barwell. Polysticks. JRM 22:3 (1990) 165-175. Polysticks are formed of unit lengths on the square lattice. There are: 1, 2, 5, 16, 55 polysticks formed with 1, 2, 3, 4, 5 unit lengths. He forms 5 x 5 squares with one 4-stick omitted, but he permits pieces to cross. He doesn't consider the triangular or hexagonal cases. See also Blyth, 1921, for a related puzzle. Cf Benjamin, above, and Wiezorke & Haubrich, below.

General Symmetrics (Douglas Engel) produced a version of polysticks, ©1991, with 4 3-sticks and 3 4-sticks to make a 3 x 3 square array with no crossing of pieces.

Nob Yoshigahara. Puzzlart. Tokyo, 1992. Ip-pineapple (pineapple delight), pp. 78-81. Imagine a cylindrical solution of the 6 x 10 pentomino rectangle and wrap it around a cylinder, giving each cell a depth of one along the radius. Hence each cell is part of an annulus. He reduces the dimensions along the short side to make the cells look like tenths of a slice of pineapple. Nob constructed and example for Toyo Glass's puzzle series and it was later found to have a unique solution.

Kate Jones, proposer; P. J. Torbijn, Jacques Haubrich, solvers. Problem 1961 -- Rhombiominoes. JRM 24:2 (1992) 144-146 & 25:3 (1993) 223-225. A rhombiomino or polyrhomb is a polyomino formed using rhombi instead of squares. There are 20 pentarhombs. Fit them into a 10 x 10 rhombus. Various other questions. Haubrich found many solutions. See Lancaster, 1918.

Bernard Wiezorke & Jacques Haubrich. Dr. Dragon's polycons. CFF 33 (Feb 1994) 6-7. Polycons (for connections) are the same as the polysticks described by Barwell in 1990, above. Authors describe a Taiwanese version on sale in late 1993, using 10 of the 4-sticks suitably shortened so they fit into the grooves of a 4 x 4 board -- so crossings are not permitted. (An n x n board has n+1 lines of n edges in each direction.) They fit 15 of the 4-sticks onto a 5 x 5 board and determine all solutions.

CFF 35 (Dec 1994) 4 gives a number of responses to the article. Brain Barwell wrote that he devised them as a student at Oxford, c1970, but did not publish until 1990. He expected someone to say it had been done before, but no one has done so. He also considered using the triangular and hexagonal lattice. He had just completed a program to consider fitting 15 of the 4-sticks onto a 5 x 5 board and found over 180,000 solutions, with slightly under half having no crossings, confirming the results of Wiezorke & Haubrich.

Dario Uri also wrote that he had invented the idea in 1984 and called them polilati (polyedges). Giovanni Ravesi wrote about them in Contromossa (Nov 1984) 23 -- a defunct magazine.

Chris Roothart. Polylambdas. CFF 34 (Oct 1994) 26-28. A lambda is a 30o-60o-90o triangle. These may be joined along corresponding legs, but not along hypotenuses. For n = 1, 2, 3, 4, 5, there are 1, 4, 4, 11, 12 n-lambdas. He gives some problems using various sets of these pieces.

Richard Guy. Letters of 29 May and 13 Jun 1996. He is interested in using the 19 one-sided hexiamonds. Hexagonal rings of hexagons contain 1, 6, 12 hexagons, so the hexagon with three hexagons on a side has 19 hexagons. If these hexagons are considered to comprise six equilateral triangles, we have a board with 19 x 6 triangles. O'Beirne asked for the number of ways to fill this board with the one-sided hexiamonds. Guy has collected over 4200 solutions. A program by Marc Paulhus found 907 solutions in eight hours, from which it initially estimated that there are about 30,000 solutions. The second letter gives the final results -- there are 124,518 solutions. This is modulo the 12 symmetries of the hexagon. In 1999, Knuth found 124,519 and Paulhus has rerun his program and found this number.

Ferdinand Lammertink. Polyshapes. Parts 1 and 2. The author, Hengelo, Netherlands, 1996 & 1997. Part 1 deals with two dimensional puzzles. Good survey of the standard polyform shapes and many others.

Hilarie Korman. Pentominoes: A first player win. IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1997??, ??NYS - described in William Hartston; What mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2. This studies the game proposed by Golomb -- players alternately place one of the pentominoes on the chess board, aligned with the squares and not overlapping the previous pieces, with the last one able to play being the winner. She used a Sun IPC Sparcstation for five days, examining about 22 x 109 positions to show the game is a first player win.

Nob Yoshigahara found in 1994 that the smallest box which can be packed with W-pentacubes is 5 x 6 x 6. In 1997, Yoshya (Wolf) Shindo found that one can pack the 6 x 10 x 10 with Z-pentacubes, but it is not known if this is the smallest such box. These were the last unsolved problems as to whether a box could be packed with a planar pentacube (= solid pentomino).

Marcel Gillen & Georges Philippe. Twinform 462 Puzzles in one. Solutions for Gillen's puzzle exchange at IPP17, 1997, 32pp + covers. Take 6 of the pentominoes and place them in a 7 x 5 rectangle, then place the other six to make the same shape on top of the first shape. There are 462 (= BC(12,6)/2) possible puzzles and all of them have solutions. Taking F, T, U, W, X, Z for the first layer, there is just one solution; all other cases have multiple solutions, totalling 22,873 solutions, but only one solution for each case is given here.)

Richard K. Guy. O'Beirne's hexiamond. In: The Mathemagican and Pied Puzzler; ed. by Elwyn Berlekamp & Tom Rodgers, A. K. Peters, Natick, Massachusetts, 1999, pp. 85-96. He relates that O'Beirne discovered the 19 one-sided hexiamonds in c1959 and found they would fill a hexagonal shape in Nov 1959 and in Jan 1960 he found a solution with the hexagonal piece in the centre. He gives Paulhus's results (see Guy's letters of 1996), broken down in various ways. He gives the number of double-sided (i.e. one can turn them over) and single-sided n-iamonds for n = 1, ..., 7. Cf Ellard, 1982, for many more values for the double-sided case.

n 1 2 3 4 5 6 7

double 1 1 1 3 4 12 24

single 1 1 1 4 6 19 44

In 1963, Conway and Mike Guy considered looking for 'symmetric' solutions for filling the hexagonal shape with the 19 one-sided hexiamonds. A number of these are described.

Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the author. Available as: . In this he introduces a new technique for backtrack programming which runs faster (although it takes more storage) and is fairly easy to adapt to different problems. In this approach, there is a symmetry between pieces and cells. He applies it to several polyshape problems, obtaining new, or at least unknown, results. He extends Scott's 1958 results to get 16146 ways to pack the 8 x 8 with the 12 pentominoes and the 2 x 2. He describes Fletcher's 1965 work. He extends Haselgrove's 1974 work and finds 212 ways to fit 15 Y-pentominoes in a 15 x 15. Describes Torbijn's 1969 work and Paulhus' 1996 work on hexiamonds, correcting the latter's number to 124,519. He then looks for the most symmetric solutions for filling the hexagonal shape with the 19 one-sided hexiamonds, in the sense discussed by Guy (1999). He then considers the 18 one-sided pentominoes (cf Meeus (1973)) and tries the 9 x 10, but finds it would take a few months on his computer (a 500 MHz Pentium III), so he's abandoned it for now. He then considers polysticks, citing an actual puzzle version that I've not seen. He adapts his program to them. He considers the 'welded tetrasticks' which have internal junction points. There are six of these and ten if they are taken as one-sided. The ten can be placed in a 4 x 4 grid. There are 15 unwelded, one-sided, tetrasticks, but they do not form a square, nor indeed any nice shape. He considers all 25 one-sided tetrasticks and asks if they can be fit into what he calls an Aztec Diamond, which is the shape looking like a square tilted 45o on the square lattice. The rows contain 1, 3, 5, 7, 9, 7, 5, 3, 1 cells. He thinks an exhaustive search is beyond present computing power.

G. P. Jelliss. Prob. 48 -- Aztec tetrasticks. G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss first discusses Benjamin's work on polysticks (see at 1946-1948 above) and Barwell's rediscovery of them (see above). He then describes Knuth's Dancing Links and gives the Aztec Diamond problem. Jelliss has managed to get all but one of the polysticks into the shape, but feels it is impossible to get them all in.

Harry L. Nelson. Solid pentomino storage, Question and answer. 1p HO at G4G5, 2002. 1: Can one put all the solid pentominoes into a cube of edge 4.5? What is the smallest cube into which they can all be placed? He gives 2 solutions to 1 and a solution due to Wei-Hwa Huang for a cube of edge 4.405889..., which is conjectured to be minimal. In fact, one edge of the packing is actually 4, so the volume is less than (4.405889...)3. This leads me to ask what is the smallest volume of a cuboid, with edges less than 5, that contains all the solid pentominoes. In Summer 2002, Harry gave me a set of solid pentominoes in a box with a list of various rectangles and boxes to fit them into: 3 x 22; 3 x 21; 3 x 20; 4 x 16; 4 x 15; 5 x 13; 5 x 12; 6 x 11; 6 x 10; 7 x 9; 8 x 8; 2 x 4 x 8; 2 x 5 x 7; 2 x 5 x 6; 2 x 6 x 6; 3 x 4 x 6; 3 x 4 x 5; 3 x 5 x 5; 4 x 4 x 5; the given box: 4.4 x 4.4 x 4.9.)

6.F.1. OTHER CHESSBOARD DISSECTIONS

See S&B, pp. 12-14. See also 6.F.5 for dissections of uncoloured boards.

Jerry Slocum. Compendium of Checkerboard Puzzles. Published by the author, 1983. Outlines the history and shows all manufactured versions known then to him: 33 types in 61 versions. The first number in Slocum's numbers is the number of pieces.

Jerry Slocum & Jacques Haubrich. Compendium of Checkerboard Puzzles. 2nd ed., published by Slocum, 1993. 90 types in 161 versions, with a table of which pieces are in which puzzles, making it much easier to see if a given puzzle is in the list or not. This gives many more pictures of the puzzle boxes and also gives the number of solutions for each puzzle and sometimes prints all of them. The Slocum numbers are revised in the 2nd ed. and I use the 2nd ed. numbers below. (There was a 3rd ed. in 1997, with new numbering of 217 types in 376 versions. NYR. Haubrich is working on an extended version with Les Barton providing information.)

Henry Luers. US Patent 231,963 -- Game Apparatus or Sectional Checker Board. Applied: 7 Aug 1880; patented: 7 Sep 1880. 1p + 1p diagrams. 15: 01329. Slocum 15.5.1. Manufactured as: Sectional Checker Board Puzzle, by Selchow & Righter. Colour photo of the puzzle box cover is on the front cover of the 1st ed. of Slocum's booklet. B&W photo is on p. 14 of S&B.

?? UK patent application 16,810. 1892. Not granted, so never published. I have spoken to the UK Patent Office and they say the paperwork for ungranted applications is destroyed after about three to five years. (Edward Hordern's collection has an example with this number on it, by Feltham & Co. In the 2nd ed., the cover is reproduced and it looks like the number may be 16,310, but that number is for a locomotive vehicle.) 14: 00149. Slocum 14.20.1. Manufactured as: The Chequers Puzzle, by Feltham & Co.

Hoffmann. 1893. Chap. III, no. 16: The chequers puzzle, pp. 97-98 & 129-130 = Hoffmann-Hordern, pp. 88-89, with photos. 14: 00149. Slocum 14.20.1. Says it is made by Messrs. Feltham, who state it has over 50 solutions. He gives two solutions. Photo on p. 89 of a example by Feltham & Co., dated 1880-1895.

At the end of the solution, he says Jacques & Son are producing a series of three "Peel" puzzles, which have coloured squares which have to be arranged so the same colour is not repeated in any row or column. Photo on p. 89 shows an example, 9: 023, with the trominoes all being L-trominoes. This makes a 5 x 5 square, but the colours have almost faded into indistinguishability.

Montgomery Ward & Co. Catalog No 57, Spring & Summer, 1895. Facsimile by Dover, 1969, ??NX. P. 237 describes item 25470: "The "Wonder" Puzzle. The object is to place 18 pieces of 81 squares together, so as to form a square, with the colors running alternately. It can be done in several different ways."

Dudeney. Problem 517 -- Make a chessboard. Weekly Dispatch (4 & 18 Oct 1903), both p. 10. 8: 00010 12111 001. Slocum 8.3.1.

Benson. 1904. The chequers puzzle, pp. 202-203. As in Hoffmann, with only one solution.

Dudeney. The Tribune (20 & 24 Dec 1906) both p. 1. ??NX Dissecting a chessboard. Dissect into maximum number of different pieces. Gets 18: 2,1,4,10,0, 0,0,1. Slocum 18.1, citing later(?) Loyd versions.

Loyd. Sam Loyd's Puzzle Magazine (Apr-Jul 1908) -- ??NYS, reproduced in: A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; no. 58, p. 52. = Cyclopedia, 1914, pp. 221 & 368, 250 & 373. = MPSL2, prob. 71, pp. 51 & 145. = SLAHP: Dissecting the chessboard, pp. 19 & 87. Cut into maximum number of different pieces -- as in Dudeney, 1906.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The rug, pp. 7-13 & 65. 14: 00149. Not in Slocum.

Loyd. A battle royal. Cyclopedia, 1914, pp. 97 & 351 (= MPSL1, prob. 51, pp. 49 & 139). Same as Dudeney's prob. 517 of 1903.

Dudeney. AM. 1917. Prob. 293: The Chinese chessboard, pp. 87 & 213-214. Same as Loyd, p. 221.

Western Puzzle Works, 1926 Catalogue. No. 79: "Checker Board Puzzle, in 16 pieces", but the picture only shows 14 pieces. 14: 00149. Picture doesn't show any colours, but assuming the standard colouring of a chess board, this is the same as Slocum 14.15.

John Edward Fransen. US Patent 1,752,248 -- Educational Puzzle. Applied: 19 Apr 1929; patented: 25 Mar 1930. 1p + 1p diagrams. 'Cut thy life.' 11: 10101 43001. Slocum 11.3.1.

Emil Huber-Stockar. Patience de l'echiquier. Comptes-Rendus du Premier Congrès International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 93-94. 15: 01329. Slocum 15.5. Says there must certainly be more than 1000 solutions.

Emil Huber-Stockar. L'echiquier du diable. Comptes-Rendus du Deuxième Congrès International de Récréation Mathématique, Paris, 1937. Librairie du "Sphinx", Bruxelles, 1937, pp. 64-68. Discusses how one solution can lead to many others by partial symmetries. Shows several solutions containing about 40 altogether. Note at end says he has now got 5275 solutions. This article is reproduced in Sphinx 8 (1938) 36-41, but without the extra pages of diagrams. At the end, a note says he has 5330 solutions. Ibid., pp. 75-76 says he has got 5362 solutions and ibid. 91-92 says he has 5365. By use of Bayes' theorem on the frequency of new solutions, he estimates c5500 solutions. Haubrich has found 6013. Huber-Stocker intended to produce a book of solutions, but he died in May 1939 [Sphinx 9 (1939) 97].

F. Hansson. Sam Loyd's 18-piece dissection -- Art. 48 & probs. 4152-4153. Fairy Chess Review 4:3 (Nov 1939) 44. Cites Loyd's Puzzles Magazine. Asserts there are many millions of solutions! He determines the number of chequered handed n-ominoes for n = 1, 2, ..., 8 is 2, 1, 4, 10, 36, 110, 392, 1371. The first 17 pieces total 56 squares. Considers 8 ways to dissect the board into 18 different pieces. Problems ask for the number of ways to choose the pieces in each of these ways and for symmetrical solutions. Solution in 4:6 (Jun 1940) 93-94 (??NX of p. 94) says there are a total of 3,309,579 ways to make the choices.

C. Dudley Langford. Note 2864: A chess-board puzzle. MG 43 (No. 345) (Oct 1959) 200. 15: 01248. Not in Slocum. Two diagrams followed by the following text. "The pieces shown in the diagrams can be arranged to form a square with either side uppermost. If the squares of the underlying grid are coloured black and white alternately, with each white square on the back of a black square, then there is at least one more way of arranging them as a chess-board by turning some of the pieces over." I thought this meant that the pieces were double-sided with the underside having the colours being the reverse of the top and the two diagrams were two solutions for this set of pieces. Jacques Haubrich has noted that the text is confusing and that the second diagram is NOT using the set of double-sided pieces which are implied by the first diagram. We are not sure if the phrasing is saying there are two different sets of pieces and hence two problems or if we are misinterpreting the description of the colouring.

B. D. Josephson. EDSAC to the rescue. Eureka 24 (Oct 1961) 10-12 & 32. Uses the EDSAC computer to find two solutions of a 12 piece chessboard dissection. 12: 00025 41. Slocum 12.9.

Leonard J. Gordon. Broken chessboards with unique solutions. G&PJ 10 (1989) 152-153. Shows Dudeney's problem has four solutions. Finds other colourings which give only one solution. Notes some equivalences in Slocum.

6.F.2. COVERING DELETED CHESSBOARD WITH DOMINOES

See also 6.U.2.

There is nothing on this in Murray.

Pál Révész. Op. cit. in 5.I.1. 1969. On p. 22, he says this problem comes from John [von] Neumann, but gives no details.

Max Black. Critical Thinking, op. cit. in 5.T. 1946 ed., pp. 142 & 394, ??NYS. 2nd ed., 1952, pp. 157 & 433. He simply gives it as a problem, with no indication that he invented it.

H. D. Grossman. Fun with lattice points: 14 -- A chessboard puzzle. SM 14 (1948) 160. (The problem is described with 'his clever solution' from M. Black, Critical Thinking, pp. 142 & 394.)

S. Golomb. 1954. Op. cit. in 6.F.

M. Gardner. The mutilated chessboard. SA (Feb 1957) = 1st Book, pp. 24 & 28.

Gamow & Stern. 1958. Domino game. Pp. 87-90.

Robert S. Raven, proposer; Walter P. Targoff, solver. Problem 85 -- Deleted checkerboard. In: L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 52 & 227.

R. E. Gomory. (Solution for deletion of any two squares of opposite colour.) In: M. Gardner, SA (Nov 1962) = Unexpected, pp. 186-187. Solution based on a rook's tour. (I don't know if this was ever published elsewhere.)

Michael Holt. What is the New Maths? Anthony Blond, London, 1967. Pp. 68 & 97. Gives the 4 x 4 case as a problem, but doesn't mention that it works on other boards. (I include this as I haven't seen earlier examples in the educational literature.)

David Singmaster. Covering deleted chessboards with dominoes. MM 48 (1975) 59-66. Optimum extension to n-dimensions. For an n-dimensional board, each dimension must be ( 2. If the board has an even number of cells, then one can delete any n-1 white cells and any n-1 black cells and still cover the board with dominoes (i.e. 2 x 1 x 1 x ... x 1 blocks). If the board has an odd number of cells, then let the corner cells be coloured black. One can then delete any n black cells and any n-1 white cells and still cover the board with dominoes.

I-Ping Chu & Richard Johnsonbaugh. Tiling deficient boards with trominoes. MM 59:1 (1986) 34-40. (3,n) = 1 and n ( 5 imply that an n x n board with one cell deleted can be covered with L trominoes. Some 5 x 5 boards with one cell deleted can be tiled, but not all can.

6.F.3. DISSECTING A CROSS INTO Zs AND Ls

The L pieces are not always drawn carefully, and in some cases the unit pieces are not all square. I have enlarged and measured those which are not clear and approximated them as n-ominoes.

Minguet. 1733. Pp. 119-121 (1755: 85-86; 1822: 138-139; 1864: 116-117). The problem has two parts. The first is a cross into 5 pieces: L-tetromino, 2 Z-pentominoes, L-hexomino, Z-hexomino. The two hexominoes are like the corresponding pentominoes lengthened by one unit. Similar to Les Amusemens, but one Z is longer and one L is shorter. The diagram shows 8 L and Z shaped pieces formed from squares, but it is not clear what the second part of the problem is doing -- either a piece or a label is erroneous or missing. Says one can make different figures with the pieces.

Les Amusemens. 1749. P. xxxi. Cross into 3 Z pentominoes and 2 L pieces. Like Minguet, but the Ls are much lengthened and are approximately a L-heptomino and an L-octomino.

Catel. Kunst-Cabinet. 1790. Das mathematische Kreuz, p. 10 & fig. 27 on plate I. As in Les Amusemens, but the Ls are approximately a 9-omino and a 10-omino.

Bestelmeier. 1801. Item 274 -- Das mathematische Kreuz. Cross into 6 pieces, but the picture has an erroneous extra line. It should be the reversal of the picture in Catel.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has the dissection of the cross into 3 Z pentominoes and two L pieces. I don't have a copy of this, but my sketch looks like the Ls are a tetromino and a pentomino, or possibly a pentomino and a hexomino.

Manuel des Sorciers. 1825. Pp. 204-205, art. 21. ??NX. Dissect a cross into three Zs and two Ls. My notes don't indicate the size of the Ls.

Boy's Own Book. 1843 (Paris): 435 & 440, no. 3. As in Les Amusemens, but with the Ls apparently intended to be a pentomino and a hexomino. = Boy's Treasury, 1844, pp. 424-425 & 428. = de Savigny, 1846, pp. 353 & 357, no. 2, except the solution has been redrawn with some slight changes and so the proportions are less clear.

Family Friend 3 (1850) 330 & 351. Practical puzzle, No. XXI. As in Les Amusemens.

Magician's Own Book. 1857. Prob. 31: Another cross puzzle, pp. 276 & 299. As in Les Amusemens.

Landells. Boy's Own Toy-Maker. 1858. P. 152. As in Les Amusemens.

Book of 500 Puzzles. 1859. Prob. 31: Another cross puzzle, pp. 90 & 113. As in Les Amusemens. = Magician's Own Book.

Indoor & Outdoor. c1859. Part II, p. 127, prob. 5: The puzzle of the cross. As in Les Amusemens.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 24, pp. 399 & 439. Identical to Magician's Own Book.

Boy's Own Conjuring book. 1860. Prob. 30: Another cross puzzle, pp. 239 & 263. = Magician's Own Book, 1857.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 584-2, pp. 286 & 404. 4  Z pentominoes to make a (Greek) cross. (Also entered in 6.F.5.)

Prob. 584-8, pp. 287 & 405. 3 Z pentominoes, L tetromino and L pentomino to make a Greek cross. Despite specifically asking for a Greek cross, the answer is a standard Latin cross with height : width = 4 : 3.

Mittenzwey. 1880. Prob. 173-174, pp. 33 & 85; 1895?: 198-199, pp. 38 & 87; 1917: 198-199, pp. 35 & 84. The first is 3 Z pentominoes, L tetromino and L pentomino to make a cross. The second is 4  Z pentominoes to make a (Greek) cross. (Also entered in 6.F.5.)

Cassell's. 1881. P. 93: The magic cross. = Manson, 1911, p. 139. Same pattern as Les Amusemens, but one end of the Zs is decidedly longer than the other and the middle 'square' of the Zs is decidedly not square. The Ls are approximately a pentomino and a heptomino, But the middle 'square' of the Zs is almost a domino and that makes the Zs into heptominoes, with the Ls being a hexomino and a nonomino.

S&B, p. 20, shows a 7 piece cross dissection, Jeu de La Croix, into 3 Zs, 2 Ls and 2 straights, from c1890. The Zs are pentominoes, with the centre 'square' lengthened a bit. The Ls appear to be a heptomino and an octomino and the straights appear to be a hexomino and a tetromino. Cf Hoffmann-Hordern for a version without the straight pieces.

Handy Book for Boys and Girls. Showing How to Build and Construct All Kinds of Useful Things of Life. Worthington, NY, 1892. Pp. 320-321: The cross puzzle. As in Cassell's.

Hoffmann. 1893. Chap. III, no. 29: Another cross puzzle, pp. 103 & 136 = Hoffmann-Hordern, pp. 100-101, with photo. States that the two Ls are the same shape, but the solution is as in Les Amusemens, with the Ls approximately a hexomino and a heptomino. Hordern has corrected the problem statement. Photo on p. 100 shows an ivory version, dated 1850-1900, of the same proportions. Hordern Collection, p. 65, shows two wood versions, La Croix Brisée and Jeu de la Croix, dated 1880-1905, both with Ls being approximately a heptomino and an octomino.

Benson. 1904. The Latin cross puzzle, p. 200. As in Hoffmann, but the solution is longer, as in Les Amusemens.

Wehman. New Book of 200 Puzzles. 1908. Another cross puzzle, p. 32. As in Les Amusemens, with the Ls being a pentomino and a hexomino.

S. Szabo. US Patent 1,263,960 -- Puzzle. Filed: 20 Oct 1917; patented: 23 Apr 1918. 1p + 1p diagrams. As in Les Amusemens, with even longer Ls, approximately a 10-omino and an 11-omino.

6.F.4. QUADRISECT AN L-TROMINO, ETC.

See also 6.AW.1 & 4.

Mittenzwey and Collins quadrisect a hollow square obtained by removing a 2 x 2 from the centre of a 4 x 4.

Bile Beans quadrisects a 5 x 5 after deleting corners and centre.

Minguet. 1733. Pp. 114-115 (1755: 80; 1822: 133-134; 1864: 111-112). Quadrisect L-tromino.

Alberti. 1747. Art. 30: Modo di dividere uno squadro di carta e di legno in quattro squadri equali, p. ?? (131) & fig. 56, plate XVI, opp. p. 130.

Les Amusemens. 1749. P. xxx. L-tromino ("gnomon") into 4 congruent pieces.

Vyse. Tutor's Guide. 1771? Prob. 9, 1793: p. 305, 1799: p. 317 & Key p. 358. Refers to the land as a parallelogram though it is drawn rectangular.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has an entry "8½ a Prob of figure" followed by the L-tromino. 8½ b is the same with a mitre and there are other dissection problems adjacent -- see 6.F.3, 6.AQ, 6.AW.1, 6.AY, so it seems clear that he knew this problem.

Jackson. Rational Amusement. 1821. Geometrical Puzzles, no. 3, pp. 23 & 83 & plate I, fig. 2.

Manuel des Sorciers. 1825. Pp. 203-204, art. 20. ??NX. Quadrisect L-tromino.

Family Friend 2 (1850) 118 & 149. Practical Puzzle -- No. IV. Quadrisect L-tromino of land with four trees.

Family Friend 3 (1850) 150 & 181. Practical puzzle, No. XV. 15/16 of a square with 10 trees to be divided equally. One tree is placed very close to another, cf Magician's Own Book and Hoffmann, below.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 8, p. 179 (1868: 190). Land in the shape of an L-tromino to be cut into four congruent parts, each with a cherry tree.

Magician's Own Book. 1857.

Prob. 3: The divided garden, pp. 267 & 292. 15/16 of a square to be divided into five (congruent) parts, each with two trees. The missing 1/16 is in the middle. One tree is placed very close to another, cf Family Friend 3, above, and Hoffmann below.

Prob. 22: Puzzle of the four tenants, pp. 273 & 296. Same as Parlour Pastime, but with apple trees. (= Illustrated Boy's Own Treasury, 1860, No. 10, pp. 397 & 437.)

Prob. 28: Puzzle of the two fathers, pp. 275-276 & 298. Each father wants to divide 3/4 of a square. One has L-tromino, other has the mitre shape. See 6.AW.1.

Landells. Boy's Own Toy-Maker. 1858.

P. 144. = Magician's Own Book, prob. 3.

Pp. 148-149. = Magician's Own Book, prob. 27.

Book of 500 Puzzles. 1859.

Prob. 3: The divided garden, pp. 81 & 106. Identical to Magician's Own Book.

Prob. 22: Puzzle of the four tenants, pp. 87 & 110. Identical to Magician's Own Book.

Prob. 28: Puzzle of the two fathers, pp. 89-90 & 112. Identical to Magician's Own Book. See also 6.AW.1.

Charades, Enigmas, and Riddles. 1860: prob. 28, pp. 59 & 63; 1862: prob. 29, pp. 135 & 141; 1865: prob. 573, pp. 107 & 154. Quadrisect L-tromino, attributed to Sir F. Thesiger.

Boy's Own Conjuring book. 1860.

Prob. 3: The divided garden, pp. 229 & 255. Identical to Magician's Own Book.

Prob. 21: Puzzle of the four tenants, pp. 235 & 260. Identical to Magician's Own Book.

Prob. 27: Puzzle of the two fathers, pp. 237-238 & 262. Identical to Magician's Own Book.

Illustrated Boy's Own Treasury. 1860. Prob. 21, pp. 399 & 439. 15/16 of a square to be divided into five (congruent) parts, each with two trees. c= Magician's Own Book, prob. 3.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 175, p. 88. L-tromino into four congruent pieces, each with two trees. The problem is given in terms of the original square to be divided into five parts, where the father gets a quarter of the whole in the form of a square and the four sons get congruent pieces.

Hanky Panky. 1872. The divided orchards, p. 130. L-tromino into 4 congruent pieces, each with two trees.

Boy's Own Book. The divided garden. 1868: 675. = Magician's Own Book, prob. 3.

Mittenzwey. 1880.

Prob. 192, pp. 36 & 89; 1895?: 217, pp. 40 & 91; 1917: 217, pp. 37 & 87. Cut 1 x 1 out of the centre of a 4 x 4. Divide the rest into five parts of equal area with four being congruent. He cuts a 2 x 2 out of the centre, which has a 1 x 1 hole in it, then divides the rest into four L-trominoes.

Prob. 213, pp. 38 & 90; 1895?: 238, pp. 42 & 92; 1917: 238, pp. 39 & 88. Usual quadrisection of an L-tromino.

Prob. 214, pp. 38 & 90; 1895?: 239, pp. 42 & 92; 1917: 239, pp. 39 & 88. Square garden with mother receiving 1/4 and the rest being divided into four congruent parts.

Cassell's. 1881. P. 90: The divided farm. = Manson, 1911, pp. 136-137. = Magician's Own Book, prob. 3.

Lemon. 1890.

The divided garden, no. 259, pp. 38 & 107. = Magician's Own Book, prob. 3.

Geometrical puzzle, no. 413, pp. 55 & 113 (= Sphinx, no. 556, pp. 76 & 116). Quadrisect L-tromino.

Hoffmann. 1893. Chap. X, no. 41: The divided farm, pp. 352-353 & 391 = Hoffmann-Hordern, p. 250. = Magician's Own Book, prob. 3. [One of the trees is invisible in the original problem, but Hoffmann-Hordern has added it, in a more symmetric pattern than in Magician's Own Book.]

Loyd. Origin of a famous puzzle -- No. 18: An ancient puzzle. Tit-Bits 31 (13 Feb & 6 Mar 1897) 363 & 419. Nearly 50 years ago he was told of the quadrisection of 3/4 of a square, but drew the mitre shape instead of the L-tromino. See 6.AW.1.

Clark. Mental Nuts. 1897, no. 73; 1904, no. 31. Dividing the land. Quadrisect an L-tromino. 1904 also has the mitre -- see 6.AW.1.

Benson. 1904. The farmer's puzzle, p. 196. Quadrisect an L-tromino.

Wehman. New Book of 200 Puzzles. 1908.

The divided garden, p. 17. = Magician's Own Book, prob. 3

Puzzle of the two fathers, p. 43. = Magician's Own Book, prob. 28.

Puzzle of the four tenants, p. 46. = Magician's Own Book, prob. 22.

Dudeney. Some much-discussed puzzles. Op. cit. in 2. 1908. Land in shape of an L-tromino to be quadrisected. He says this is supposed to have been invented by Lord Chelmsford (Sir F. Thesiger), who died in 1878 -- see Charades, Enigmas, and Riddles (1860). But cf Les Amusemens.

M. Adams. Indoor Games. 1912. The clever farmer, pp. 23-25. Dissect L-tromino into four congruent pieces.

Blyth. Match-Stick Magic. 1921. Dividing the inheritance, pp. 20-21. Usual quadrisection of L-tromino set out with matchsticks.

Collins. Book of Puzzles. 1927. The surveyor's puzzle, pp. 2-3. Quadrisect 3/4 of a square, except the deleted 1/4 is in the centre, so we are quadrisecting a hollow square -- cf Mittenzwey,

The Bile Beans Puzzle Book. 1933.

No. 22: Paper squares. Quadrisect a P-pentomino into P-pentominoes. One solution given, I find another. Are there more? How about quadrisecting into congruent pentominoes? Which pentominoes can be quadrisected into four copies of themself?

No. 41: Five lines. Consider a 5 x 5 square and delete the corners and centre. Quadrisect into congruent pentominoes. One solution given. I find three more. Are there more? One can extend this to consider quadrisecting the 5 x 5 with just the centre removed into congruent hexominoes. I find seven ways.

Depew. Cokesbury Game Book. 1939. A plot of ground, p. 227. 3/4 of XX     

a square to be quadrisected, but the shape is as shown at the right. XXX  

X XX

XXXX

Ripley's Puzzles and Games. 1966. Pp. 18 & 19, item 8. Divide an L-tromino into eight congruent pieces.

F. Göbel. Problem 1771: The L-shape dissection problem. JRM 22:1 (1990) 64-65. The L-tromino can be dissected into 2, 3, or 4 congruent parts. Can it be divided into 5 congruent parts?

Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Apple-eating monsters, pp. 40 & 63. Trisect into equal parts, the shape consisting of a 2 x 4 rectangle with a 1 x 1 square attached to one of the central squares of the long side. [Actually, this can be done with the square attached to any of the squares, though if it is attached to the end of the long side, the resulting pieces are straight trominoes.]

6.F.5. OTHER DISSECTIONS INTO POLYOMINOES

Catel. Kunst-Cabinet. 1790.

Das Zakk- und Hakenspiel, p. 10 & fig. 11 on plate 1. 4 Z-pentominoes and 4  L-tetrominoes make a 6 x 6 square.

Die zwolf Winkelhaken, p. 11 & fig. 26 on plate 1. 8 L-pentominoes and 4  L-hexominoes make a 8 x 8 square.

Bestelmeier. 1801. Item 61 -- Das Zakken und Hakkenspiel. As in Catel, p. 10, but not as regularly drawn. Text copies some of Catel.

Manuel des Sorciers. 1825. Pp. 203-204, art. 20. ??NX Use four L-trominoes to make a 3 x 4 rectangle or a 4 x 4 square with four corners deleted.

Family Friend 3 (1850) 90 & 121. Practical puzzle -- No. XIII. 4 x 4 square, with 12 trees in the corners, centres of sides and four at the centre of the square, to be divided into 4 congruent parts each with 3 trees. Solution uses 4 L-tetrominoes. The same problem is repeated as Puzzle 17 -- Twelve-hole puzzle in (1855) 339 with solution in (1856) 28.

Magician's Own Book. 1857. Prob. 14: The square and circle puzzle, pp. 270 & 295. Same as Family Friend. = Book of 500 Puzzles, 1859, prob. 14, pp. 84 & 109. = Boy's Own Conjuring book, 1860, prob. 13, pp. 231-232 & 257. c= Illustrated Boy's Own Treasury, 1860, prob. 8, pp. 396 & 437. c= Hanky Panky, 1872, A square of four pieces, p. 117.

Landells. Boy's Own Toy-Maker. 1858. Pp. 146-147. Identical to Family Friend.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 584-2, pp. 286 & 404. 4  Z-pentominoes to make a Greek cross. (Also entered in 6.F.3.)

Prob. 584-3, pp. 286 & 404. 4 L-tetrominoes to make a square.

Prob. 584-5, pp. 286 & 404. 8  L-pentominoes and 4  L-hexominoes make a 8 x 8 square. Same as Catel, but diagram is inverted.

Prob. 584-7, pp. 287 & 405. 4 Z-pentominoes and 4  L-tetrominoes make a 6 x 6 square. Same as Catel, but diagram is inverted.

Mittenzwey. 1880.

Prob. 174, pp. 33 & 85; 1895?: 199, pp. 38 & 87; 1917: 199, pp. 35 & 84. 4  Z pentominoes to make a (Greek) cross. (Also entered in 6.F.5.)

Prob. 186, pp. 35 & 88; 1895?: 211, pp. 40 & 90; 1917: 211, pp. 36 & 87. 4 x 4 square into 4 L-tetrominoes.

Prob. 187, pp. 35 & 88; 1895?: 212, pp. 40 & 90; 1917: 212, pp. 36 & 87. 6 x 6 square into 4 Z-pentominoes and 4  L-tetrominoes, as in Catel, p. 10.

Prob. 215, pp. 38 & 90; 1895?: 240, pp. 42 & 92; 1917: 240, pp. 39 & 88. Square garden with 12 trees quadrisected into four L-tetrominoes.

S&B, p. 20, shows a 7 piece cross dissection into 3 Zs, 2 Ls and 2 straights, from c1890.

Hoffmann. 1893. Chap. X, no. 37: The orchard puzzle, pp. 350 & 390 = Hoffmann-Hordern, pp. 247, with photo. Same as Family Friend 3. Photo on p. 247 shows St. Nicholas Puzzle Card, © 1892 in the USA.

Tom Tit, vol 3. 1893. Les quatre Z et des quatre L, pp. 181-182. = K, No. 27: The four Z's and the four L's, pp. 70-71. = R&A, Squaring the L's and Z's, p. 102. 6 x 6 square as in Catel, p. 10.

Sphinx. 1895. The Maltese cross, no. 181, pp. 28 & 103. Make a Maltese cross (actually a Greek cross of five equal squares) from 4 P-pentominoes. Also: quadrisect a P-pentomino.

Wehman. New Book of 200 Puzzles. 1908. The square and circle puzzle, p. 5. = Family Friend.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The Zoltan's orchard, pp. 24-28 & 64. = Family Friend.

Anon. Prob. 84. Hobbies 31 (No. 799) (4 Feb 1911) 443. Use at least one each of: domino; L-tetromino; P and X pentominoes to make the smallest possible square Due to ending of this puzzle series, no solution ever appeared. I find numerous solutions for 5 x 5, 6 x 6, 8 x 8, of which the first is easily seen to be the smallest possibility.

A. Neely Hall. Carpentry & Mechanics for Boys. Lothrop, Lee & Shepard, Boston, nd [1918]. The square puzzle, pp. 20-21. 7 x 7 square cut into 1 straight tromino, 1  L-tetromino and 7 L-hexominoes.

Collins. Book of Puzzles. 1927. The surveyor's puzzle, pp. 2-3. Quadrisect 3/4 of a square, except the deleted 1/4 is in the centre, so we are quadrisecting a hollow square.

Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared in 1908, but versions continued until the 1950s. This looks like 1930s?? 4  Z-pentominoes and 4  L-tetrominoes make a 6 x 6 square and a 4 x 9 rectangle.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. All square, pp. 42 & 129. Make a 6 x 6 square from the staircase hexomino, 2 Y-pentominoes, an N-tetromino, an L-tetromino and 3 T-tetrominoes. None of the pieces is turned over in the solution, though this restriction is not stated.

6.G. SOMA CUBE

Piet Hein invented the Soma Cube in 1936. (S&B, pp. 40-41.) ??Is there any patent??

M. Gardner. SA (Sep 1958) = 2nd Book, Chap 6.

Richard K. Guy. Loc. cit. in 5.H.2, 1960. Pp. 150-151 discusses cubical solutions -- 234 found so far. He proposes the 'bath' shape -- a 5 x 3 x 2 cuboid with a 3 x 1 x 1 hole in the top layer. In a 1985 letter, he said that O'Beirne had introduced the Soma to him and his family. in 1959 and they found 234 solutions before Mike Guy went to Cambridge -- see below.

P. Hein, et al. Soma booklet. Parker Bros., 1969, 56pp. Asserts there are 240 simple solutions and 1,105,920 total solutions, found by J. H. Conway & M. J. T. Guy with a a computer (but cf Gardner, below) and by several others. [There seem to be several versions of this booklet, of various sizes.]

Thomas V. Atwater, ed. Soma Addict. 4 issues, 1970-1971, produced by Parker Brothers. (Gardner, below, says only three issues appeared.) ??NYS -- can anyone provide a set or photocopies??

M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. States there are 240 solutions for the cube, obtained by many programs, but first found by J. H. Conway & M. J. T. Guy in 1962, who did not use a computer, but did it by hand "one wet afternoon". Richard Guy's 1985 letter notes that Mike Guy had a copy of the Guy family's 234 solutions with him.

SOMAP ??NYS -- ??details. (Schaaf III 52)

Winning Ways, 1982, II, 802-803 gives the SOMAP.

Jon Brunvall et al. The computerized Soma Cube. Comp. & Maths. with Appl. 12B:1/2 (1986 [Special issues 1/2 & 3/4 were separately printed as: I. Hargittai, ed.; Symmetry -- Unifying Human Understanding; Pergamon, 1986.] 113-121. They cite Gardner's 2nd Book which says the number of solutions is unknown and they use a computer to find them.

6.G.1. OTHER CUBE DISSECTIONS

See also 6.N, 6.U.2, 6.AY.1 and 6.BJ. The predecessors of these puzzles seem to be the binomial and trinomial cubes showing (a+b)3 and (a+b+c)3. I have an example of the latter from the late 19C. Here I will consider only cuts parallel to the cube faces -- cubes with cuts at angles to the faces are in 6.BJ. Most of the problems here involve several types of piece -- see 6.U.2 for packing with one kind of piece.

Catel. Kunst-Cabinet. 1790. Der algebraische Würfel, p. 6 & fig 50 on plate II. Shows a binomial cube: (a + b)3 = a3 + 3a2b + 3ab2 + b3.

Bestelmeier. 1801. Item 309 is a binomial cube, as in Catel. "Ein zerschnittener Würfel, mit welchem die Entstehung eines Cubus, dessen Seiten in 2 ungleiche Theile a + b getheilet ist, gezeigt ist."

Hoffmann. 1893. Chap. III, no. 39: The diabolical cube, pp. 108 & 142 = Hoffmann-Hordern, pp. 108-109, with photos. 6: 0, 1, 1, 1, 1, 1, 1, i.e. six pieces of volumes 2, 3, 4, 5, 6, 7. Photos on p. 108 shows Cube Diabolique and its box, by Watilliaux, dated 1874-1895.

J. G.-Mikusiński. French patent. ??NYS -- cited by Steinhaus.

H. Steinhaus. Mikusiński's Cube. Mathematical Snapshots. Not in Stechert, 1938, ed. OUP, NY: 1950: pp. 140-142 & 263; 1960, pp. 179-181 & 326; 1969 (1983): pp. 168-169 & 303.

John Conway. In an email of 7 Apr 2000, he says he developed the dissection of the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 in c1960 and then adapted it to the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13 1 x 2 x 4 and the 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2. He says his first publication of it was in Winning Ways, 1982 (cf below).

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. Discusses Hoffmann's Diabolical Cube and Mikusiński's cube. Says he has 8 solutions for the first and that there are just 2 for the second. The Addendum reports that Wade E. Philpott showed there are just 13 solutions of the Diabolical Cube. Conway has confirmed this. Gardner briefly describes the solutions. Gardner also shows the Lesk Cube, designed by Lesk Kokay (Mathematical Digest [New Zealand] 58 (1978) ??NYS), which has at least 3 solutions.

D. A. Klarner. Brick-packing puzzles. JRM 6 (1973) 112-117. Discusses 3 x 3 x 3 into 3  1 x 1 x 1 and 6 1 x 2 x 2 attributed to Slothouber-Graatsma; Conway's 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2; Conway's 5 x 5 x 5 into 3 1 x 1 x 3, 1 2 x 2 x 2, 1  1 x 2 x 2 and 13 1 x 2 x 4. Because of the attribution to Slothouber & Graatsma and not knowing the date of Conway's work, I had generally attributed the 3 x 3 x 3 puzzle to them and Stewart Coffin followed this in his book. However, it now seems that it really is Conway's invention and I must apologize for misleading people.

Leisure Dynamics, the US distributor of Impuzzables, a series of 6 3 x 3 x 3 cube dissections identified by colours, writes that they were invented by Robert Beck, Custom Concepts Inc., Minneapolis. However, the Addendum to Gardner, above, says they were designed by Gerard D'Arcey.

Winning Ways. 1982. Vol. 2, pp. 736-737 & 801. Gives the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 and the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13  1 x 2 x 4, which is called Blocks-in-a-Box. No mention of the other 5 x 5 x 5. Mentions Foregger & Mather, cf in 6.U.2.

Michael Keller. Polycube update. World Game Review 4 (Feb 1985) 13. Reports results of computer searches for solutions. Hoffmann's Diabolical Cube has 13; Mikusinski's Cube has 2; Soma Cube has 240; Impuzzables: White -- 1; Red -- 1; Green -- 16; Blue -- 8; Orange -- 30; Yellow -- 1142.

Michael Keller. Polyform update. World Game Review 7 (Oct 1987) 10-13. Says that Nob Yoshigahara has solved a problem posed by O'Beirne: How many ways can 9  L-trominoes make a cube? Answer is 111. Gardner, Knotted, chap. 3, mentioned this. Says there are solutions with n L-trominoes and 9-n straight trominoes for n ( 1 and there are 4 solutions for n = 0. Says the Lesk Cube has 4 solutions. Says Naef's Gemini Puzzle was designed by Toshiaki Betsumiya. It consists of the 10 ways to join two 1 x 2 x 2 blocks.

H. J. M. van Grol. Rik's Cube Kit -- Solid Block Puzzles. Analysis of all 3 x 3 x 3 unit solid block puzzles with non-planar 4-unit and 5-unit shapes. Published by the author, The Hague, 1989, 16pp. There are 3 non-planar tetracubes and 17 non-planar pentacubes. A 3 x 3 x 3 cube will require the 3 non-planar tetracubes and 3 of the non-planar pentacubes -- assuming no repeated pieces. He finds 190 subsets which can form cubes, in 1 to 10 different ways.

Nob Yoshigahara. (Title in Japanese: (Puzzle in Wood)). H. Tokuda, Sowa Shuppan, Japan, 1987. Pp. 68-69 is a 3^3 designed by Nob -- 6: 01005.

6.G.2. DISSECTION OF 63 INTO 33, 43 AND 53, ETC.

H. W. Richmond. Note 1672: A geometrical problem. MG 27 (No. 275) (Jul 1943) 142. AND Note 1704: Solution of a geometrical problem (Note 1672). MG 28 (No. 278) (Feb 1944) 31-32. Poses the problem of making such a dissection, then gives a solution in 12 pieces: three 1 x 3 x 3; 4 x 4 x 4; four 1 x 5 x 5; 1 x 4 x 4; two 1 x 1 x 2 and a V-pentacube.

Anon. [= John Leech, according to Gardner, below]. Two dissection problems, no. 2. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951) 23. Asks for such a dissection using at most 10 pieces. Gives an 8 piece solution due to R. F. Wheeler. [Cundy & Rollett; Mathematical Models; 2nd ed., pp. 203-205, say Eureka is the first appearance they know of this problem. See Gardner, below, for the identity of Leech.]

Richard K. Guy. Loc. cit. in 5.H.2, 1960. Mentions the 8 piece solution.

J. H. Cadwell. Some dissection problems involving sums of cubes. MG 48 (No. 366) (Dec 1964) 391-396. Notes an error in Cundy & Rollett's account of the Eureka problem. Finds examples for 123 + 13 = 103 + 93 with 9 pieces and 93 = 83 + 63 + 13 with 9 pieces.

J. H. Cadwell. Note 3278: A three-way dissection based on Ramanujan's number. MG 54 (No. 390) (Dec 1970) 385-387. 7 x 13 x 19 to 103 + 93 and 123 + 13 using 12 pieces.

M. Gardner. SA (Oct 1973) c= Knotted, chap. 16. He says that the problem was posed by John Leech. He gives Wheeler's initials as E. H. ?? He says that J. H. Thewlis found a simpler 8-piece solution, further simplified by T. H. O'Beirne, which keeps the 4 x 4 x 4 cube intact. This is shown in Gardner. Gardner also shows an 8-piece solution which keeps the 5 x 5 x 5 intact, due to E. J. Duffy, 1970. O'Beirne showed that an 8-piece dissection into blocks is impossible and found a 9-block solution in 1971, also shown in Gardner.

Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1984. Section 24.1, pp. 118-120 gives Wheeler's solution and admires it.

Richard K. Guy, proposer; editors & Charles H. Jepson [should be Jepsen], partial solvers. Problem 1122. CM 12 (1987) 50 & 13 (1987) 197-198. Asks for such dissections under various conditions, of which (b) is the form given in Eureka. Eight pieces is minimal in one case and seems minimal in two other cases. Eleven pieces is best known for the first case, where the pieces must be blocks, but this appears to be the problem solved by O'Beirne in 1971, reported in Gardner, above.

Charles H. Jepsen. Additional comment on Problem 1122. CM 14 (1988) 204-206. Gives a ten piece solution of the first case.

Chris Pile. Cube dissection. M500 134 (Aug 1993) 2-3. He feels the 1 x 1 x 2 piece occurring in Cundy & Rollett is too small and he provides another solution with 8 pieces, the smallest of which contains 8 unit cubes. Asks how uniform the piece sizes can be.

6.G.3. DISSECTION OF A DIE INTO NINE 1 x 1 x 3

Hoffmann. 1893. Chap. III, no. 17: The "Spots" puzzle, pp. 98-99 & 130-131 = Hoffmann-Hordern, pp. 90-91, with photo. Says it is made by Wolff & Son. Photo on p. 91 shows an example made by E. Wolff & Son, London.

Benson. 1904. The spots puzzle, pp. 203-204. As in Hoffmann.

Collins. Book of Puzzles. 1927. Pp. 131-134: The dissected die puzzle. The solution is different than Hoffmann's.

Rohrbough. Puzzle Craft. 1932. P. 21 shows a dissected die, but with no text. The picture is the same as in Hoffmann's solution.

Slocum. Compendium. Shows Diabolical Dice from Johnson Smith catalogue, 1935.

Harold Cataquet. The Spots puzzle revisited. CFF 33 (Feb 1994) 20-21. Brief discussion of two versions.

David Singmaster. Comment on the "Spots" puzzle. 29 Sep 1994, 2pp. Letter in response to the above. I note that there is no standard pattern for a die other than the opposite sides adding to seven. There are 23 = 8 ways to orient the spots forming 2, 3, and 6. There are two handednesses, so there are 16 dice altogether. (This was pointed out to me perhaps 10 years before by Richard Guy and Ray Bathke. I have since collected examples of all 16 dice.) However, Ray Bathke showed me Oriental dice with the two spots of the 2 placed horizontal or vertically rather than diagonally, giving another 16 dice (I have 5 types), making 32 dice in all. A die can be dissected into 9 1 x 1 x 3 pieces in 6 ways if the layers have to alternate in direction, or in 21 ways in general. I then pose a number of questions about such dissections.

6.G.4. USE OF OTHER POLYHEDRAL PIECES

S&B. 1986. P. 42 shows Stewart Coffin's 'Pyramid Puzzle' using pieces made from truncated octahedra and his 'Setting Hen' using pieces made from rhombic dodecahedra. Coffin probably devised these in the 1960s -- perhaps his book has some details of the origins of these ideas. ??check.

Mark Owen & Matthew Richards. A song of six splats. Eureka 47 (1987) 53-58. There are six ways to join three truncated octahedra. For reasons unknown, these are called 3-splats. They give various shapes which can and which cannot be constructed from the six 3-splats.

6.H. PICK'S THEOREM

Georg Pick. Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag (NS) 19 (1899) 311-319. Pp. 311-314 gives the proof, for an oblique lattice. Pp. 318-319 gives the extension to multiply connected and separated regions. Rest relates to number theory. [I have made a translation of the material on Pick's Theorem.]

Charles Howard Hinton. The Fourth Dimension. Swan Sonnenschein & Co., London, 1906. Metageometry, pp. 46-60. [This material is in Speculations on the Fourth Dimension, ed. by R. v. B. Rucker; Dover, 1980, pp. 130-141. Rucker says the book was published in 1904, so my copy may be a reprint??] In the beginning of this section, he draws quadrilateral shapes on the square lattice and determines the area by counting points, but he counts I + E/2 + C/4, which works for quadrilaterals but is not valid in general.

H. Steinhaus. O mierzeniu pól płaskich. Przegląd Matematyczno-Fizyczny 2 (1924) 24-29. Gives a version of Pick's theorem, but doesn't cite Pick. (My thanks to A. Mąkowski for an English summary of this.)

H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938, pp. 16-17 & 132. OUP, NY: 1950: pp. 76-77 & 260 (note 77); 1960: pp. 99-100 & 324 (note 95); 1969 (1983): pp. 96-97 & 301 (note 107). In 1938 he simply notes the theorem and gives one example. In 1950, he outlines Pick's argument. He refers to Pick's paper, but in "Ztschr. d. Vereins 'Lotos' in Prag". Steinhaus also cites his own paper, above.

J. F. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957) 378-395. Deals with the failure of the obvious form of Pick's theorem in 3-D and finds a valid generalization.

Ivan Niven & H. S. Zuckerman. Lattice points and polygonal area. AMM 74 (1967) 1195-1200. Straightforward proof. Mention failure for tetrahedra.

D. W. De Temple & J. M. Robertson. The equivalence of Euler's and Pick's theorems. MTr 67 (1974) 222-226. ??NYS.

W. W. Funkenbusch. From Euler's formula to Pick's formula using an edge theorem. AMM 81 (1974) 647-648. Easy proof though it could be easier.

R. W. Gaskell, M. S. Klamkin & P. Watson. Triangulations and Pick's theorem. MM 49 (1976) 35-37. A bit roundabout.

Richard A. Gibbs. Pick iff Euler. MM 49 (1976) 158. Cites DeTemple & Robertson and observes that both Pick and Euler can be proven from a result on triangulations.

John Reay. Areas of hex-lattice polygons, with short sides. Abstracts Amer. Math. Soc. 8:2 (1987) 174, #832-51-55. Gives a formula for the area in terms of the boundary and interior points and the characteristic of the boundary, but it is an open question to determine when this formula gives the actual area.

6.I. SYLVESTER'S PROBLEM OF COLLINEAR POINTS

If a set of non-collinear points in the plane is such that the line through any two points of the set contains a third point of the set, then the set is infinite.

J. J. Sylvester. Question 11851. The Educational Times 46 (NS, No. 383) (1 Mar 1893) 156.

H. J. Woodall & editorial comment. Solution to Question 11851. Ibid. (No. 385) (1 May 1893) 231. A very spurious solution.

(The above two items appear together in Math. Quest. with their Sol. Educ. Times 59 (1893) 98-99.)

E. Melchior. Über Vielseite der projecktiven Ebene. Deutsche Math. 5 (1940) 461-475. Solution, but in a dual form.

P. Erdös, proposer; R. Steinberg, solver & editorial comment giving solution of T. Grünwald (later = T. Gallai). Problem 4065. AMM 50 (1943) 65 & 51 (1944) 169-171.

L. M. Kelly. (Solution.) In: H. S. M. Coxeter; A problem of collinear points; AMM 55 (1948) 26-28. Kelly's solution is on p. 28.

G. A. Dirac. Note 2271: On a property of circles. MG 36 (No. 315) (Feb 1952) 53-54. Replace 'line' by 'circle' in the problem. He shows this is true by inversion. He asks for an independent proof of the result, even for the case when two, three are replaced by three, four.

D. W. Lang. Note 2577: The dual of a well-known theorem. MG 39 (No. 330) (Dec 1955) 314. Proves the dual easily.

H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 4.7: Sylvester's problem of collinear points, pp. 65-66. Sketches history and gives Kelly's proof.

W. O. J. Moser. Sylvester's problem, generalizations and relatives. In his: Research Problems in Discrete Geometry 1981, McGill University, Montreal, 1981. Section 27, pp. 27-1 -- 27-14. Survey with 73 references. (This problem is not in Part 1 of the 1984 ed. nor in the 1986 ed.)

6.J. FOUR BUGS AND OTHER PURSUIT PROBLEMS

The general problem becomes too technical to remain recreational, so I will not try to be exhaustive here.

Arthur Bernhart.

Curves of pursuit. SM 20 (1954) 125-141.

Curves of pursuit -- II. SM 23 (1957) 49-65.

Polygons of pursuit. SM 24 (1959) 23-50.

Curves of general pursuit. SM 24 (1959) 189-206.

Extensive history and analysis. First article covers one dimensional pursuit, then two dimensional linear pursuit. Second article deals with circular pursuit. Third article is the 'four bugs' problem -- analysis of equilateral triangle, square, scalene triangle, general polygon, Brocard points, etc. Last article includes such variants as variable speed, the tractrix, miscellaneous curves, etc.

Mr. Ash, proposer; editorial note saying there is no solver. Ladies' Diary, 1748-47 = T. Leybourn, II: 15-17, quest. 310, with 'Solution by ΦIΛΟΠΟΝΟΣ, taken from Turner's Exercises, where this question was afterwards proposed and answered ...' A fly is constrained to move on the periphery of a circle. Spider starts 30o away from the fly, but walks across the circle, always aiming at the fly. If she catches the fly 180o from her starting point, find the ratio of their speeds. ΦIΛΟΠΟΝΟΣ solves the more general problem of finding the curve when the spider starts anywhere.

Carlile. Collection. 1793. Prob. CV, p. 62. A dog and a duck are in a circular pond of radius 40 and they swim at the same speed. The duck is at the edge and swims around the circumference. The dog starts at the centre and always swims toward the duck, so the dog and the duck are always on a radius. How far does the dog swim in catching the duck? He simply gives the result as 20π. Letting R be the radius of the pond and V be the common speed, I find the radius of the dog, r, is given by r = R sin Vt/R. Since the angle, θ, of both the duck and the dog is given by θ = Vt/R, the polar equation of the dog's path is r = R sin θ and the path is a semicircle whose diameter is the appropriate radius perpendicular to the radius to the duck's initial position.

Cambridge Math. Tripos examination, 5 Jan 1871, 9 to 12. Problem 16, set by R. K. Miller. Three bugs in general position, but with velocities adjusted to make paths similar and keep the triangle similar to the original.

Lucas. (Problem of three dogs.) Nouvelle Correspondance Mathématique 3 (1877) 175-176. ??NYS -- English in Arc., AMM 28 (1921) 184-185 & Bernhart.

H. Brocard. (Solution of Lucas' problem.) Nouv. Corr. Math. 3 (1877) 280. ??NYS -- English in Bernhart.

Pearson. 1907. Part II, no. 66: A duck hunt, pp. 66 & 172. Duck swims around edge of pond; spaniel starts for it from the centre at the same speed.

A. S. Hathaway, proposer and solver. Problem 2801. AMM 27 (1920) 31 & 28 (1921) 93-97. Pursuit of a prey moving on a circle. Morley's and other solutions fail to deal with the case when the velocities are equal. Hathaway resolves this and shows the prey is then not caught.

F. V. Morley. A curve of pursuit. AMM 28 (1921) 54-61. Graphical solution of Hathaway's problem.

R. C. Archibald [Arc.] & H. P. Manning. Remarks and historical notes on problems 19 [1894], 160 [1902], 273 [1909] & 2801 [1920]. AMM 28 (1921) 91-93.

W. W. Rouse Ball. Problems -- Notes: 17: Curves of pursuit. AMM 28 (1921) 278-279.

A. H. Wilson. Note 19: A curve of pursuit. AMM 28 (1921) 327.

Editor's note to Prob. 2 (proposed by T. A. Bickerstaff), National Mathematics Magazine (1937/38) 417 cites Morley and Archibald and adds that some authors credit the problem to Leonardo da Vinci -- e.g. MG (1930-31) 436 -- ??NYS

Nelson F. Beeler & Franklyn M. Branley. Experiments in Optical Illusion. Ill. by Fred H. Lyon. Crowell, 1951, An illusion doodle, pp. 68-71, describes the pattern formed by four bugs starting at the corners of a square, drawing the lines of sight at (approximately) regular intervals. Putting several of the squares together, usually with alternating directions of motion, gives a pleasant pattern which is now fairly common. They call this 'Huddy's Doodle', but give no source.

J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. 'Lion and man', pp. 135-136 (114-117). The 1986 ed. adds three diagrams and revises the text somewhat. I quote from it. "A lion and a man in a closed circular arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal?" This was "invented by R. Rado in the late thirties" and "swept the country 25 years later". [The 1953 ed., says Rado didn't publish it.] The correct solution "was discovered by Professor A. S. Besicovitch in 1952". [The 1953 ed. says "This has just been discovered ...; here is the first (and only) version in print."]

C. C. Puckette. The curve of pursuit. MG 37 (No. 322) (Dec 1953) 256-260. Gives the history from Bouguer in 1732. Solves a variant of the problem.

R. H. Macmillan. Curves of pursuit. MG 40 (No. 331) (Feb 1956) 1-4. Fighter pursuing bomber flying in a straight line. Discusses firing lead and acceleration problems.

Gamow & Stern. 1958. Homing missiles. Pp. 112-114.

Howard D. Grossman, proposer; unspecified solver. Problem 66 -- The walk around. In: L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 40 & 203-205. Four bugs -- asserts Grossman originated the problem.

I. J. Good. Pursuit curves and mathematical art. MG 43 (No. 343) (Feb 1959) 34-35. Draws tangent to the pursuit curves in an equilateral triangle and constructs various patterns with them. Says a similar but much simpler pattern was given by G. B. Robison; Doodles; AMM 61 (1954) 381-386, but Robison's doodles are not related to pursuit curves, though they may have inspired Good to use the pursuit curves.

J. Charles Clapham. Playful mice. RMM 10 (Aug 1962) 6-7. Easy derivation of the distance travelled for n bugs at corners of a regular n-gon. [I don't see this result in Bernhart.]

C. G. Paradine. Note 3108: Pursuit curves. MG 48 (No. 366) (Dec 1964) 437-439. Says Good makes an error in Note 3079. He shows the length of the pursuit curve in the equilateral triangle is ⅔ of the side and describes the curve as an equiangular spiral. Gives a simple proof that the length of the pursuit curve in the regular n-gon is the side divided by (1 - cos 2π/n).

M. S. Klamkin & D. J. Newman. Cyclic pursuit or "The three bugs problem". AMM 78 (1971) 631-639. General treatment. Cites Bernhart's four SM papers and some of the history therein.

P. K. Arvind. A symmetrical pursuit problem on the sphere and the hyperbolic plane. MG 78 (No. 481) (Mar 1994) 30-36. Treats the n bugs problems on the surfaces named.

Barry Lewis. A mathematical pursuit. M500 170 (Oct 1999) 1-8. Starts with equilateral triangular case, giving QBASIC programs to draw the curves as well as explicit solutions. Then considers regular n-gons. Then considers simple pursuit, one beast pursuing another while the other moves along some given path. Considers the path as a straight line or a circle. For the circle, he asserts that the analytic solution was not determined until 1926, but gives no reference.

6.K. DUDENEY'S SQUARE TO TRIANGLE DISSECTION

Dudeney. Weekly Dispatch (6 Apr, 20 Apr, 4 May, 1902) all p. 13.

Dudeney. The haberdasher's puzzle. London Mag. 11 (No. 64) (Nov 1903) 441 & 443. (Issue with solution not found.)

Dudeney. Daily Mail (1 & 8 Feb 1905) both p. 7.

Dudeney. CP. 1907. Prob. 25: The haberdasher's puzzle, pp. 49-50 & 178-180.

Western Puzzle Works, 1926 Catalogue. No. 1712 -- unnamed, but shows both the square and the triangle. Apparently a four piece puzzle.

M. Adams. Puzzle Book. 1939. Prob. C.153: Squaring a triangle, pp. 162 & 189. Asserts that Dudeney's method works for any triangle, but his example is close to equilateral and I recall that this has been studied and only certain shapes will work??

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 40-41. Extends to dissecting a quadrilateral to a specified triangle and gives a number of related problems.

6.L. CROSSED LADDERS

Two ladders are placed across a street, each reaching from the base of the house on one side to the house on the other side.

The simple problem gives the heights a, b that the ladders reach on the walls. If the height of the crossing is c, we easily get 1/c = 1/a + 1/b. NOTATION -- this problem will be denoted by (a, b).

The more common and more complex problem is where the ladders have lengths a and b, the height of their crossing is c and one wants the width d of the street. If the heights of the ladder ends are x, y, this situation gives x2 - y2 = a2 - b2 and 1/x + 1/y = 1/c which leads to a quartic and there seems to be no simple solution. NOTATION -- this will be denoted (a, b, c).

Mahavira. 850. Chap. VII, v. 180-183, pp. 243-244. Gives the simple version with a modification -- each ladder reaches from the top of a pillar beyond the foot of the other pillar. The ladder from the top of pillar Y (of height y) extends by m beyond the foot of pillar X and the ladder from the top of pillar X (of height x) reaches n beyond the foot of pillar Y. The pillars are d apart. Similar triangles then yield: (d+m+n)/c = (d+n)/x + (d+m)/y and one can compute the various distances along the ground. He first does problems with m = n = 0, which are the simple version of the problem, but since d is given, he also asks for the distances on the ground.

v. 181. (16, 16) with d = 16.

v. 182. (36, 20) with d = 12.

v. 183. x, y, d, m, n = 12, 15, 4, 1, 4.

Bhaskara II. Lilavati. 1150. Chap. VI, v. 160. In Colebrooke, pp. 68-69. (10, 15). (= Bijaganita, ibid., chap. IV, v. 127, pp. 205-206.)

Fibonacci. 1202. Pp. 397-398 (S: 543-544) looks like a crossed ladders problem but is a simple right triangle problem.

Pacioli. Summa. 1494. Part II.

F. 56r, prob. 48. (4, 6).

F. 60r, prob. 64. (10, 15).

Hutton. A Course of Mathematics. 1798? Prob. VIII, 1833: 430; 1857: 508. A ladder 40 long in a roadway can reach 33 up one side and, from the same point, can reach 21 up the other side. How wide is the street? This is actually a simple right triangle problem.

Victor Katz reports that Hutton's problem, with values 60; 37, 23 appears in a notebook of Benjamin Banneker (1731-1806).

Loyd. Problem 48: A jubilee problem. Tit-Bits 32 (21 Aug, 11 & 25 Sep 1897) 385, 439 & 475. Given heights of the ladder ends above ground and the width of the street, find the height of the intersection. However one wall is tilted and the drawing has it covered in decoration so one may interpret the tilt in the wrong way.

Jno. A. Hodge, proposer; G. B. M. Zerr, solver. Problem 131. SSM 8 (1908) 786 & 9 (1909) 174-175. (100, 80, 10).

W. V. N. Garretson, proposer; H. S. Uhler, solver. Problem 2836. AMM 27 (1920) & 29 (1922) 181. (40, 25, 15).

C. C. Camp, proposer; W. J. Patterson & O. Dunkel, solvers. Problem 3173. AMM 33 (1926) 104 & 34 (1927) 50-51. General solution.

Morris Savage, proposer; W. E. Batzler, solver. Problem 1194. SSM 31 (1931) 1000 & 32 (1932) 212. (100, 80, 10).

S. A. Anderson, proposer; Simon Vatriquant, solver. Problem E210. AMM 43 (1936) 242 & 642-643. General solution in integers.

C. R. Green, proposer; C. W. Trigg, solver. Problem 1498. SSM 37 (1937) 484 & 860-861. (40, 30, 15). Trigg cites Vatriquant for smallest integral case.

A. A. Bennett, proposer; W. E. Buker, solver. Problem E433. AMM 47 (1940) 487 & 48 (1941) 268-269. General solution in integers using four parameters.

J. S. Cromelin, proposer; Murray Barbour, solver. Problem E616 -- The three ladders. AMM 51 (1944) 231 & 592. Ladders of length 60 & 77 from one side. A ladder from the other side crosses them at heights 17 & 19. How long is the third ladder and how wide is the street?

Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and Enthusiasts. (Blakiston, 1946); revised ed., Dover, 1954. Prob. 103: The extension ladder, pp. 58-59 & 176-178. Complex problem with three ladders.

Arthur Labbe, proposer; various solvers. Problem 25 -- The two ladders. Sep 1947 [date given in Graham's second book, cited at 1961]. In: L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 18 & 116-118. (20, 30, 8).

M. Y. Woodbridge, proposer and solver. Problem 2116. SSM 48 (1948) 749 & 49 (1949) 244-245. (60, 40, 15). Asks for a trigonometric solution. Trigg provides a list of early references.

Robert C. Yates. The ladder problem. SSM 51 (1951) 400-401. Gives a graphical solution using hyperbolas.

G. A. Clarkson. Note 2522: The ladder problem. MG 39 (No. 328) (May 1955) 147-148. (20, 30, 10). Let A = ((a2 - b2) and set x = A sec t, y = A tan t. Then cos t + cot t = A and he gets a trigonometrical solution. Another method leads to factoring the quartic in terms of a constant k whose square satisfies a cubic.

L. A. Graham. The Surprise Attack in Mathematical Problems. Dover, 1968. Problem 6: Searchlight on crossed ladders, pp. 16-18. Says they reposed Labbe's Sep 1947 problem in Jun 1961. Solution by William M. Dennis which is the same trigonometric method as Clarkson.

H. E. Tester. Note 3036: The ladder problem. A solution in integers. MG 46 (No. 358) (Dec 1962) 313-314. A four parameter, incomplete, solution. He finds the example (119, 70, 30).

A. Sutcliffe. Complete solution of the ladder problem in integers. MG 47 (No. 360) (May 1963) 133-136. Three parameter solution. First few examples are: (119, 70, 30); (116, 100, 35); (105, 87, 35). Simpler than Anderson and Bennett/Buker.

Alan Sutcliffe, proposer; Gerald J. Janusz, solver. Problem 5323 -- Integral solutions of the ladder problem. AMM 72 (1965) 914 & 73 (1966) 1125-1127. Can the distance f between the tops of the ladders be integral? (80342, 74226, 18837) has x = 44758, y = 32526, d = 66720, f = 67832. This is not known to be the smallest example.

Anon. A window cleaner's problem. Mathematical Pie 51 (May 1967) 399. From a point in the road, a ladder can reach 30 ft up on one side and 40 ft up on the other side. If the two ladder positions are at right angles, how wide is the road?

J. W. Gubby. Note 60.3: Two chestnuts (re-roasted). MG 60 (No. 411) (Mar 1976) 64-65. 1.  Given heights of ladders as a, b, what is the height c of their intersection? Solution: 1/c = 1/a + 1/b or c = ab/(a+b). 2. The usual ladder problem -- he finds a quartic.

J. Jabłkowski. Note 61:11: The ladder problem solved by construction. MG 61 (No. 416) (Jun 1977) 138. Gives a 'neusis' construction. Cites Gubby.

Birtwistle. Calculator Puzzle Book. 1978. Prob. 83, A second ladder problem, pp. 58-59 & 115-118. (15, 20, 6). Uses xy as a variable to simplify the quartic for numerical solution and eventually gets 11.61.

See: Gardner, Circus, p. 266 & Schaaf for more references. ??follow up.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. The tangled ladders, pp. 43-44 & 116. (30, 20, 10). Gives answer 12.311857... with no explanation.

6.L.1. LADDER OVER BOX

A ladder of length L is placed to just clear a box of width w and height h at the base of a wall. How high does the ladder reach? Denote this by (w, h, L). Letting x be the horizontal distance of the foot and y be the vertical distance of the top of the ladder, measured from the foot of the wall, we get x2 + y2 = L2 and (x-w)(y-h) = wh, which gives a quartic in general. But if w = h, then use of x + y as a variable reduces the quartic to a quadratic. In this case, the idea is old -- see e.g. Simpson.

The question of determining shortest ladder which can fit over a box of width w and height h is the same as determining the longest ladder which will pass from a corridor of width w into another corridor of width h. See Huntington below and section 6.AG.

Simpson. Algebra. 1745. Section XVIII, prob. XV, p. 250 (1790: prob. XIX, pp. 272-273). "The Side of the inscribed Square BEDF, and the Hypotenuse AC of a right-angled Triangle ABC being given; to determine the other two Sides of the Triangle AB and BC." Solves "by considering x + y as one Quantity".

Pearson. 1907. Part II, no. 102: Clearing the wall, p. 103. For (15, 12, 52), the ladder reaches 48.

D. John Baylis. The box and ladder problem. MTg 54 (1971) 24. (2, 2, 10). Finds the quartic which he solves by symmetry. Editorial note in MTg 57 (1971) 13 says several people wrote to say that use of similar triangles avoids the quartic.

Birtwistle. Math. Puzzles & Perplexities. 1971. The ladder and the box problem, pp. 44-45. = Birtwistle; Calculator Puzzle Book; 1978; Prob. 53: A ladder problem, pp. 37 & 96-98. (3, 3, 10). Solves by using x + y - 6 as a variable.

Monte Zerger. The "ladder problem". MM 60:4 (1987) 239-242. (4, 4, 16). Gives a trigonometric solution and a solution via two quadratics.

Oliver D. Anderson. Letter. MM 61:1 (1988) 63. In response to Zerger's article, he gives a simpler derivation.

Tom Heyes. The old box and ladder problem -- revisited. MiS 19:2 (Mar 1990) 42-43. Uses a graphic calculator to find roots graphically and then by iteration.

A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5. Does usual problem, getting a quartic. Then finds the shortest ladder. [This turns out to be the same as the longest ladder one can get around a corner from corridors of widths w and h, so this problem is connected to 6.AG.]

David Singmaster. Integral solutions of the ladder over box problem. In preparation. Easily constructs all the primitive integral examples from primitive Pythagorean triples. E.g. for the case of a square box, i.e. w = h, if X, Y, Z is a primitive Pythagorean triple, then the corresponding primitive solution has w = h = XY, x = X (X + Y), y = Y (X + Y), L = Z (X + Y), and remarkably, x - h = X2, y - w = Y2.

6.M. SPIDER & FLY PROBLEMS

These involve finding the shortest distance over the surface of a cube or cylinder. I've just added the cylindrical form -- see Dudeney (1926), Perelman and Singmaster. The shortest route from a corner of a cube or cuboid to a diagonally opposite corner must date back several centuries, but I haven't seen any version before 1937! I don't know if other shapes have been done -- the regular (and other) polyhedra and the cone could be considered.

Two-dimensional problems are in 10.U.

Loyd. The Inquirer (May 1900). Gives the Cyclopedia problem. ??NYS -- stated in a letter from Shortz.

Dudeney. Problem 501 -- The spider and the fly. Weekly Dispatch (14 & 28 Jun 1903) both p. 16. 4 side version.

Dudeney. Breakfast table problems, No. 320 -- The spider and the fly. Daily Mail (18 & 21 Jan 1905) both p. 7. Same as the above problem.

Dudeney. Master of the breakfast table problem. Daily Mail (1 & 8 Feb 1905) both p. 7. Interview with Dudeney in which he gives the 5 side version.

Ball. MRE, 4th ed., 1905, p. 66. Gives the 5 side version, citing the Daily Mail of 1 Feb 1905. He says he heard a similar problem in 1903 -- presumably Dudeney's first version. In the 5th ed., 1911, p. 73, he attributes the problem to Dudeney.

Dudeney. CP. 1907. Prob. 75: The spider and the fly, pp. 121-122 & 221-222. 5 side version with discussion of various generalizations.

Dudeney. The world's best problems. 1908. Op. cit. in 2. P. 786 gives the five side version.

Sidney J. Miller. Some novel picture puzzles -- No. 6. Strand Mag. 41 (No. 243) (Mar 1911) 372 & 41 (No. 244) (Apr 1911) 506. Contest between two snails. Better method uses four sides, similar to Dudeney's version, but with different numbers.

Loyd. The electrical problem. Cyclopedia, 1914, pp. 219 & 368 (= MPSL2, prob. 149, pp. 106 & 169 = SLAHP: Wiring the hall, pp. 72 & 114). Same as Dudeney's first, four side, version. (In MPSL2, Gardner says Loyd has simplified Dudeney's 5 side problem. More likely(?) Loyd had only seen Dudeney's earlier 4 side problem.)

Dudeney. MP. 1926. Prob. 162: The fly and the honey, pp. 67 & 157. (= 536, prob. 325, pp. 112 & 313.) Cylindrical problem.

Perelman. FFF. 1934. The way of the fly. 1957: Prob. 68, pp. 111-112 & 117-118; 1979: Prob. 72, pp. 136 & 142-144. MCBF: Prob. 72, pp. 134 & 141-142. Cylindrical form, but with different numbers and arrangement than Dudeney's MP problem.

Haldeman-Julius. 1937. No. 34: The louse problem, pp. 6 & 22. Room 40 x 20 x 10 with louse at a corner wanting to go to a diagonally opposite corner. Problem sent in by J. R. Reed of Emmett, Idaho. Answer is 50!

M. Kraitchik. Mathematical Recreations, 1943, op. cit. in 4.A.2, chap. 1, prob. 7, pp. 17-21. Room with 8 equal routes from spider to fly. (Not in his Math. des Jeux.)

Sullivan. Unusual. 1943. Prob. 10: Why not fly? Find shortest route from a corner of a cube to the diagonally opposite corner.

William R. Ransom. One Hundred Mathematical Curiosities. J. Weston Walch, Portland, Maine, 1955. The spider problem, pp. 144-146. There are three types of path, covering 3, 4 and 5 sides. He determines their relative sizes as functions of the room dimensions.

Birtwistle. Math. Puzzles & Perplexities. 1971.

Round the cone, pp. 144 & 195. What is the shortest distance from a point P around a cone and back to P? Answer is "An ellipse", which doesn't seem to answer the question. If the cone has height H, radius R and P is l from the apex, then the slant height L is ((R2 + H2), the angle of the opened out cone is θ = 2πR/L and the required distance is 2l sin θ/2.

Spider circuit, pp. 144 & 198. Spider is at the midpoint of an edge of a cube. He wants to walk on each of the faces and return. What is his shortest route? Answer is "A regular hexagon. (This may be demonstrated by putting a rubber band around a cube.)"

David Singmaster. The spider spied her. Problem used as: More than one way to catch a fly, The Weekend Telegraph (2 Apr 1984). Spider inside a glass tube, open at both ends, goes directly toward a fly on the outside. When are there two equally short paths? Can there be more than two shortest routes?

Yoshiyuki Kotani has posed the following general and difficult problem. On an a x b x c cuboid, which two points are furthest apart, as measured by an ant on the surface? Dick Hess has done some work on this, but I believe that even the case of square cross-section is not fully resolved.

6.N. DISSECTION OF A 1 x 1 x 2 BLOCK TO A CUBE

W. F. Cheney, Jr., proposer; W. R. Ransom; A. H. Wheeler, solvers. Problem E4. AMM 39 (1932) 489; 40 (1933) 113-114 & 42 (1934) 509-510. Ransom finds a solution in 8 pieces; Wheeler in 7.

Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1964. Section 24.2, p. 120 gives a variant of Wheeler's solution.

Michael Goldberg. A duplication of the cube by dissection and a hinged linkage. MG 50 (No. 373) (Oct 1966) 304-305. Shows that a hinged version exists with 10 pieces. Hanegraaf, below, notes that there are actually 12 pieces here.

Anton Hanegraaf. The Delian Altar Dissection. Polyhedral Dissections, Elst, Netherlands, 1989. Surveys the problem, gives a 6 piece solution and a 7 piece hinged solution.

6.O. PASSING A CUBE THROUGH AN EQUAL OR SMALLER CUBE --

PRINCE RUPERT'S PROBLEM

The projection of a unit cube along a space diagonal is a regular hexagon of side (2/(3. The largest square inscribable in this hexagon has edge (6 - (2 = 1.03527618. By passing the larger cube on a slant to the space diagonal, one can get the larger cube having edge 3(2/4 = 1.06066172.

There are two early attributions of this. Wallis attributes it to Prince Rupert, but Hennessy says Philip Ronayne of Cork invented it. I have discovered a possible connection. Prince Rupert of the Rhine (1619-1682), nephew of Charles I, was a major military figure of his time, becoming commander-in-chief of Charles I's armies in the 1640s. In 1648-1649, he was admiral of the King's fleet and was blockaded with 16 ships in Kinsale Harbor for 20 months. Kinsale is about 20km south of Cork.

Ronayne wrote an Algebra, of which only a second edition of 1727 is in the BL. Schrek has investigated the family histories and says Ronayne lived in the early 18C. This would seem to make him too young to have met Rupert. Perhaps Rupert invented the problem while in Kinsale and this was conveyed to Ronayne some years later. Does anyone know the dates of Ronayne or of the 1st ed (Schrek only located the BL example of the 2nd ed)? I cannot find anything on him in Wallis, May, Poggendorff, DNB, but Google has turned up a reference to a 1917 history of the family which Schrek cites, but I have not yet tried to find this.

Hennessy's article says a little about Daniel Voster and details are in Wallis's . His father, Elias (1682 - >1728) wrote an Arithmetic, of which Wallis lists 30 editions. The BL lists one as late as 1829. The son, Daniel (1705 - >1760) was a schoolmaster and instrument maker who edited later versions of his father's arithmetic. The 1750 History of Cork quoted by Hennessy says the author had seen the cubes with Daniel. Hennessy conjectures that his example was made specially, perhaps under the direction of a mathematician. It seems likely that Daniel knew Ronayne and made this example for him.

John Wallis. Perforatio cubi, alterum ipsi aequalem recipiens. (De Algebra Tractatus; 1685; Chap. 109) = Opera Mathematica, vol. II, Oxford, 1693, pp. 470-471, ??NYS. Cites Rupert as the source of the equal cube version. (Latin and English in Schrek.) Scriba, below, found an errata slip in Wallis's copy of his Algebra in the Bodleian. This corrects the calculations, but was published in the Opera, p. 695.

Ozanam-Montucla. 1778. Percer un cube d'une ouverture, par laquelle peut passer un autre cube égal au premier. Prob. 30 & fig. 53, plate 7, 1778: 319-320; 1803: 315-316; 1814: 268-269. Prob. 29, 1840: 137. Equal cubes with diagonal movement.

J. H. van Swinden. Grondbeginsels der Meetkunde. 2nd ed., Amsterdam, 1816, pp. 512-513, ??NYS. German edition by C. F. A. Jacobi, as: Elemente der Geometrie, Friedrich Frommann, Jena, 1834, pp. 394-395. Cites Rupert and Wallis and gives a simple construction, saying Nieuwland has found the largest cube which can pass through a cube.

Peter Nieuwland. (Finding of maximum cube which passes through another). In: van Swinden, op. cit., pp. 608-610; van Swinden-Jacobi, op. cit. above, pp. 542-544, gives Nieuwland's proof.

Cundy and Rollett, p. 158, give references to Zacharias (see below) and to Cantor, but Cantor only cites Hennessy.

H. Hennessy. Ronayne's cubes. Phil. Mag. (5) 39 (Jan-Jun 1895) 183-187. Quotes, from Gibson's 'History of Cork', a passage taken from Smith's 'History of Cork', 1st ed., 1750, vol. 1, p. 172, saying that Philip Ronayne had invented this and that a Daniel Voster had made an example, which may be the example owned by Hennessy. He gives no reference to Rupert. He finds the dimensions.

F. Koch & I. Reisacher. Die Aufgabe, einen Würfel durch einen andern durchzuschieben. Archiv Math. Physik (3) 10 (1906) 335-336. Brief solution of Nieuwland's problem.

M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften. Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und Minima. Die isoperimetrische Aufgabe. Pp. 1133-1134. Attributes it to Prince Rupert, following van Swinden. Cites Wallis & Ronayne, via Cantor, and Nieuwland, via van Swinden.

U. Graf. Die Durchbohrung eines Würfels mit einem Würfel. Zeitschrift math. naturwiss. Unterricht 72 (1941) 117. Nice photos of a model made at the Technische Hochschule Danzig. Larger and better versions of the same photos can be found in: W. Lietzmann & U. Graf; Mathematik in Erziehung und Unterricht; Quelle & Meyer, Leipzig, 1941, vol. 2, plate 3, opp. p. 168, but I can't find any associated text for it.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 12: Curios [sic] cubes, p. 14. First says it can be done with equal cubes and then a larger can pass through a smaller. Claims that the larger cube can be about 1.1, but this is due to an error -- he thinks the hexagon has the same diameter as the cube itself.

H. D. Grossman, proposer; C. S. Ogilvy & F. Bagemihl, solvers. Problem E888 -- Passing a cube through a cube of same size. AMM 56 (1949) 632 ??NYS & 57 (1950) 339. Only considers cubes of the same size, though Bagemihl's solution permits a slightly larger cube. No references.

D. J. E. Schrek. Prince Rupert's problem and its extension by Pieter Nieuwland. SM 16 (1950) 73-80 & 261-267. Historical survey, discussing Rupert, Wallis, Ronayne, van Swinden & Nieuwland. Says Ronayne is early 18C.

M. Gardner. SA (Nov 1966) = Carnival, pp. 41-54. The largest square inscribable in a cube is the cross section of the maximal hole through which another cube can pass.

Christoph J. Scriba. Das Problem des Prinzen Ruprecht von der Pfalz. Praxis der Mathematik 10 (1968) 241-246. ??NYS -- described by Scriba in an email to HM Mailing List, 20 Aug 1999. Describes the correction to Wallis's work and considers the problem for the tetrahedron and octahedron.

6.P. GEOMETRICAL VANISHING

Gardner. MM&M. 1956. Chap. 7 & 8: Geometrical Vanishing -- Parts I & II, pp. 114-155. Best extensive discussion of the subject and its history.

Gardner. SA (Jan 1963) c= Magic Numbers, chap. 3. Discusses application to making an extra bill and Magic Numbers adds citations to several examples of people trying it and going to jail.

Gardner. Advertising premiums to beguile the mind: classics by Sam Loyd, master puzzle-poser. SA (Nov 1971) = Wheels, Chap. 12. Discusses several forms.

S&B, p. 144, shows several versions.

6.P.1. PARADOXICAL DISSECTIONS OF THE CHESSBOARD BASED

ON FIBONACCI NUMBERS

Area 63 version: AWGL, Dexter, Escott, White, Loyd, Ahrens, Loyd Jr., Ransom.

(W. Leybourn. Pleasure with Profit. 1694. ?? I cannot recall the source of this reference and think it may be an error. I have examined the book and find nothing relevant in it.)

Loyd. Cyclopedia, 1914, pp. 288 & 378. 8 x 8 to 5 x 13 and to an area of 63. Asserts Loyd presented the first of these in 1858. Cf Loyd Jr, below.

O. Schlömilch. Ein geometrisches Paradoxon. Z. Math. Phys., 13 (1868) 162. 8 x 8 to 5 x 13. (This article is only signed Schl. Weaver, below, says this is Schlömilch, and this seems right as he was a co-editor at the time. Coxeter (SM 19 (1953) 135-143) says it is V. Schlegel, apparently confusing it with the article below.) Doesn't give any explanation, leaving it as a student exercise.

F. J. Riecke. Op. cit. in 4.A.1. Vol. 3, 1873. Art. 16: Ein geometrisches Paradoxon. Quotes Schlömilch and explains the paradox.

G. H. Darwin. Messenger of Mathematics 6 (1877) 87. 8 x 8 to 5 x 13 and generalizations.

V. Schlegel. Verallgemeinerung eines geometrischen Paradoxons. Z. Math. Phys. 24 (1879) 123-128 & Plate I. 8 x 8 to 5 x 13 and generalizations.

Mittenzwey. 1880. Prob. 299, pp. 54 & 105; 1895?: 332, pp. 58 & 106-107; 1917: 332, pp. 53 & 101. 8 x 8 to 5 x 13. Clear explanation.

The Boy's Own Paper. No. 109, vol. III (12 Feb 1881) 327. A puzzle. 8 x 8 to 5 x 13 without answer.

Richard A. Proctor. Some puzzles. Knowledge 9 (Aug 1886) 305-306. "We suppose all the readers ... know this old puzzle." Describes and explains 8 x 8 to 5 x 13. Gives a different method of cutting so that each rectangle has half the error -- several typographical errors.

Richard A. Proctor. The sixty-four sixty-five puzzle. Knowledge 9 (Oct 1886) 360-361. Corrects the above and explains it in more detail.

Will Shortz has a puzzle trade card with the 8 x 8 to 5 x 13 version, c1889.

Ball. MRE, 1st ed., 1892, pp. 34-36. 8 x 8 to 5 x 13 and generalizations. Cites Darwin and describes the examples in Ozanam-Hutton (see Ozanam-Montucla in 6.P.2). In the 5th ed., 1911, p. 53, he changes the Darwin reference to Schlömilch. In the 7th ed., 1917, he only cites the Ozanam-Hutton examples.

Clark. Mental Nuts. 1897, no. 33; 1904, no. 41; 1916, no. 43. Four peculiar drawings. 8 x 8 to 5 x 13.

Carroll-Collingwood. 1899. Pp. 316-317 (Collins: 231 and/or 232 (lacking in my copy)) = Carroll-Wakeling II, prob. 7: A geometrical paradox, pp. 12 & 7. 8 x 8 to 5 x 13. Carroll may have stated this as early as 1888. Wakeling says the papers among which this was found on Carroll's death are now in the Parrish Collection at Princeton University and suggests Schlömilch as the earliest version.

AWGL (Paris). L'Echiquier Fantastique. c1900. Wooden puzzle of 8 x 8 to 5 x 13 and to area 63. ??NYS -- described in S&B, p. 144.

Walter Dexter. Some postcard puzzles. Boy's Own Paper (14 Dec 1901) 174-175. 8 x 8 to 5 x 13 and to area 63.

C. A. Laisant. Initiation Mathématique. Georg, Geneva & Hachette, Paris, 1906. Chap. 63: Un paradoxe: 64 = 65, pp. 150-152.

Wm. F. White. In the mazes of mathematics. A series of perplexing puzzles. III. Geometric puzzles. The Open Court 21 (1907) 241-244. Shows 8 x 8 to 5 x 13 and a two-piece 11 x 13 to area 145.

E. B. Escott. Geometric puzzles. The Open Court 21 (1907) 502-505. Shows 8 x 8 to area 63 and discusses the connection with Fibonacci numbers.

William F. White. Op. cit. in 5.E. 1908. Geometric puzzles, pp. 109-117. Partly based on above two articles. Gives 8 x 8 to 5 x 13 and to area 63. Gives an extension which turns 12 x 12 into 8 x 18 and into area 144, but turns 23 x 23 into 16 x 33 and into area 145. Shows a puzzle of Loyd: three-piece 8 x 8 into 7 x 9.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. 5 x 5 into four pieces that make a 3 x 8.

M. Adams. Indoor Games. 1912. Is 64 equal to 65? Pp. 345-346 with fig. on p. 344.

Loyd. Cyclopedia. 1914. See entry at 1858.

W. Ahrens. Mathematische Spiele. Teubner, Leipzig. 3rd ed., 1916, pp. 94-95 & 111-112. The 4th ed., 1919, and 5th ed., 1927, are identical with the 3rd ed., but on different pages: pp. 101-102 & pp. 118-119. Art. X. 65 = 64 = 63 gives 8 x 8 to 5 x 13 and to area 63. The area 63 case does not appear in the 2nd ed., 1911, which has Art. V. 64 = 65, pp. 107 & 118-119 and this material is not in the 1st ed. of 1907.

Tom Tit?? In Knott, 1918, but I can't find it in Tom Tit. No. 3: The square and the rectangle: 64 = 65!, pp. 15-16. Clearly explained.

Hummerston. Fun, Mirth & Mystery. 1924. A puzzling paradox, pp. 44 & 185. Usual 8 x 8 to 5 x 13, but he erases the chessboard lines except for the cells the cuts pass through, so one way has 16 cells, the other way has 17 cells. Reasonable explanation.

Collins. Book of Puzzles. 1927. A paradoxical puzzle, pp. 4-5. 8 x 8 to 5 x 13. Shades the unit cells that the lines pass through and sees that one way has 16 cells, the other way has 17 cells, but gives only a vague explanation.

Loyd Jr. SLAHP. 1928. A paradoxical puzzle, pp. 19-20 & 90. Gives 8 x 8 to 5 x 13. "I have discovered a companion piece ..." and gives the 8 x 8 to area 63 version. But cf AWGL, Dexter, etc. above.

W. Weaver. Lewis Carroll and a geometrical paradox. AMM 45 (1938) 234-236. Describes unpublished work in which Carroll obtained (in some way) the generalizations of the 8 x 8 to 5 x 13 in about 1890-1893. Weaver fills in the elementary missing arguments.

W. R. Ransom, proposer; H. W. Eves, solver. Problem E468. AMM 48 (1941) 266 & 49 (1942) 122-123. Generalization of the 8 x 8 to area 63 version.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 23: Summat for nowt?, pp. 27-28. 8 x 8 to 5 x 13, clearly drawn.

Warren Weaver. Lewis Carroll: Mathematician. Op. cit. in 1. 1956. Brief mention of 8 x 8 to 5 x 13. John B. Irwin's letter gives generalizations to other consecutive triples of Fibonacci numbers (though he doesn't call them that). Weaver's response cites his 1938 article, above.

6.P.2. OTHER TYPES

In several early examples, the authors appear unaware that area has vanished!

Pacioli. De Viribus. c1500. Ff. 189v - 191r. Part 2. LXXIX. Do(cumento). un tetragono saper lo longare con restregnerlo elargarlo con scortarlo (a tetragon knows lengthening and contraction, enlarging with shortening ??) = Peirani 250-252. Convert a 4 x 24 rectangle to a 3 x 32 using one cut into two pieces. Pacioli's

description is cryptic but seems to have two cuts, making d c

three pieces. There is a diagram at the bottom of f. 190v, badly k f e

redrawn on Peirani 458. Below this is a inserted note which Peirani

252 simply mentions as difficult to read, but can make sense. The g  

points are as laid out at the right. abcd is the original 4 x 24 h a o b

rectangle. g is one unit up from a and e is one unit down from c.

Cut from c to g and from e parallel to the base, meeting cg at f. Then move cdg to fkh and move fec to hag. Careful rereading of Pacioli seems to show he is using a trick! He cuts from e to f to g. then turns over the upper piece and slides it along so that he can continue his cut from g to h, which is where f to c is now. This gives three pieces from a single cut! Pacioli clearly notes that the area is conserved.

Although not really in this topic, I have put it here as it seems to be a predecessor of this topic and of 6.AY.

Sebastiano Serlio. Libro Primo d'Architettura. 1545. This is the first part of his Architettura, 5 books, 1537-1547, first published together in 1584. I have seen the following editions.

With French translation by Jehan Martin, no publisher shown, Paris, 1545, f. 22.r. ??NX

1559. F. 15.v.

Francesco Senese & Zuane(?) Krugher, Venice, 1566, f. 16.r. ??NX

Jacomo de'Franceschi, Venice, 1619, f. 16.r.

Translated into Dutch by Pieter Coecke van Aelst as: Den eerstē vijfsten boeck van architecturē; Amsterdam, 1606. This was translated into English as: The Five Books of Architecture; Simon Stafford, London, 1611 = Dover, 1982. The first Booke, f. 12v.

3 x 10 board is cut on a diagonal and slid to form a 4 x 7 table with 3 x 1 left over, but he doesn't actually put the two leftover pieces together nor notice the area change!

Pietro Cataneo. L'Architettura di Pietro Cataneo Senese. Aldus, Venice, 1567. ??NX. Libro Settimo.

P. 164, prop. XXVIIII: Come si possa accresciere una stravagante larghezza. Gives a correct version of Serlio's process.

P. 165, prop. XXX: Falsa solutione del Serlio. Cites p. xxii of Serlio. Carefully explains the error in Serlio and says his method is "insolubile, & mal pensata".

Schwenter. 1636. Part 15, ex. 14, p. 541: Mit einem länglichten schmahlen Brett / für ein bräites Fenster einen Laden zu machen. Cites Gualtherus Rivius, Architectur. Discusses Serlio's dissection as a way of making a 4 x 7 from a 3 x 10 but doesn't notice the area change.

Gaspar Schott. Magia Universalis. Joh. Martin Schönwetter, Bamberg, Vol. 3, 1677. Pp. 704-708 describes Serlio's error in detail, citing Serlio. ??NX of plates.

I have a vague reference to the 1723 ed. of Ozanam, but I have not seen it in the 1725 ed. -- this may be an error for the 1778 ed. below.

Minguet. 1755. Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Same as Hooper. Not in 1733 ed.

Vyse. Tutor's Guide. 1771? Prob. 8, 1793: p. 304, 1799: p. 317 & Key p. 358. Lady has a table 27 square and a board 12 x 48. She cuts the board into two 12 x 24 rectangles and cuts each rectangle along a diagonal. By placing the diagonals of these pieces on the sides of her table, she makes a table 36 square. Note that 362 = 1296 and 272 + 12 x 48 = 1305. Vyse is clearly unaware that area has been created. By dividing all lengths by 3, one gets a version where one unit of area is lost. Note that 4, 8, 9 is almost a Pythagorean triple.

William Hooper. Rational Recreations. 1774. Op. cit. in 4.A.1. Vol. 4, pp. 286-287: Recreation CVI -- The geometric money. 3 x 10 cut into four pieces which make a 2 x 6 and a 4 x 5. (The diagram is shown in Gardner, MM&M, pp. 131-132.) (I recently saw that an edition erroneously has a 3 x 6 instead of a 2 x 6 rectangle. This must be the 1st ed. of 1774, as it is correct in my 2nd ed. of 1782.)

Ozanam-Montucla. 1778. Transposition de laquelle semble résulter que le tout peut être égal à la partie. Prob. 21 & fig. 127, plate 16, 1778: 302-303 & 363; 1803: 298-299 & 361; 1814: 256 & 306; 1840: omitted. 3 x 11 to 2 x 7 and 4 x 5. Remarks that M. Ligier probably made some such mistake in showing 172 = 2 x 122 and this is discussed further on the later page.

E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Nouvelle éd. La Librairie, Rue S. André-des-Arcs[sic], Paris, Year 7 [1799]. Vol. 2, Deuxième récréation: Or géométrique -- construction, pp. 41-42 & plate 6, opp. p. 37. Same as Hooper.

Manuel des Sorciers. 1825. Pp. 202-203, art. 19. ??NX Same as Hooper.

The Boy's Own Book. The geometrical money. 1828: 413; 1828-2: 419; 1829 (US): 212; 1855: 566-567; 1868: 669. Same as Hooper.

Magician's Own Book. 1857. Deceptive vision, pp. 258-259. Same as Hooper. = Book of 500 Puzzles, 1859, pp. 72-73.

Illustrated Boy's Own Treasury. 1860. Optics: Deceptive vision, p. 445. Same as Hooper. Identical to Book of 500 Puzzles.

Wemple & Company (New York). The Magic Egg Puzzle. ©1880. S&B, p. 144. Advertising card, the size of a small postcard, but with ads for Rogers Peet on the back. Starts with 9 eggs. Cut into four rectangles and reassemble to make 6, 7, 8, 10, 11, 12 eggs.

R. March & Co. (St. James's Walk, Clerkenwell). 'The Magical Egg Puzzle', nd [c1890]. (I have a photocopy.) Four rectangles which produce 6, 7, ..., 12 eggs. Identical to the Wemple version, but with Wemple's name removed. I only have a photocopy of the front of this and I don't know what's on the back. I also have a photocopy of the instructions.

Loyd. US Patent 563,778 -- Transformation Picture. Applied: 11 Mar 1896; patented: 14 Jul 1896. 1p + 1p diagrams. Simple rotating version using 8 to 7 objects.

Loyd. Get Off the Earth. Puzzle notices in the Brooklyn Daily Eagle (26 Apr - 3 May 1896), printing individual Chinamen. Presenting all of these at an office of the newspaper gets you an example of the puzzle. Loyd ran discussions on it in his Sunday columns until 3 Jan 1897 and he also sold many versions as advertising promotions. S&B, p. 144, shows several versions.

Loyd. Problem 17: Ye castle donjon. Tit-Bits 31 (6 & 27 Feb & 6 & 20 Mar 1897) 343, 401, 419 & 455. = Cyclopedia, 1914, The architect's puzzle, pp. 241 & 372. 5 x 25 to area 124.

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. Discusses and shows Get Off the Earth.

Ball. MRE, 4th ed., 1905, pp. 50-51: Turton's seventy-seven puzzle. Additional section describing Captain Turton's 7 x 11 to 7 x 11 with one projecting square, using bevelled cuts. This is dropped from the 7th ed., 1917.

William F. White. 1907 & 1908. See entries in 6.P.1.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Gives "Get Off the Earth" on p. 785.

Loyd. Teddy and the Lions. Gardner, MM&M, p. 123, says he has seen only one example, made as a promotional item for the Eden Musee in Manhattan. This has a round disc, but two sets of figures -- 7 natives and 7 lions which become 6 natives and 8 lions.

Dudeney. A chessboard fallacy. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 676 (= AM, prob. 413, pp. 141 & 247). (There is a solution in Strand Mag. 39 (No. 229) (Jan 1910) ??NYS.) 8 x 8 into 3 pieces which make a 9 x 7.

Fun's Great Baseball Puzzle. Will Shortz gave this out at IPP10, 1989, as a colour photocopy, 433 x 280 mm (approx. A3). ©1912 by the Press Publishing Co (The New York World). I don't know if Fun was their Sunday colour comic section or what. One has to cut it diagonally and slide one part along to change from 8 to 9 boys.

Loyd. The gold brick puzzle. Cyclopedia, 1914, pp. 32 & 342 (= MPSL1, prob. 24, pp. 22 & 129). 24 x 24 to 23 x 25.

Loyd. Cyclopedia. 1914. "Get off the earth", p. 323. Says over 10 million were sold. Offers prizes for best answers received in 1909.

Loyd Jr. SLAHP. 1928. "Get off the Earth" puzzle, pp. 5-6. Says 'My "Missing Chinaman Puzzle"' of 1896. Gives a simple and clear explanation.

John Barnard. The Handy Boy's Book. Ward, Lock & Co., London, nd [c1930?]. Some interesting optical illusions, pp. 310-311. Shows a card with 11 matches and a diagonal cut so that sliding it one place makes 10 matches.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 24: A chessboard fallacy, pp. 28-29. 8 x 8 cut with a diagonal of a 8 x 7 region, then pieces slid and a triangle cut off and moved to the other end to make a 9 x 7. Clear illustration.

Mel Stover. From 1951, he devised a number of variations of both Get off the Earth (perhaps the best is his Vanishing Leprechaun) and of Teddy and the Lions (6 men and 4 glasses of beer become 5 men and 5 glasses). I have examples of some of these from Stover and I have looked at his notebooks, which are now with Mark Setteducati. See Gardner, MM&M, pp. 125-128.

Gardner. SA (May 1961) c= NMD, chap. 11. Mentioned in Workout, chap. 27. Describes his adaptation of a principle of Paul Curry to produce The Disappearing Square puzzle, where 16 or 17 pieces seem to make the same square. The central part of the 17 piece version consists of five equal squares in the form of a Greek cross. The central part of the 16 piece version has four of the squares in the shape of a square. This has since been produced in several places.

Ripley's Puzzles and Games. 1966. P. 60. Asserts that when you cut a 2½ x 4½ board into six right triangles with legs 1½ and 2½, then they assemble into an equilateral triangle of edge 5. This has an area loss of about 4%.

John Fisher. John Fisher's Magic Book. Muller, London, 1968.

Financial Wizardry, pp. 18-19. 7 x 8 region with £ signs marking the area. A line cuts off a triangle of width 7 and height 2 at the top. The rest of the area is divided by a vertical into strips of widths 4 and 3, with a small rectangle 3 by 1 cut from the bottom of the width 3 strip. When the strips are exchanged, one unit of area is lost and one £ sign has vanished.

Try-Angle, pp. 126-127. This is one of Curry's triangles -- see Gardner, MM&M, p. 147.

Alco-Frolic!, pp. 148-149. This is a form of Stover's 6 & 4 to 5 & 5 version.

D. E. Knuth. Disappearances. In: The Mathematical Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth, 1981. P. 264. An eight line poem which rearranges to a seven line poem.

Dean Clark. A centennial tribute to Sam Loyd. CMJ 23:5 (Nov 1992) 402-404. Gives an easy circular version with 11 & 12 astronauts around the earth and a 15 & 16 face version with three pieces, a bit like the Vanishing Leprechaun.

6.Q. KNOTTING A STRIP TO MAKE A REGULAR PENTAGON

Urbano d'Aviso. Trattato della Sfera e Pratiche per Uso di Essa. Col modeo di fare la figura celeste, opera cavata dalli manoscritti del. P. Bonaventura Cavalieri. Rome, 1682. ??NYS cited by Lucas (1895) and Fourrey.

Dictionary of Representative Crests. Nihon Seishi Monshō Sōran (A Comprehensive Survey of Names and Crests in Japan), Special issue of Rekishi Dokuhon (Readings in History), Shin Jinbutsu Oraisha, Tokyo, 1989, pp. 271-484. Photocopies of relevant pages kindly sent by Takao Hayashi.

Crests 3504 and 3506 clearly show a strip knotted to make a pentagon. 3507 has two such knots and 3508 has five. I don't know the dates, but most of these crests are several centuries old.

Lucas. RM2, 1883, pp. 202-203.

Tom Tit.

Vol. 2, 1892. L'Étoile à cinq branches, pp. 153-154. = K, no. 5: The pentagon and the five pointed star, pp. 20-21. He adds that folding over the free end and holding the knot up to the light shows the pentagram.

Vol. 3, 1893. Construire d'un coup de poing un hexagone régulier, pp. 159-161. = K, no. 17: To construct a hexagon by finger pressure, pp. 49-51. Pressing an appropriate size Möbius strip flat gives a regular hexagon.

Vol. 3, 1893. Les sept pentagones, pp. 165-166. = K, no. 19: The seven pentagons, pp. 54-55. By tying five pentagons in a strip, one gets a larger pentagon with a pentagonal hole in the middle.

Somerville Gibney. So simple! The hexagon, the enlarged ring, and the handcuffs. The Boy's Own Paper 20 (No. 1012) (4 Jun 1898) 573-574. As in Tom Tit, vol. 3, pp. 159-161.

Lucas. L'Arithmétique Amusante. 1895. Note IV: Section II: Les Jeux de Ruban, Nos. 1 & 2: Le nœud de cravate & Le nœud marin, pp. 220-222. Cites d'Aviso and says he does both the pentagonal and hexagonal knots, but Lucas only shows the pentagonal one.

E. Fourrey. Procédés Originaux de Constructions Géométriques. Vuibert, Paris, 1924. Pp. 113 & 135-139. Cites Lucas and cites d'Aviso as Traité de la Sphère and says he gives the pentagonal and hexagonal knots. Fourrey shows and describes both, also giving the pictures on his title page.

F. V. Morley. A note on knots. AMM 31 (1924) 237-239. Cites Knott's translation of Tom Tit. Says the process generalizes to (2n+3)-gons by using n loops. Gets even-gons by using two strips. Discusses using twisted strips.

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 64-65 gives square (a bit trivial), pentagon, hexagon, heptagon and octagon. Even case need two strips.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, pp. 16-17: Polygons constructed by tying paper knots. Shows how to tie square, pentagon, hexagon, heptagon and octagon.

James K. Brunton. Polygonal knots. MG 45 (No. 354) (Dec 1961) 299-302. All regular n-gons, n > 4, can be obtained, except n = 6 which needs two strips. Discusses which can be made without central holes.

Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. On pp. 47-48, he calls this the 'golden knot' (Le "nœud doré") and describes how to make it.

6.R. GEOMETRIC FALLACIES

General surveys of such fallacies can be found in the following. See also: 6.P, 10.A.1.

These fallacies are actually quite profound as the first two point out some major gaps in Euclid's axioms -- the idea of a point being inside a triangle really requires notions of order of points on a line and even the idea of continuity, i.e. the idea of real numbers.

Ball. MRE. 1st ed., 1892, pp. 31-34, two examples, discussed below. 3rd ed., 1896, pp. 39-46 = 4th ed., 1905, pp. 41-48, seven examples. 5th ed., 1911, pp. 44-52 = 11th ed., 1939, pp. 76-84, nine example.

Walther Lietzmann. Wo steckt der Fehler? Teubner, Stuttgart, (1950), 3rd ed., 1953. (Strens/Guy has 3rd ed., 1963(?).) (There are 2nd ed, 1952??; 5th ed, 1969; 6th ed, 1972. MG 54 (1970) 182 says the 5th ed appears to be unchanged from the 3rd ed.) Chap. B: V, pp. 87-99 has 18 examples.

(An earlier version of the book, by Lietzmann & Trier, appeared in 1913, with 2nd ed. in 1917. The 3rd ed. of 1923 was divided into two books: Wo Steckt der Fehler? and Trugschlüsse. There was a 4th ed. in 1937. The relevant material would be in Trugschlüsse, but I have not seen any of the relevant books, though E. P. Northrop cites Lietzmann, 1923, three times -- ??NYS.)

E. P. Northrop. Riddles in Mathematics. 1944. Chap. 6, 1944: 97-116, 232-236 & 249-250; 1945: 91-109, 215-219 & 230-231; 1961: 98-115, 216-219 & 229. Cites Ball, Lietzmann (1923), and some other individual items.

V. M. Bradis, V. L. Minkovskii & A. K. Kharcheva. Lapses in Mathematical Reasoning. (As: [Oshibki v Matematicheskikh Rassuzhdeniyakh], 2nd ed, Uchpedgiz, Moscow, 1959.) Translated by J. J. Schorr-Kon, ed. by E. A. Maxwell. Pergamon & Macmillan, NY, 1963. Chap. IV, pp. 123-176. 20 examples plus six discussions of other examples.

Edwin Arthur Maxwell. Fallacies in Mathematics. CUP, (1959), 3rd ptg., 1969. Chaps. II-V, pp. 13-36, are on geometric fallacies.

Ya. S. Dubnov. Mistakes in Geometric Proofs. (2nd ed., Moscow?, 1955). Translated by Alfred K. Henn & Olga A. Titelbaum. Heath, 1963. Chap 1-2, pp. 5-33. 10 examples.

А. Г. Конфорович. [A. G. Konforovich]. (Математичні Софізми і Парадокси [Matematichnī Sofīzmi ī Paradoksi] (In Ukrainian). Радянська Школа [Radyans'ka Shkola], Kiev, 1983.) Translated into German by Galina & Holger Stephan as: Konforowitsch, Andrej Grigorjewitsch; Logischen Katastrophen auf der Spur – Mathematische Sophismen und Paradoxa; Fachbuchverlag, Leipzig, 1990. Chap. 4: Geometrie, pp. 102-189 has 68 examples, ranging from the type considered here up through fractals and pathological curves.

S. L. Tabachnikov. Errors in geometrical proofs. Quantum 9:2 (Nov/Dec 1998) 37-39 & 49. Shows: every triangle is isosceles (6.R.1); the sum of the angles of a triangle is 180o without use of the parallel postulate; a rectangle inscribed in a square is a square; certain approaching lines never meet (6.R.3); all circles have the same circumference (cf Aristotle's Wheel Paradox in 10.A.1); the circumference of a wheel is twice its radius; the area of a sphere of radius R is π2R2.

6.R.1. EVERY TRIANGLE IS ISOSCELES

This is sometimes claimed to have been in Euclid's lost Pseudaria (Fallacies).

Ball. MRE, 1st ed., 1892, pp. 33-34. On p. 32, Ball refers to Euclid's lost Fallacies and presents this fallacy and the one in 6.R.2: "I do not know whether either of them has been published previously." In the 3rd ed., 1896, pp. 42-43, he adds the heading: To prove that every triangle is isosceles. In the 5th ed., 1911, p. 45, he adds a note that he believes these two were first published in his 1st ed. and notes that Carroll was fascinated by them and they appear in The Lewis Carroll Picture Book (= Carroll-Collingwood) -- see below.

Mathesis (1893). ??NYS. [Cited by Fourrey, Curiosities Geometriques, p. 145. Possibly Mathesis (2) 3 (Oct 1893) 224, cited by Ball in MRE, 3rd ed, 1896, pp. 44-45, cf in Section 6.R.4.]

Carroll-Collingwood. 1899. Pp. 264-265 (Collins: 190-191). = Carroll-Wakeling II, prob. 27: Every triangle has a pair of equal sides!, pp. 43 & 27. Every triangle is isosceles. Carroll may have stated this as early as 1888. Wakeling's solution just suggests making an accurate drawing. Carroll-Gardner, p. 65, mentions this and says it was not original with Carroll.

Ahrens. Mathematische Spiele. Teubner. Alle Dreiecke sind gleichschenklige. 2nd ed., 1911, chap. X, art. VI, pp. 108 & 119-120. 3rd ed., 1916, chap. IX, art. IX, pp. 92-93 & 109-111. 4th ed., 1919 & 5th ed., 1927, chap IX, art. IX, pp. 99-101 & 116-118.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 15: To prove all triangles are equilateral, pp. 16-17. Clear exposition of the fallacy.

See Read in 6.R.4 for a different proof of this fallacy.

6.R.2. A RIGHT ANGLE IS OBTUSE

Ball. MRE, 1st ed., 1892, pp. 32-33. See 6.R.1. In the 3rd ed., 1896, pp. 40-41, he adds the heading: To prove that a right angle is equal to an angle which is greater than a right angle.

Mittenzwey. 1895?. Prob. 331, pp. 58 & 106; 1917: 331, pp. 53 & 101.

Carroll-Collingwood. 1899. Pp. 266-267 (Collins 191-192). An obtuse angle is sometimes equal to a right angle. Carroll-Gardner, p. 65, mentions this and says it was not original with Carroll.

H. E. Licks. 1917. Op. cit. in 5.A. Art. 82, p. 56.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 16: To prove one right angle greater than another right angle, pp. 18-19. "Here again, if you take the trouble to draw an accurate diagram, you will find that the "construction" used for the alleged proof is impossible."

E. A. Maxwell. Note 2121: That every angle is a right angle. MG 34 (No. 307) (Feb 1950) 56-57. Detailed demonstration of the error.

6.R.3. LINES APPROACHING BUT NOT MEETING

Proclus. 5C. A Commentary on the First Book of Euclid's Elements. Translated by Glenn R. Morrow. Princeton Univ. Press, 1970. Pp. 289-291. Gives the argument and tries to refute it.

van Etten/Henrion/Mydorge. 1630. Part 2, prob. 7: Mener une ligne laquelle aura inclination à une autre ligne, & ne concurrera jamais contre l'Axiome des paralelles, pp. 13-14.

Schwenter. 1636. To be added.

Ozanam-Montucla. 1778. Paradoxe géométrique des lignes .... Prob. 70 & fig. 116-117, plate 13, 1778: 405-407; 1803: 411-413; 1814: 348-350. Prob. 69, 1840: 180-181. Notes that these arguments really produce a hyperbola and a conchoid. Hutton adds that a great many other examples might be found.

E. P. Northrop. Riddles in Mathematics. 1944. 1944: 209-211 & 239; 1945: 195-197 & 222; 1961: 197-198 & 222. Gives the 'proof' and its fallacy, with a footnote on p. 253 (1945: 234; 1961: 233) saying the argument "has been attributed to Proclus."

Jeremy Gray. Ideas of Space. OUP, 1979. Pp. 37-39 discusses Proclus' arguments in the context of attempts to prove the parallel postulate.

6.R.4. OTHERS

Ball. MRE, 3rd ed, 1896, pp. 44-45. To prove that, if two opposite sides of a quadrilateral are equal, the other two sides must be parallel. Cites Mathesis (2) 3 (Oct 1893) 224 -- ??NYS

Cecil B. Read. Mathematical fallacies & More mathematical fallacies. SSM 33 (1933) 575-589 & 977-983. There are two perpendiculars from a point to a line. Part of a line is equal to the whole line. Every triangle is isosceles (uses trigonometry). Angle trisection (uses a marked straightedge).

P. Halsey. Class Room Note 40: The ambiguous case. MG 43 (No. 345) (Oct 1959) 204-205. Quadrilateral ABCD with angle A = angle C and AB = CD. Is this a parallelogram?

6.S. TANGRAMS, ET AL.

GENERAL HISTORIES.

Hoffmann. 1893. Chap III, pp. 74-90, 96-97, 111-124 & 128 = Hoffmann-Hordern, pp. 62-79 & 86-87 with several photos. Describes Tangrams and Richter puzzles at some length. Lots of photos in Hordern. Photos on pp. 67, 71, 75, 87 show Richter's: Anchor (1890-1900, = Tangram), Tormentor (1898), Pythagoras (1892), Cross Puzzle (1892), Circular Puzzle (1891), Star Puzzle (1899), Caricature (1890-1900, = Tangram) and four non-Richter Tangrams in Tunbridge ware, ivory, mother-of-pearl and tortoise shell. Hordern Collection, pp. 45-57 & 60, (photos on pp. 46, 49, 50, 52, 54, 56, 60) shows different Richter versions of Tormentor (1880-1900), Pythagoras (1880-1900), Circular Puzzle (1880-1900), Star Puzzle (1880-1900) and has a wood non-Richter version instead of the ivory version in the last photo.

Ronald C. Read. Tangrams -- 330 Puzzles. Dover, 1965. The Introduction, pp. 1-6, is a sketch of the history. Will Shortz says this is the first serious attempt to counteract the mythology created by Loyd and passed on by Dudeney. Read cannot get back before the early 1800s and notes that most of the Loyd myth is historically unreasonable. However, Read does not pursue the early 1800s history in depth and I consider van der Waals to be the first really serious attempt at a history of the subject.

Peter van Note. Introduction. IN: Sam Loyd; The Eighth Book of Tan; (Loyd & Co., 1903); Dover, 1968, pp. v-viii. Brief debunking of the Loyd myth.

Jan van der Waals. History & Bibliography. In: Joost Elffers; Tangram; (1973), Penguin, 1976. Pp. 9-27 & 29-31. Says the Chinese term "ch'i ch'ae" dates from the Chu era (-740/-330), but the earliest known Chinese book is 1813. The History reproduces many pages from early works. The Bibliography cites 8 versions of 4 Chinese books (with locations!) from 1813 to 1826 and 18 Western books from 1805 to c1850. The 1805, and several other references, now seem to be errors.

S&B. 1986. Pp. 22-33 discusses loculus of Archimedes, Chie no Ita, Tangrams and Richter puzzles.

Alberto Milano. Due giochi di società dell'inizio dell'800. Rassegna di Studi e di Notizie 23 (1999) 131-177. [This is a publication by four museums in the Castello Sforzesco, Milan: Raccolta delle Stampe Achille Bertarelli; Archivio Fotografico; Raccolte d'Arte Applicata; Museo degli Strumenti Musicali. Photocopy from Jerry Slocum.] This surveys early books on tangrams, some related puzzles and the game of bell and hammer, with many reproductions of TPs and problems.

Jerry Slocum. The Tangram Book. (With Jack Botermans, Dieter Gebhardt, Monica Ma, Xiaohe Ma, Harold Raizer, Dic Sonneveld and Carla van Spluntern.) ©2001 (but the first publisher collapsed), Sterling, 2003. This is the long awaited definitive history of the subject! It will take me sometime to digest and summarize this, but a brief inspection shows that much of the material below needs revision!

Recent research by Jerry Slocum, backed up by The Admired Chinese Puzzle, indicates that the introduction of tangrams into Europe was done by a person or persons in Lord Amherst's 1815-1817 embassy to China, which visited Napoleon on St. Helena on its return voyage. If so, then the conjectural dating of several items below needs to be amended. I have amended my discussion accordingly and marked such dates with ??. Although watermarking of paper with the correct date was a legal requirement at the time, paper might have been stored for some time before it was printed on, so watermark dates only give a lower bound for the date of printing. I have seen several further items dated 1817, but it is conceivable that some material may have been sent back to Europe or the US a few years earlier -- cf Lee.

On 2 Nov 2003, I did the following brief summary of Slocum's work in a letter to an editor. I've made a few corrections and added a citation to the following literature.

Tangrams. The history of this has now been definitely established in Jerry Slocum's new book: The Tangram Book; ©2001 (but the first publisher collapsed), Sterling, 2003. This history has been extremely difficult to unravel because Sam Loyd deliberately obfuscated it in 1903, claiming the puzzle went back to 2000 BC, because the only previous attempt at a history had many errors, and because much of the material doesn't survive, or only a few examples survive. The history covers a wide range in both time and location, as evidenced by the presence of seven co-authors from several countries.

Briefly, the puzzle, in the standard form, dates from about 1800, in China. It is attributed to Yang-cho-chü-shih, but this is a pseudonym, meaning 'dim-witted recluse', and no copies of his work are known. The oldest known example of the game is one dated 1802 in a museum near Philadelphia -- see Lee, below. The oldest known book on the puzzle had a preface by Sang-hsia-ko [guest under the mulberry tree] dated June 1813 and a postscript by Pi-wu-chü-shih dated July 1813. This is only known from a Japanese facsimile of it made in 1839. This book was republished, with a book of solutions, in two editions in 1815 -- one with about four problems per page, the other with about eleven. The latter version was the ancestor of many 19C books, both in China and the west. Another 2 volume version appeared later in 1815. Sang-hsia-ko explicitly says "The origin of the Tangram lies within the Pythagorean theorem".

In 1816, several ships brought copies of the eleven problems per page books to the US, England and Europe. The first western publication of the puzzle is in early 1817 when J. Leuchars of 47 Piccadilly registered a copyright and advertised sets for sale. But the craze was really set off by the publication of The Fashionable Chinese Puzzle and its Key by John and Edward Wallis and John Wallis Jr in March 1817. This included a poem with a note that the game was "the favourite amusement of Ex-Emperor Napoleon". This went through many printings, with some (possibly the first) versions having nicely coloured illustrations. By the end of the year, there were many other books, including examples in France, Italy and the USA.

Dic Sonneveld, one of the co-authors of Slocum's book, managed to locate the tangram and books that had belonged to Napoleon in the Château de Malmaison, outside Paris, but there is no evidence that Napoleon spent much time playing with it. St. Helena was a regular stop for ships in the China trade. Napoleon is recorded as having bought a chess set from one ship and several notables are recorded as having presented Napoleon with gifts of Chinese objects. A diplomatic letter of Jan 1817 records sending an example of the game from St. Helena to Prince Metternich, but this example has not been traced.

The first American book was Chinese Philosophical and Mathematical Trangram by James Coxe, appearing in Philadelphia in August 1817. The word 'trangram' meaning 'an odd, intricately contrived thing' according to Johnson's Dictionary, was essentially obsolete by 1817, but was still in some use in the US. The earliest known use of the word 'tangram' is in Thomas Hill's Geometrical Puzzles for the Young, Boston, 1848. One suspects that he was influenced by Coxe's book, but he may have known that 'T'ang' is the Cantonese word for 'Chinese'. Hill later became President of Harvard University and was an active promoter and inventor of games for classroom use. In 1864, the word was in Webster's Dictionary.

However, the above is the story of the seven-piece tangram that we know today. There is a long background to this, dating back to the 3rd century BC, when Archimedes wrote a letter to Eratosthenes describing a fourteen piece puzzle, known as the Stomachion or Loculus of Archimedes. The few surviving texts are not very clear and there are two interpretations -- in one the standard arrangement of the pieces is a square and in the other it is a rectangle twice as wide as high. There are six (at least) references to the puzzle in the classical world, the last being in the 6th century. The puzzle was used to make a monstrous elephant, a brutal boar, a ship, a sword, etc., etc. The puzzle then disappears, and no form of it appears in the Arabic world, which has always surprised me, given the Arabic interest in patterns.

Further, several eastern predecessors of the tangrams are known. The earliest is a Japanese version of 1742 by Ganriken (or Granreiken) which has seven pieces, attributed (as were many things) to Sei Shonagon, a 10th century courtesan famous for her ingenuity. By the end of the 18th century, three other dissection/arrangement puzzles appeared in Japan, with 15, 19 and 19 pieces, including some semi-circles. An 1804 print by Utamaro shows courtesans playing with some version of the puzzle -- only two copies of this print have been located.

But the basic puzzle idea has its roots in Chinese approaches to the Theorem of Pythagoras and similar geometric proofs by dissection and rearrangement which date back to the 3rd century (and perhaps earlier). But the tangram did not develop directly from these ideas. From the 12th century, there was a Chinese tradition of making "Banquet Tables" in the form of several pieces that could be arranged in several ways. The first known Chinese book on furniture, by Huang Po-ssu in 1194, describes a Banquet Table formed of seven rectangular pieces: two long, two medium and three short. In 1617, Ko Shan described 'Butterfly Wing" tables with 13 pieces, including isosceles right triangles, right trapeziums and isosceles trapeziums. In 1856, a Chinese scholar noted the resemblance of these tables with the tangram and a modern Chinese historian of mathematics has observed that half of the butterfly arrangement can be easily transformed into the tangrams. No examples of these tables have survived, but tables (and serving dishes) in the tangram pattern exist and are probably still being made in China.

SPECIFIC ITEMS

Kanchusen. Wakoku Chiekurabe. 1727. Pp. 9 & 28-29: a simple dissection puzzle with 8 pieces. The problem appears to consist of a mitre comprising ¾ of a unit square; 4 isosceles right triangles of hypotenuse 1 and 3 isosceles right triangles of side ½, but the solution shows that all the triangles are the same size, say having hypotenuse 1, and the mitre shape is actually formed from a rectangle of size 1 x (2.

"Ganriken" [pseud., possibly of Fan Chu Sen]. Sei Shōnagon Chie-no-Ita (The Ingenious Pieces by Sei Shōnagon.) (In Japanese). Kyoto Shobo, Aug 1742, 18pp, 42 problems and solutions. Reproduced in a booklet, ed. by Kazuo Hanasaki, Tokyo, 1984, as pp. 19-36. Also reproduced in a booklet, transcribed into modern Japanese, with English pattern names and an English abstract, by Shigeo Takagi, 1989. This uses a set of seven pieces different than the Tangram. S&B, p. 22, shows these pieces. Sei Shōnagon (c965-c1010) was a famous courtier, author of The Pillow Book and renowned for her intelligence. The Introduction is signed Ganriken. S&B say this is probably Fan Chu Sen, but Takagi says the author's real name is unknown.

Utamaro. Interior of an Edo house, from the picture-book: The Edo Sparrows (or Chattering Guide), 1786. Reproduced in B&W in: J. Hillier; Utamaro -- Colour Prints and Paintings; Phaidon Press, Oxford, (1961), 2nd ed., 1979, p. 27, fig. 15. I found this while hunting for the next item. This shows two women contemplating some pieces but it is hard to tell if it is a tangram-type puzzle, or if perhaps they are cakes. Hiroko and Mike Dean tell me that they are indeed cooking cakes.

Utamaro. Woodcut. 1792. Shows two courtesans working on a tangram puzzle. Van der Waals dated this as 1780, but Slocum has finally located it, though he has only been able to find two copies of it! The courtesans are clearly doing a tangram-like puzzle with 12(?) pieces -- the pieces are a bit piled up and one must note that one of the courtesans is holding a piece. They are looking at a sheet with 10 problem figures on it.

Early 19C books from China -- ??NYS -- cited by Needham, p. 111.

Jean Gordon Lee. Philadelphians and the China Trade 1784-1844. Philadelphia Museum of Art, 1984, pp. 122-124. (Photocopy from Jerry Slocum.) P. 124, item 102, is an ivory tangram in a cardboard box, inscribed on the bottom of the box: F. Waln April 4th 1802. Robert Waln was a noted trader with China and this may have been a present for his third son Francis (1799-1822). This item is in the Ryerss Museum, a city museum in Philadelphia in the country house called Burholme which was built by one of Robert Waln's sons-in-law.

A New Invented Chinese Puzzle, Consisting of Seven Pieces of Ivory or Wood, Viz. 5 Triangles, 1 Rhomboid, & 1 Square, which may be so placed as to form the Figures represented in the plate. Paine & Simpson, Boro'. Undated, but the paper is watermarked 1806. This consists of two 'volumes' of 8 pages each, comprising 159 problems with no solutions. At the end are bound in a few more pages with additional problems drawn in -- these are direct copies of plates 21, 26, 22, 24, and 28 (with two repeats from plate 22) of The New and Fashionable Chinese Puzzle, 1817. Bound in plain covers. This is in Edward Hordern's collection and he provided a photocopy. Dalgety also has a copy.

Ch'i Ch'iao t'u ho-pi (= Qiqiao tu hebi) (Harmoniously combined book of tangram problems OR Seven clever pieces). 1813. (Bibliothek Leiden 6891, with an 1815 edition at British Library 15257 d 13.) van der Waals says it has 323 examples. The 1813 seems to be the earliest Chinese tangram book of problems, with the 1815 being the solutions. Slocum says there was a solution book in 1815 and that the problem book had a preface by Sang-hsia K'o (= Sang-xia-ke), which was repeated in the solution book with the same date. Milano mentions this, citing Read and van der Waals/Elffers, and says an example is on the BL. A version of this appears to have been the book given to Napoleon and to have started the tangram craze in Europe. I have now received a photocopy from Peter Rasmussen & Wei Zhang which is copied from van der Waals' copy from BL 15257 d 13. It has a cover, 6 preliminary pages and 28 plates with 318 problems. The pages are larger than the photocopies of 1813/1815 versions in the BL that Slocum gave me, which have 334 problems on 86 pages, but I see these are from 15257 d 5 and 14. I have a version of the smaller page format from c1820s which has 334 problems on 84pp, apparently lacking its first sheet. The problems are not numbered, but given Chinese names. They are identical to those appearing in Wallis's Fashionable Chinese Puzzle, below, except the pages are in different order, two pages are inverted, Wallis replaces Chinese names by western numbers and draws the figures a bit more accurately. Wallis skips one number and adds four new problems to get 323 problems - van der Waals seems to have taken 323 from Wallis.

Shichi-kou-zu Gappeki [The Collection of Seven-Piece Clever Figures]. Hobunkoku Publishing, Tokyo, 1881. This is a Japanese translation of an 1813 Chinese book "recognized as the earliest of existing Tangram book", apparently the previous item. [The book says 1803, but Jerry Slocum reports this is an error for 1813!] Reprinted, with English annotations by Y. Katagiri, from N. Takashima's copy, 1989. 129 problems (but he counts 128 because he omits one after no. 124), all included in my version of the previous item, no solutions.

Anonymous. A Grand Eastern Puzzle. C. Davenporte & Co. Registered on 24 Feb 1817, hence the second oldest English (and European?) tangram book [Slocum, p. 71.] It is identical to Ch'i Ch'iao t'u ho-pi, 1815, above, except that plates 25 and 27 have been interchanged. It appears to be made by using Chinese pages and putting a board cover on it. On the front cover is the only English text:

A

Grand Eastern Puzzle

----------

THE following Chineze Puzzle is recommended

to the Nobility, Gentry, and others, being superior to

any hitherto invented for the Amusement of the Juvenile

World, to whom it will afford unceasing recreation and

information; being formed on Geometrical principles, it

may not be considered as trifling to those of mature

years, exciting interest, because difficult and instructive,

imperceptibly leading the mind on to invention and per-

severence. -- The Puzzle consists of five triangles, a

square, and a rhomboid, which may be placed in upwards

of THREE HUNDRED and THIRTY Characters, greatly re-

sembling MEN, BEASTS, BIRDS, BOATS, BOTTLES, GLASS-

ES, URNS, &c. The whole being the unwearied exertion

of many years study and application of one of the Lite-

rati of China, and is now offered to the Public for their

patronage and support.

ENTERED AT STATIONERS HALL

----

Published and sold by

C. DAVENPORTE and Co.

No. 20, Grafton Street, East Euston Square.

The Fashionable Chinese Puzzle. Published by J. & E. Wallis, 42, Skinner Street and J. Wallis Junr, Marine Library, Sidmouth, nd [Mar 1817]. Photocopy from Jerry Slocum. This has an illustrated cover, apparently a slip pasted onto the physical cover. This shows a Chinese gentleman holding a scroll with the title. There is a pagoda in the background, a bird hovering over the scroll and a small person in the foreground examining the scroll. Slocum's copy has paper watermarked 1816.

PLUS

A Key to the New and Fashionable Chinese Puzzle, Published by J. and E. Wallis, 42, Skinner Street, London, Wherein is explained the method of forming every Figure contained in That Pleasing Amusement. Nd [Mar 1817]. Photocopy from the Bodleian Library, Oxford, catalogue number Jessel e.1176. TP seems to made by pasting in the cover slip and has been bound in as a left hand page. ALSO a photocopy from Jerry Slocum. In the latter copy, the apparent TP appears to be a paste down on the cover. The latter copy does not have the Stanzas mentioned below. Slocum's copy has paper watermarked 1815; I didn't check this at the Bodleian.

NOTE. This is quite a different book than The New and Fashionable Chinese Puzzle published by Goodrich in New York, 1817.

Bound in at the beginning of the Fashionable Chinese Puzzle and the Bodleian copy of the Key is: Stanzas, Addressed to Messrs. Wallis, on the Ingenious Chinese Puzzle, Sold by them at the Juvenile Repository, 42, Skinner Street. In the Key, this is on different paper than the rest of the booklet. The Stanzas has a footnote referring to the ex-Emperor Napoleon as being in a debilitated state. (Napoleon died in 1821, which probably led to the Bodleian catalogue's date of c1820 for the entire booklet - but see below. Then follow 28 plates with 323 numbered figures (but number 204 is skipped), solved in the Key. In the Bodleian copy of the Key, these are printed on stiff paper, on one side of each sheet, but arranged as facing pairs, like Chinese booklets.

[Philip A. H. Brown; London Publishers and Printers c. 1800-1870; British Library, 1982, p. 212] says the Wallis firm is only known to have published under the imprint J. & E. Wallis during 1813 and Ruth Wallis showed me another source giving 1813?-1814. This led me to believe that the booklets originally appeared in 1813 or 1814, but that later issues or some owner inserted the c1820 sheet of Stanzas, which was later bound in and led the Bodleian to date the whole booklet as c1820. Ruth Wallis showed me a source that states that John Wallis (Jun.) set up independently of his father at 186 Strand in 1806 and later moved to Sidmouth. Finding when he moved to Sidmouth might help date this publication more precisely, but it may be a later reissue. However, Slocum has now found the book advertised in the London Times in Mar 1817 and says this is the earliest Western publication of tangrams, based on the 1813/1815 Chinese work. Wallis also produced a second book of problems of his own invention and some copies seem to be coloured.

In AM, p. 43, Dudeney says he acquired the copy of The Fashionable Chinese Puzzle which had belonged to Lewis Carroll. He says it was "Published by J. and E. Wallis, 42 Skinner Street, and J. Wallis, Jun., Marine Library, Sidmouth" and quotes the Napoleon footnote, so this copy had the Stanzas included. This copy is not in the Strens Collection at Calgary which has some of Dudeney's papers.

Van der Waals cites two other works titled The Fashionable Chinese Puzzle. An 1818 edition from A. T. Goodridge [sic], NY, is in the American Antiquarian Society Library (see below) and another, with no details given, is in the New York Public Library. Could the latter be the Carroll/Dudeney copy?

Toole Stott 823 is a copy with the same title and imprint as the Carroll/Dudeney copy, but he dates it c1840. This version is in two parts. Part I has 1 leaf text + 26 col. plates -- it seems clear that col. means coloured, a feature that is not mentioned in any other description of this book -- perhaps these were hand-coloured by an owner. Unfortunately, he doesn't give the number of puzzles. I wonder if the last two plates are missing from this?? Part II has 1 leaf text + 32 col. plates, giving 252 additional figures. The only copy cited was in the library of J. B. Findlay -- I have recently bought a copy of the Findlay sale catalogue, ??NYR.

Toole Stott 1309 is listed with the title: Stanzas, .... J. & F. [sic] Wallis ... and Marine Library, Sidmouth, nd [c1815]. This has 1 leaf text and 28 plates of puzzles, so it appears that the Stanzas have been bound in and the original cover title slip is lost or was not recognised by Toole Stott. The date of c1815 is clearly derived from the Napoleon footnote but 1817 would have been more reasonable, though this may be a later reissue. Again only one copy is cited, in the library of Leslie Robert Cole.

Plates 1-28 are identical to plates 1-28 of The Admired Chinese Puzzle, but in different order. The presence of the Chinese text in The Admired Chinese Puzzle made me think the Wallis version was later than it.

Comparison of the Bodleian booklet with the first 27 plates of Giuoco Cinese, 1818?, reveals strong similarities. 5 plates are essentially identical, 17 plates are identical except for one, two or three changes and 3 plates are about 50% identical. I find that 264 of the 322 figures in the Wallis booklet occur in Giuoco Cinese, which is about 82%. However, even when the plates are essentially identical, there are often small changes in the drawings or the layout.

Some of the plates were copied by hand into the Hordern Collection's copy of A New Invented Chinese Puzzle, c1806??.

The Admired Chinese Puzzle A New & Correct Edition From the Genuine Chinese Copy. C. Taylor, Chester, nd [1817]. Paper is clearly watermarked 1812, but the Prologue refers to the book being brought from China by someone in Lord Amherst's embassy to China, which took place in 1815-1817 and which visited Napoleon on St. Helena on its return. Slocum dates this to after 17 Aug 1817, when Amherst's mission returned to England and this seems to be the second western book on tangrams. Not in Christopher, Hall, Heyl or Toole Stott -- Slocum says there is only one copy known in England! It originally had a cover with an illustration of two Chinese, titled The Chinese Puzzle, and one of the men holds a scroll saying To amuse and instruct. The Chinese text gives the title Ch'i ch'iao t'u ho pi (Harmoniously combined book of tangram problems). I have a photocopy of the cover from Slocum. Prologue facing TP; TP; two pp in Chinese, printed upside down, showing the pieces; 32pp of plates numbered at the upper left (sometimes with reversed numbers), with problems labelled in Chinese, but most of the characters are upside down! The plates are printed with two facing plates alternating with two facing blank pages. Plate 1 has 12 problems, with solution lines lightly indicated. Plates 2 - 28 contain 310 problems. Plates 29-32 contain 18 additional "caricature Designs" probably intended to be artistic versions of some of the abstract tangram figures. The Prologue shows faint guide lines for the lettering, but these appear to be printed, so perhaps it was a quickly done copperplate. The text of the Prologue is as follows.

This ingenious geometrical Puzzle was introduced into this Kingdom from China.

The following sheets are a correct Copy from the Chinese Publication, brought to England by a Gentleman of high Rank in the suit [sic] of Lord Amherst's late Embassy. To which are added caricature Designs as an illustration, every figure being emblematical of some Being or Article known to the Chinese.

The plates are identical to the plates in The Fashionable Chinese Puzzle above, but in different order and plate 4 is inverted and this version is clearly upside down.

Sy Hall. A New Chinese Puzzle, The Above Consists of Seven Pieces of Ivory or Wood, viz. 5 Triangles, 1 Rhomboid, and 1 Square, which will form the 292 Characters, contained in this Book; Observing the Seven pieces must be used to form each Character. NB. This Edition has been corrected in all its angles, with great care and attention. Engraved by Sy Hall, 14 Bury Street, Bloomsbury. 31 plates with 292 problems. Slocum, the Hordern Collection and BL have copies. I have a photocopy from a version from Slocum which has no date but is watermarked 1815. Slocum's recent book [The Tangram Book, pp. 74-75] shows a version of the book with the publisher's name as James Izzard and a date of 1817. Sy probably is an abbreviation of Sydney (or possibly Stanley?).

(The BL copy is watermarked IVY MILL 1815 and is bound with a large folding Plate 2 by Hall, which has 83 tinted examples with solution lines drawn in (by hand??), possibly one of four sheets giving all the problems in the book. However there is no relationship between the Plate and the book -- problems are randomly placed and often drawn in different orientation. I have a photocopy of the plate on two A3 sheets and a copy of a different plate with 72 problems, watermarked J. Green 1816.)

A New Chinese Puzzle. Third Edition: Universally allowed to be the most correct that has been published. 1817. Dalgety has a copy.

A New Chinese Puzzle Consisting of Seven Pieces of Ivory or Wood, The Whole of which must be used, and will form each of the CHARACTERS. J. Buckland, 23 Brook Street, Holborn, London. Paper watermarked 1816. (Dalgety has a copy, ??NYS.)

Miss D. Lowry. A Key to the Only Correct Chinese Puzzle Which has Yet Been Published, with above a Hundred New Figures. No. 1. Drawn and engraved by Miss Lowry. Printed by J. Barfield, London, 1817. The initial D. is given on the next page. Edward Hordern's collection has a copy.

W. Williams. New Mathematical Demonstrations of Euclid, rendered clear and familiar to the minds of youth, with no other mathematical instruments than the triangular pieces commonly called the Chinese Puzzle. Invented by Mr. W. Williams, High Beech Collegiate School, Essex. Published by the author, London, 1817. [Seen at BL.]

Enigmes Chinoises. Grossin, Paris, 1817. ??NYS -- described and partly reproduced in Milano. Frontispiece facing the TP shows an oriental holding a banner which has the pieces and a few problems on it. This is a small book, with five or six figures per page. The figures seem to be copied from the Fashionable Chinese Puzzle, but some figures are not in that work. Milano says this is cited as the first French usage of the term 'tangram', but this does not appear in Milano's photos and it is generally considered that Loyd introduced the word in the 1850s. Milano's phrasing might be interpreted as saying this is the first French work on tangrams.

Chinesische-Raethsel. Produced by Daniel Sprenger with designs by Matthaeus Loder, Vienna, c1818. ??NYS -- mentioned by Milano.

Chinesisches Rätsel. Enigmes chinoises. Heinrich Friedrich Muller (or Mueller), Vienna, c1810??. ??NYS (van der Waals). This is probably a German edition of the above and should be dated 1817 or 1818. However, Milano mentions a box in the Historisches Museum der Stadt Wien, labelled Grosse Chinesische Raethsel, produced by Mueller and dated 1815-1820.

Passe-temps Mathématique, ou Récréation à l'ile Sainte-Hélène. Ce jeu qui occupé à qu'on prétend, les loisirs du fameux exilé à St.-Hélène. Briquet, Geneva, 1817. 21pp. [Copy advertised by Interlibrum, Vaduz, in 1990.]

The New and Fashionable Chinese Puzzle. A. T. Goodrich & Co., New York, 1817. TP, 1p of Stanzas (seems like there should be a second page??), 32pp with 346 problems. Slocum has a copy.

[Key] to the Chinese Philosophical Amusements. A. T. Goodrich & Co., New York, 1817. TP, 2pp of stanzas (the second page has the Napoleon footnote and a comment which indicates it is identical to the material in the problem book), Index to the Key to the Chinese Puzzle, 80pp of solutions as black shapes with white spacing. Slocum has a copy.

NOTE. This is quite a different book than The Fashionable Chinese Puzzle published in London by Wallis in 1817.

Slocum writes: "Although the Goodrich problem book has the same title as the British book by Wallis and Goodrich has the "Stanzas" poem (except for the first 2 paragraphs which he deleted) the problem books have completely different layouts and Goodrich's solution book largely copies Chinese books."

Il Nuovo e Dilettevole Giuoco Chinese. Bardi, Florence, 1817. ??NYS -- mentioned by Milano.

Buonapartes Geliefkoosste Vermaack op St. Helena, op Chineesch Raadsel. 1er Rotterdam by J. Harcke. Prijs 1 - 4 ??. 2e Druck te(?) Rotterdam. Ter Steendrukkery van F. Scheffers & Co. Nanco Bordewijk has recently acquired this and Slocum has said it is a translation of one of the English items in c1818. I have just a copy of the cover, and it uses many fancy letters which I don't guarantee to have read correctly.

Recueil des plus jolis Jeux de Sociéte, dans lequel on trouve les gravures d'un grand nombre d'énigmes chinoises, et l'explication de ce nouveau jeu. Chez Audot, Librairie, Paris, 1818. Pp. 158-162: Le jeu des énigmes chinoises. This is a short introduction, saying that the English merchants in Japan have sent it back to their compatriots and it has come from England to France. This is followed by 11 plates. The first three are numbered. The first shows the pieces formed into a rectangle. The others have 99 problems, with 7 shown solved (all six of those on plate 2 and one (the square) on the 10th plate.)

Das grosse chinesische Rätselspiel für die elegante Welt. Magazin für Industrie (Leipzig) (1818). ??NYS (van der Waals). Jerry Slocum informs me that 'Magazin' here denotes a store, not a periodical, and that this is actually a game version with a packet of 50 cards of problems, occurring in several languages, from 1818. I have acquired a set of the cards which lacks one card (no. 17), in a card box with labels in French and Dutch pasted on. One side has: Nouvelles / ENIGMES / Chinoises / en Figures et en Paysages with a dancing Chinaman below. The other side has: Chineesch / Raadselspel, / voor / de Geleerde Waereld / in / 50 Beelaachlige / Figuren. with two birds below. Both labels are printed in red, with the dancing Chinaman having some black lines. The cards are 82 x 55 mm and are beautifully printed with coloured pictures of architectonic, anthropomorphic and zoomorphic designs in appropriate backgrounds. The first card has four shapes, three of which show the solution with dotted lines. All other cards have just one problem shape. The reverses have a simple design. Slocum says the only complete set he has seen is in the British Library. I have scanned the cards and the labels.

Gioco cinese chiamato il rompicapo. Milan, 1818. ??NYS (van der Waals). Fratelli Bettali, Milan, nd, of which Dalgety has a copy.

Al Gioco Cinese Chiamato Il Rompicapo Appendice di Figure Rappresentanti ... Preceduta da un Discorso sul Rompicapo e sulla Cina intitolato Passatempo Preliminare scritto dall'Autore Firenze All'Insegna dell'Ancora 1818. 64pp + covers. The cover or TP has an almond shape with the seven shapes inside. Pp. 3-43 are text -- the Passatempo Preliminare and an errata page. 12 plates. The first is headed Alfabeto in fancy Gothic. Plates 1-3 give the alphabet (J and W are omitted). Plate 4 has the positive digits. Plates 5-12 have facing pages giving the names of the figures (rather orientalized) and contain 100 problems. Hence a total of 133 problems, no solutions. The Hordern collection has a copy and I have a photocopy from it. This has some similarities to Giuoco Cinese. Described and partly reproduced in Milano.

Al Gioco Cinese chiamato il Rompicapo Appendice. Pietro & Giuseppe Vallardi, Milan, 1818. Possibly another printing of the item above. ??NYS -- described in Milano, who reproduces plates 1 & 2, which are identical to the above item, but with a simpler heading. Milano says the plates are identical to those in the above item.

Nuove e Dilettevole Giuoco Chinese. Milano presso li Frat. Bettalli Cont. del Cappello N. 4031. Dalgety has a copy. It is described and two pages are reproduced in Milano from an example in the Raccolta Bertarelli. Milano dates it as 1818. Cover illustration is the same as The Fashionable Chinese Puzzle, with the text changed. But it is followed by some more text: Questa ingegnosa invenzione è fondata sopra principi Geometrici, e consiste in 7 pezzi cioè 5. triangoli, un quadrato ed un paralellogrammo i quali possono essere combinati in modo da formare piu di 300 figure curiose. The second photo shows a double page identical to pp. 3-4 of The Fashionable Chinese Puzzle, except that the page number on p. 4 was omitted in printing and has been written in. (Quaritch's catalogue 646 (1947) item 698 lists this as Nuovo e dilettevole Giuoco Chinese, from Milan, [1820?])

Nuove e Dilettevole Giuoco Chinese. Bologna Stamperia in pietra di Bertinazzi e Compag. ??NYS -- described and partly reproduced in Milano from an example in the Raccolta Bertarelli. Identical to the above item except that it is produced lithographically, the text under the cover illustration has been redrawn, the page borders, the page numbers and the figure numbers are a little different. Milano's note 5 says the dating of this is very controversial. Apparently the publisher changed name in 1813, and one author claims the book must be 1810. Milano opts for 1813? but feels this is not consistent with the above item. From Slocum's work and the examples above, it seems clear it must be 1818?

Supplemento al nuovo giuoco cinese. Fratelli Bettalli, Milan, 1818. ??NYS -- described in Milano, who says it has six plates and the same letters and digits as Al Gioco Cinese Chiamato Il Rompicapo Appendice.

Giuoco Cinese Ossia Raccolta di 364. Figure Geometrica [last letter is blurred] formate con un Quadrato diviso in 7. pezzi, colli quali si ponno formare infinite Figure diversi, come Vuomini[sic], Bestie, Ucelli[sic], Case, Cocchi, Barche, Urne, Vasi, ed altre suppelletili domestiche: Aggiuntovi l'Alfabeto, e li Numeri Arabi, ed altre nuove Figure. Agapito Franzetti alle Convertite, Rome, nd [but 1818 is written in by hand]. Copy at the Warburg Institute, shelf mark FMH 4050. TP & 30 plates. It has alternate openings blank, apparently to allow you to draw in your solutions, as an owner has done in a few cases. The first plate shows the solutions with dotted lines, otherwise there are no solutions. There is no other text than on the TP, except for a florid heading Alfabeto on plate XXVIII. The diagrams have no numbers or names. The upper part of the TP is a plate of three men, intended to be Orientals, in a tent? The one on the left is standing and cutting a card marked with the pieces. The man on the right is sitting at a low table and playing with the pieces. He is seated on a box labelled ROMPI CAPO. A third man is seated behind the table and watching the other seated man. On the ground are a ruler, dividers and right angle. The Warburg does not know who put the date 1818 in the book, but the book has a purchase note showing it was bought in 1913. James Dalgety has the only other copy known. Sotheby's told him that Franzetti was most active about 1790, but Slocum finds Sotheby's is no longer very definite about this. I thought it possible that a page was missing at the beginning which gave a different form of the title, but Dalgety's copy is identical to this one. Mario Velucchi says it is not listed in a catalogue of Italian books published in 1800-1900. The letters and numbers are quite different to those shown in Elffers and the other early works that I have seen, but there are great similarities to The New and Fashionable Chinese Puzzle, 1817 (check which??), and some similarities to Al Gioco Cinese above. I haven't counted the figures to verify the 364. Mentioned in Milano, based on the copy I sent to Dario Uri.

Jeu du Casse Tete Russe. 1817? ??NYS -- described and partly reproduced in Milano from an example in the Raccolta Bertarelli but which has only four cards. Here the figures are given anthropomorphic or architectonic shapes. There are four cards on one coloured sheet and each card has a circle of three figures at the top with three more figures along the bottom. Each card has the name of the game at the top of the circle and "les secrets des Chinois dévoliés" and "casse tête russe" inside and outside the bottom of the circle. The figures are quite different than in the following item.

Nuovo Giuoco Russo. Milano presso li Frat. Bettalli Cont. del Cappello. [Frat. is an abbreviation of Fratelli (Brothers) and Cont. is an abbreviation of Contrade (road).] Box, without pieces, but with 16 cards of problems (one being examples) and instruction sheet (or leaflet). ??NYS - described by Milano with reproductions of the box cover and four of the cards. This example is in the Raccolta Bertarelli. Box shows a Turkish(?) man handing a box to another. On the first card is given the title and publisher in French: Le Casse-Tête Russe Milan, chez les Fr. Bettalli, Rue du Chapeau. The instruction sheet says that the Giuoco Chinese has had such success in the principal cities of Europe that a Parisian publisher has conceived another game called the Casse Tête Russe and that the Brothers Bettalli have hurried to produce it. Each card has four problems where the figures are greatly elaborated into architectonic forms, very like those in Metamorfosi, below. Undated, but Milano first gives 1815-1820, and feels this is closely related to Metamorfosi and similar items, so he concludes that it is 1818 or 1819, and this seems to be as correct as present knowledge permits. The figures are quite different than in the French version above.

Metamorfosi del Giuoco detto l'Enimma Chinese. Firenze 1818 Presso Gius. Landi Libraio sul Canto di Via de Servi. Frontispiece shows an angel drawing a pattern on a board which has the seven pieces at the top. The board leans against a plinth with the solution for making a square shown on it. Under the drawing is A. G. inv. Milano reproduces this plate. One page of introduction, headed Idea della Metamorfosi Imaginata dell'Enimma Chinese. 100 shapes, some solved, then with elegant architectonic drawings in the same shapes, signed Gherardesce inv: et inc: Milano identifies the artist as Alessandro Gherardesca (1779-1852), a Pisan architect. See S&B, pp. 24-25.

Grand Jeu du Casse Tête Français en X. Pieces. ??NYS -- described and partly reproduced in Milano, who says it comes from Paris and dates it 1818? The figures are anthropomorphic and are most similar to those in Jeu du Casse Tete Russe.

Grande Giuocho del Rompicapo Francese. Milano presso Pietro e Giuseppe Vallardi Contrada di S. Margherita No 401(? my copy is small and faint). ??NYS -- described and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in the previous item, but the figures have been redrawn rather than copied exactly.

Ch'i Ch'iao pan. c1820. (Bibliothek Leiden 6891; Antiquariat Israel, Amsterdam.) ??NYS (van der Waals).

Le Veritable Casse-tete, ou Enigmes chinoises. Canu Graveur, Paris, c1820. BL. ??NYS (van der Waals).

L'unico vero Enimma Chinese Tradotto dall'originale, pubblicato a Londra, da J. Barfield. Florence, [1820?]. [Listed in Quaritch's catalogue 646 (1947) item 699.)

A tangram appears in Pirnaisches Wochenblatt of 16 Dec 1820. ??NYS -- described in Slocum, p. 60.

Ch'i Ch'iao ch'u pien ho-pi. After 1820. (Bibliothek Leiden 6891.) ??NYS (van der Waals). 476 examples.

Nouveau Casse-Tête Français. c1820 (according to van der Waals). Reproduced in van der Waals, but it's not clear how the pages are assembled. Milano dates it a c1815 and indicates it is 16 cards, but van der Waals looks like it may have been a booklet of 16 pp with TP, example page and end page. The 16 pp have 80 problems.

Jerry Slocum has sent 2 large pages with 58 figurative shapes which are clearly the same pictures. The instructions are essentially the same, but are followed by rules for a Jeu de Patience on the second page and there is a 6 x 6 table of words on the first page headed "Morales trouvées dans les ruines de la célébres Ville de Persépolis ..." which one has to assemble into moral proverbs. It looks like these are copies of folding plates in some book of games.

Chinese Puzzle Georgina. A. & S. Josh Myers, & Co 144, Leadenhall Street, London. Ganton Litho. 81 examples on 8 plates with elegant TP. Pages are one-sided sheets, sewn in the middle, but some are upside down. Seen at BL (1578/4938).

Bestelmeier, 1823. Item 1278: Chinese Squares. It is not in the 1812 catalogue.

Slocum. Compendium. Shows the above Bestelmeier entry.

Anonymous. Ch'i ch'iao t'u ho pi (Harmoniously combined book of Tangram problems) and Ch'i ch'iao t'u chieh (Tangram solutions). Two volumes of tangrams and solutions with no title page, Chinese labels of the puzzles, in Chinese format (i.e. printed as long sheets on thin paper, accordion folded and stitched with ribbon. Nd [c1820s??], stiff card covers with flyleaves of a different paper, undoubtedly added later. 84 pages in each volume, containing 334 problems and solutions. With ownership stamp of a cartouche enclosing EWSHING, probably a Mr. E. W. Shing. Slocum says this is a c1820s reprint of the earliest Chinese tangram book which appeared in 1813 & 1815. This version omits the TP and opening text. I have a photocopy of the opening material from Slocum. The original problem book had a preface by Sang-hsia K'o, which was repeated in the solution book with the same date. Includes all the problems of Shichi-kou-zu Gappeki, qv.

New Series of Ch'i ch'iau puzzles. Printed by Lou Chen-wan, Ch'uen Liang, January 1826. ??NYS. (Copy at Dept. of Oriental Studies, Durham Univ., cited in R. C. Bell; Tangram Teasers.)

Neues chinesisches Rätselspiel für Kinder, in 24 bildlichen und alphabetischen Darstellungen. Friese, Pirna. Van der Waals, copying Santi, gives c1805, but Slocum, p. 60, reports that it first appears in Pirnaisches Wochenblatt of 19 Dec 1829, though there is another tangram in the issue of 16 Dec 1820. ??NYS.

Child. Girl's Own Book. 1833: 85; 1839: 72; 1842: 156. "Chinese Puzzles -- These consist of pieces of wood in the form of squares, triangles, &c. The object is to arrange them so as to form various mathematical figures."

Anon. Edo Chiekata (How to Learn It??) (In Japanese). Jan 1837, 19pp, 306 problems. (Unclear if this uses the Tangram pieces.) Reprinted in the same booklet as Sei Shōnagon, on pp. 37-55.

A Grand Eastern Puzzle. C. Davenport & Co., London. Nd. ??NYS (van der Waals). (Dalgety has a copy and gives C. Davenporte (??SP) and Co., No. 20, Grafton Street, East Euston Square. Chinese pages dated 1813 in European binding with label bearing the above information.)

Augustus De Morgan. On the foundations of algebra, No. 1. Transactions of the Cambridge Philosophical Society 7 (1842) 287-300. ??NX. On pp. 289, he says "the well-known toy called the Chinese Puzzle, in which a prescribed number of forms are given, and a large number of different arrangements, of which the outlines only are drawn, are to be produced."

Crambrook. 1843. P. 4, no. 4: Chinese Puzzle. Chinese Books, thirteen numbers. Though not illustrated, this seems likely to be the Tangrams -- ??

Boy's Own Book. 1843 (Paris): 439.

No. 19: The Chinese Puzzle. Instructions give five shapes and say to make one copy of some and two copies of the others. As written, this has two medium sized triangles instead of two large ones, though it is intended to be the tangrams. 11 problem shapes given, no answers. Most of the shapes occur in earlier tangram collections, particularly in A New Invented Chinese Puzzle. "The puzzle may be purchased, ..., at Mr. Wallis's, Skinner street, Snow hill, where numerous books, containing figures for this ingenious toy may also be obtained." = Boy's Treasury, 1844, pp. 426-427, no. 16. It is also reproduced, complete with the error, but without the reference to Wallis, as: de Savigny, 1846, pp. 355-356, no. 14: Le casse-tête chinois; Magician's Own Book, 1857, prob. 49, pp. 289-290; Landells, Boy's Own Toy-Maker, 1858, pp. 139-140; Book of 500 Puzzles, 1859, pp. 103-104; Boy's Own Conjuring Book, 1860, pp. 251-252; Wehman, New Book of 200 Puzzles, 1908, pp. 34-35.

No. 20: The Circassian puzzle. "This is decidedly the most interesting puzzle ever invented; it is on the same principle, but composed of many more pieces than the Chinese puzzle, and may consequently be arranged in more intricate figures. ..." No pieces or problems are shown. In the next problem, it says: "This and the Circassian puzzle are published by Mr. Wallis, Skinner-street, Snow-hill." = Boy's Treasury, 1844, p. 427, no. 17. = de Savigny, 1846, p. 356, no. 15: Le problème circassien, but the next problem omits the reference to Wallis.

Although I haven't recorded a Circassian puzzle yet -- cf in 6.S.2 -- I have just seen that the puzzle succeeding The Chinese Puzzle in Wehman, New Book of 200 Puzzles, 1908, pp. 35-36, is called The Puzzle of Fourteen which might be the Circassian puzzle. Taking a convenient size, this has two equilateral triangles of edge 1 and four each of the following: a 30o-60o-90o triangle with edges 2, 1, (3; a parallelogram with angles 60o and 120o with edges 1 and 2; a trapezium with base angles 60o and 60o, with lower and upper base edges 2 and 1, height (3/4 and slant edges 1/2 and (3/2. All 14 pieces make a rectangle 2(3 by 4.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 584-11, pp. 288 & 405: Chinesisches Verwandlungsspiel. Make a square with the tangram pieces. Shows just five of the pieces, but correctly states which two to make two copies of.

Prob. 584-16, pp. 289 & 406. Make an isosceles right triangle with the tangram pieces.

Prob. 584-18/25, pp. 289-291 & 407: Hieroglyphenspiele. Form various figures from various sets of pieces, mostly tangrams, but the given shapes have bits of writing on them so the assembled figure gives a word. Only one of the shapes is as in Boy's Own Book.

Prob. 588, pp. 298 & 410: Etliche Knackmandeln. Another tangram problem like the preceding, not equal to any in Boy's Own Book.

Adams & Co., Boston. Advertisement in The Holiday Journal of Parlor Plays and Pastimes, Fall 1868. Details?? -- photocopy sent by Slocum. P. 6: Chinese Puzzle. The celebrated Puzzle with which a hundred or more symmetrical forms can be made, with book showing the designs. Though not illustrated, this seems likely to be the Tangrams -- ??

Mittenzwey. 1880. Prob. 243-252, pp. 45 & 95-96; 1895?: 272-281, pp. 49 & 97-98; 1917: 272-282, pp. 45 & 92-93. Make a funnel, kitchen knife, hammer, hat with brim being horizontal or hanging down or turned up, church, saw, dovecote, hatchet, square, two equal squares.

J. Murray (editor of the OED). Two letters to H. E. Dudeney (9 Jun 1910 & 1 Oct 1910). The first inquires about the word 'tangram', following on Dudeney's mention of it in his "World's best puzzles" (op. cit. in 2). The second says that 'tan' has no Chinese origin; is apparently mid 19C, probably of American origin; and the word 'tangram' first appears in Webster's Dictionary of 1864. Dudeney, AM, 1917, p. 44, excerpts these letters.

F. T. Wang & C.-S. Hsiung. A theorem on the tangram. AMM 49 (1942) 596-599. They determine the 20 convex regions which 16 isosceles right triangles can form and hence the 13 ones which the Tangram pieces can form.

Mitsumasa Anno. Anno's Math Games. (Translation of: Hajimete deau sugaku no ehon; Fufkuinkan Shoten, Tokyo, 1982.) Philomel Books, NY, 1987. Pp. 38-43 & 95-96 show a simplified 5-piece tangram-like puzzle which I have not seen before. The pieces are: a square of side 1; three isosceles right triangles of side 1; a right trapezium with bases 1 and 2, altitude 1 and slant side (2. The trapezium can be viewed as putting together the square with a triangle. 19 problems are set, with solutions at the back.

James Dalgety. Latest news on oldest puzzles. Lecture to Second Meeting on the History of Recreational Mathematics, 1 Jun 1996. 10pp. In 1998, he extracted the two sections on tangrams and added a list of tangram books in his collection as: The origins of Tangram; © 1996/98; 10pp. (He lists about 30 books, eight up to 1850.) In 1993, he was buying tangrams in Hong Kong and asked what they called it. He thought they said 'tangram' but a slower repetition came out 'ta hau ban' and they wrote down the characters and said it translates as 'seven lucky tiles'. He has since found the characters in 19C Chinese tangram books. It is quite possible that Sam Loyd (qv under Murray, above) was told this name and wrote down 'tangram', perhaps adjusted a bit after thinking up Tan as the inventor.

At the International Congress on Mathematical Education, Seville, 1996, the Mathematical Association gave out The 3, 4, 5 Tangram, a cut card tangram, but in a 6 x 8 rectangular shape, so that the medium sized triangle was a 3-4-5 triangle. I modified this in Nov 1999, by stretching along a diagonal to form a rhombus with angles double the angles of a 3-4-5 triangle, so that four of the triangles are similar to 3-4-5 triangles. Making the small triangles be actually 3-4-5, all edges are integral. I made up 35 problems with these pieces. I later saw that Hans Wiezorke has mentioned this dissection in CFF, but with no problems. I distributed this as my present at G4G4, 2000.

6.S.1. LOCULUS OF ARCHIMEDES

See S&B 22. I recall there is some dispute as to whether the basic diagram should be a square or a double square.

E. J. Dijksterhuis. Archimedes. Munksgaard, Copenhagen, 1956; reprinted by Princeton Univ. Press, 1987. Pp. 408-412 is the best discussion of this topic and supplies most of the classical references.

Archimedes. Letter to Eratosthenes, c-250?. Greek palimpsest, c975, on MS no. 355, from the Cloister of Saint Sabba (= Mar Saba), Jerusalem, then at Metochion of the Holy Sepulchre, Constantinopole. [This MS disappeared in the confusion in Asia Minor in the 1920s but reappeared in 1998 when it was auctioned by Christie's in New York for c2M$. Hopefully, modern technology will allow a facsimile and an improved transcription in the near future.] Described by J. L. Heiberg (& H. G. Zeuthen); Eine neue Schrift des Archimedes; Bibliotheca Math. (3) 7 (1906-1907) 321-322. Heiberg describes the MS, but only mentions the loculus. The text is in Heiberg's edition of Archimedes; Opera; 2nd ed., Teubner, Leipzig, 1913, vol. II, pp. 415-424, where it has been restored using the Suter MSS below. Heath only mentions the problem in passing. Heiberg quotes Marius Victorinus, Atilius Fortunatianus and cites Ausonius and Ennodius.

H. Suter. Der Loculus Archimedius oder das Syntemachion des Archimedes. Zeitschr. für Math. u. Phys. 44 (1899) Supplement = AGM 9 (1899) 491-499. This is a collation from two 17C Arabic MSS which describe the construction of the loculus. They are different than the above MS. The German is included in Archimedes Opera II, 2nd ed., 1913, pp. 420-424.

Dijksterhuis discusses both of the above and says that they are insufficient to determine what was intended. The Greek seems to indicate that Archimedes was studying the mathematics of a known puzzle. The Arabic shows the construction by cutting a square, but the rest of the text doesn't say much.

Lucretius. De Rerum Natura. c-70. ii, 778-783. Quoted and discussed in H. J. Rose; Lucretius ii. 778-83; Classical Review (NS) 6 (1956) 6-7. Brief reference to assembling pieces into a square or rectangle.

Decimus Magnus Ausonius. c370. Works. Edited & translated by H. G. Evelyn White. Loeb Classical Library, ??date. Vol. I, Book XVII: Cento Nuptialis (A Nuptial Cento), pp. 370-393 (particularly the Preface, pp. 374-375) and Appendix, pp. 395-397. Refers to 14 little pieces of bone which form a monstrous elephant, a brutal boar, etc. The Appendix gives the construction from the Arabic version, via Heiberg, and forms the monstrous elephant.

Marius Victorinus. 4C. VI, p. 100 in the edition of Keil, ??NYS. Given in Archimedes Opera II, 2nd ed., 1913, p. 417. Calls it 'loculus Archimedes' and says it had 14 pieces which make a ship, sword, etc.

Ennodius. Carmina: De ostomachio eburneo. c500. In: Magni Felicis Ennodii Opera; ed. by F. Vogel, p. 340. In: Monumenta Germaniae Historica, VII (1885) 249. ??NYS. Refers to ivory pieces to be assembled.

Atilius Fortunatianus. 6C. ??NYS Given in Archimedes Opera II, p. 417. Same comment as for Marius Victorinus.

E. Fourrey. Curiositiés Géométriques. (1st ed., Vuibert & Nony, Paris, 1907); 4th ed., Vuibert, Paris, 1938. Pp. 106-109. Cites Suter, Ausonius, Marius Victorinus, Atilius Fortunatianus.

Collins. Book of Puzzles. 1927. The loculus of Archimedes, pp. 7-11. Pieces made from a double square.

6.S.2. OTHER SETS OF PIECES

See Hoffmann & S&B, cited at the beginning of 6.S, for general surveys.

See Bailey in 6.AS.1 for an 1858 puzzle with 10 pieces and The Sociable and Book of 500 Puzzles, prob. 10, in 6.AS.1 for an 11 piece puzzle.

There are many versions of this idea available and some are occasionally given in JRM.

The Richter Anchor Stone puzzles and building blocks were inspired by Friedrich Froebel (or Fröbel) (1782-1852), the educational innovator. He was the inventor of Kindergartens, advocated children's play blocks and inspired both the Richter Anchor Stone Puzzles and Milton Bradley. The stone material was invented by Otto Lilienthal (1848-1896) (possibly with his brother Gustav) better known as an aviation pioneer -- they sold the patent and their machines to F. Adolph Richter for 1000 marks. The material might better be described as a kind of fine brick which could be precisely moulded. Richter improved the stone and began production at Rudolstadt, Thüringen, in 1882; the plant closed in 1964. Anchor was the company's trademark. He made at least 36 puzzles and perhaps a dozen sets of building blocks which were popular with children, architects, engineers, etc. The Deutsches Museum in Munich has a whole room devoted to various types of building blocks and materials, including the Anchor blocks. The Speelgoed Museum 'Op Stelten' (Sint Vincentiusstraat 86, NL-4902 (or 4901) GL Oosterhout, Noord-Brabant, The Netherlands; tel: 0262 452 825; fax: 0262 452 413) has a room of Richter blocks and some puzzles. There was an Anker Museum in the Netherlands (Stichting Ankerhaus (= Anker Museum); Opaalstraat 2-4 (or Postf. 1061), NL-2400 BB Alphen aan den Rijn, The Netherlands; tel: 01720-41188) which produced replacement parts for Anker stone puzzles. Modern facsimiles of the building sets are being produced at Rudolstadt.

In 1996 I noticed the ceiling of the room to the south of the Salon of the Ambassadors in the Alcazar of Seville. This 15C? ceiling was built by workmen influenced by the Moorish tradition and has 112 square wooden panels in a wide variety of rectilineal patterns. One panel has some diagonal lines and looks like it could be used as a 10 piece tangram-like puzzle. Consider a 4 x 4 square. Draw both diagonal lines, then at two adjacent corners, draw two lines making a unit square at these corners. At the other two corners draw one of these two lines, namely the one perpendicular to their common side. This gives six isosceles right triangles of edge 1; two pentagons with three right angles and sides 1, 2, 1, (2, (2; two quadrilaterals with two right angles and sides 2, 1, (2, 2(2. Since geometric patterns and panelling are common features of Arabic art, I wonder if there are any instances of such patterns being used for a tangram-like puzzle?

Grand Jeu du Casse Tête Français en X. Pieces. ??NYS -- described and partly reproduced in Milano, who says it comes from Paris and dates it 1818? The figures are anthropomorphic and are most similar to those in Jeu du Casse Tete Russe.

Grande Giuocho del Rompicapo Francese. Milano presso Pietro e Giuseppe Vallardi Contrada di S. Margherita No 401(? my copy is small and faint). ??NYS -- described and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in the previous item, but the figures have been redrawn rather than copied exactly.

Allizeau. Les Métamorphoses ou Amusemens Géometriques Dédiée aux Amateurs Par Allizeau. A Paris chex Allizeau Quai Malaquais, No 15. ??NYS -- described and partly reproduced in Milano. This uses 15 pieces and the problems tend to be architectural forms, like towers.

Jackson. Rational Amusement. 1821. Geometrical Puzzles, nos. 20-27, pp. 27-29 & 88-89 & plate II, figs. 15-22. This is a set of 20 pieces of 8 shapes used to make a square, a right triangle, three squares, etc.

Crambrook. 1843. P. 4, no. 1: Pythagorean Puzzle, with Book. Though not illustrated, this is probably(??) the puzzle described in Hoffmann, below, which was a Richter Anchor puzzle No. 12 of the same name and is still occasionally seen. See S&B 28.

Edward Hordern's collection has a Circassian Puzzle, c1870, with many pieces, but I didn't record the shapes -- cf Boy's Own Book, 1843 (Paris), in section 6.S.

Mittenzwey. 1880.

Prob. 177-179, pp. 34 & 86; 1895?: 202-204, pp. 38-39 & 88; 1917: 202-204, pp. 35 & 84-85. Consider the ten piece version of dissecting 5 squares to one (6.AS.1). Use the pieces to make:

a squat octagon, a house gable-end, a church (no solution), etc.;

two dissimilar rectangles;

three dissimilar parallelograms, two dissimilar trapezoids. Solution says one can make many other shapes with these pieces, e.g. a trapezoid with parallel sides in the proportion 9 : 11.

Prob. 181-184, pp. 34-35 & 87-88; 1895?: 206-209, pp. 39 & 89-90; 1917: 206-209, pp. 36 & 85-86. Take six equilateral triangles of edge 2. Cut an equilateral triangle of edge 1 from the corner of each of them, giving 12 pieces. Make a hexagon in eight different ways [there are many more -- how many??] and three tangram-like shapes.

Prob. 195-196, pp. 36 & 89; 1895?: 220-221, pp. 41 & 91; 1917: 220-221, pp. 37 & 87. Use four isosceles right triangles, say of leg 1, to make a square, a 1 x 4 rectangle and an isosceles right triangle.

Nicholas Mason. US Patent 232,140 - Geometrical Puzzle-Block. Applied: 13 May 1880; patented 14 Sep 1880. 1p plus 2pp diagrams. Five squares, six units square, each cut into four pieces in the same way. Start at the midpoint of a side and cut to an opposite corner. (This is the same cut used to produce the ten piece 'Five Squares to One' puzzle.) Cut again in the triangle just formed, from the same midpoint to a point one unit from the right angle corner of the piece just made. This gives a right triangle of sides 3, 1, (10 and a triangle of sides 5, (10, 3(45. Cut again from the same midpoint across the trapezoidal piece made by the first cut, to a point five units from the corner previously cut to. This gives a triangle of sides 5, 3(5, 2(10 and a right trapezoid with sides 2,(10, 1, 6, 3. This was produced as Hill's American Geometrical Prize Puzzle in England ("Price, One Shilling.") in 1882. Harold Raizer produced a facsimile version, with facsimile box label and instructions for IPP22. The instructions have 20 problems to solve and the solutions have to be submitted by 1 May 1882.

Hoffmann. 1893. Chap. III, no. 3: The Pythagoras Puzzle, pp. 83-85 & 117-118 = Hoffmann-Hordern, pp. 69-72. This has 7 pieces and is quite like the Tangram -- see comment under Crambrook. Photo on p. 71, with different version in Hordern Collection, p. 50.

C. Dudley Langford. Note 1538: Tangrams and incommensurables. MG 25 (No. 266) (Oct 1941) 233-235. Gives alternate dissections of the square and some hexagonal dissections.

C. Dudley Langford. Note 2861: A curious dissection of the square. MG 43 (No. 345) (Oct 1959) 198. There are 5 triangles whose angles are multiples of π/8 = 22½o. He uses these to make a square.

See items at the end of 6.S.

6.T. NO THREE IN A LINE PROBLEM

See also section 6.AO.2.

Loyd. Problem 14: A crow puzzle. Tit-Bits 31 (16 Jan & 6 Feb 1897) 287 & 343. = Cyclopedia, 1914, Crows in the corn, pp. 110 & 353. = MPSL1, prob. 114, pp. 113 & 163-164. 8 queens with no two attacking and no three in any line.

Dudeney. The Tribune (7 Nov 1906) 1. ??NX. = AM, prob. 317, pp. 94 & 222. Asks for a solution with two men in the centre 2 x 2 square.

Loyd. Sam Loyd's Puzzle Magazine, January 1908. ??NYS. (Given in A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; p. 100, where it is described as the only solution with 2 pieces in the 4 central squares.)

Ahrens, MUS I 227, 1910, says he first had this in a letter from E. B. Escott dated 1 Apr 1909. (W. Moser, below, refers this to the 1st ed., 1900, but this must be due to his not having seen it.)

C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV: No. 2: Another draught puzzle, pp. 515 & 520. The problem says "no three men shall be in a line, either horizontally or perpendicularly". The solution says "no three are in a line in any direction" and the diagram shows this is indeed true.

Loyd. Picket posts. Cyclopedia, 1914, pp. 105 & 352. = MPSL2, prob. 48, pp. 34 & 136. 2 pieces initially placed in the 4 central squares.

Blyth. Match-Stick Magic. 1921. Matchstick board game, p. 73. 6 x 6 version phrased as putting "only two in any one line: horizontal, perpendicular, or diagonal." However, his symmetric solution has three in a row on lines of slope 2.

King. Best 100. 1927. No. 69, pp. 28 & 55. Problem on the 6 x 6 board -- gives a symmetric solution. Says "there are two coins on every row" including "diagonally across it", but he has three in a row on lines of slope 2.

Loyd Jr. SLAHP. 1928. Checkers in rows, pp. 40 & 98. Different solution than in Cyclopedia.

M. Adams. Puzzle Book. 1939. Prob. C.83: Stars in their courses, pp. 144 & 181. Same solution as King, but he says "two stars in each vertical row, two in each horizontal row, and two in each of the the two diagonals .... There must not be more than two stars in the same straight line", but he has three in a row on lines of slope 2.

W. O. J. Moser & J. Pach. No-three-in-line problem. In: 100 Research Problems in Discrete Geometry 1986; McGill Univ., 1986. Problem 23, pp. 23.1 -- 23.4. Survey with 25 references. Solutions are known on the n x n board for n ( 16 and for even n ( 26. Solutions with the symmetries of the square are only known for n = 2, 4, 10.

6.U. TILING

6.U.1. PENROSE PIECES

R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10 (1974) 266-272.

M. Gardner. SA (Jan 1977). Extensively rewritten as Penrose Tiles, Chaps. 1 & 2.

R. Penrose. Pentaplexity. Eureka 39 (1978) 16-22. = Math. Intell. 2 (1979) 32-37.

D. Shechtman, I. Blech, D. Gratias & J. W. Cohn. Metallic phase with long-range orientational order and no translational symmetry. Physical Rev. Letters 53:20 (12 Nov 1984) 1951-1953. Describes discovery of 'quasicrystals' having the symmetry of a Penrose-like tiling with icosahedra.

David R. Nelson. Quasicrystals. SA 255:2 (Aug 1986) 32-41 & 112. Exposits the discovery of quasicrystals. First form is now called 'Shechtmanite'.

Kimberly-Clark Corporation has taken out two patents on the use of the Penrose pattern for quilted toilet paper as the non-repetition prevents the tissue from 'nesting' on the roll. In Apr 1997, Penrose issued a writ against Kimberly Clark Ltd. asserting his copyright on the pattern and demanding damages, etc.

John Kay. Top prof goes potty at loo roll 'rip-off'. The Sun (11 Apr 1997) 7.

Patrick McGowan. It could end in tears as maths boffin sues Kleenex over design. The Evening Standard (11 Apr 1997) 5.

Kleenex art that ended in tears. The Independent (12 Apr 1997) 2.

For a knight on the tiles. Independent on Sunday (13 Apr 1997) 24. Says they exclusively reported Penrose's discovery of the toilet paper on sale in Dec 1996.

D. Trull. Toilet paper plagiarism. Parascope, 1997 -- available on content/paranormal/arti...

6.U.2. PACKING BRICKS IN BOXES

In two dimensions, it is not hard to show that a x b packs A x B if and only if a  divides either A or B; b divides either A or B; A and B are both linear combinations of a and b. E.g. 2 x 3 bricks pack a 5 x 6 box.

See also 6.G.1.

Anon. Prob. 52. Hobbies 30 (No. 767) (25 Jun 1910) 268 & 283 & (No. 770) (16 Jul 1910) 328. Use at least one of each of 5 x 7, 5 x 10, 6 x 10 to make the smallest possible square. Solution says to use 4, 4, 1, but doesn't show how. There are lots of ways to make the assembly.

Manuel H. Greenblatt ( -1972, see JRM 6:1 (Winter 1973) 69). Mathematical Entertainments. Crowell, NY, 1965. Construction of a cube, pp. 80-81. Can 1 x 2 x 4 fill 6 x 6 x 6? He asserts this was invented by R. Milburn of Tufts Univ.

N. G. de Bruijn. Filling boxes with bricks. AMM 76 (1969) 37-40. If a1 x ... x an fills A1 x ... x An and b divides k of the ai, then b divides at least k of the Ai. He previously presented the results, in Hungarian, as problems in Mat. Lapok 12, pp. 110-112, prob. 109 and 13, pp. 314-317, prob. 119. ??NYS.

D. A. Klarner. Brick-packing puzzles. JRM 6 (1973) 112-117. General survey. In this he mentions a result that I gave him -- that 2 x 3 x 7 fills a 8 x 11 x 21, but that the box cannot be divided into two packable boxes. However, I gave him the case 1 x 3 x 4 in 5 x 5 x 12 which is the smallest example of this type. Tom Lensch makes fine examples of these packing puzzles.

T. H. Foregger, proposer; Michael Mather, solver. Problem E2524 -- A brick packing problem. AMM 82:3 (Mar 1975) 300 & 83:9 (Nov 1976) 741-742. Pack 41  1 x 2 x 4 bricks in a 7 x 7 x 7 box. One cannot get 42 such bricks into the box.

6.V. SILHOUETTE AND VIEWING PUZZLES

Viewing problems must be common among draughtsmen and engineers, but I haven't seen many examples. I'd be pleased to see further examples.

2 silhouettes.

Circle & triangle -- van Etten, Ozanam, Guyot, Magician's Own Book (UK version)

Circle & square -- van Etten

Circle & rhombus -- van Etten, Ozanam

Rectangle with inner rectangle & rectangle with notch -- Diagram Group.

3 silhouettes.

Circle, circle, circle -- Madachy

Circle, cross, square -- Shortz collection (c1884), Wyatt, Perelman

Circle, oval, rectangle -- van Etten, Ozanam, Guyot, Magician's Own Book (UK version)

Circle, oval, square -- van Etten, Tradescant, Ozanam, Ozanam-Montucla, Badcock, Jackson, Rational Recreations, Endless  Amusement II, Young Man's Book

Circle, rhombus, rectangle -- Ozanam, Alberti

Circle, square, triangle -- Catel, Bestelmeier, Jackson, Boy's Own Book, Crambrook, Family Friend, Magician's Own Book, Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Riecke, Elliott, Mittenzwey, Tom Tit, Handy Book, Hoffmann, Williams, Wyatt, Perelman, Madachy. But see Note below.

Square, tee, triangle -- Perelman

4 silhouettes.

Circle, square, triangle, rectangle with curved ends -- Williams

2 views.

Antilog, Ripley's, Diagram Group;

3 views.

Madachy, Ranucci,

For the classic Circle, Square, Triangle, version, the triangle cannot be not equilateral. Consider a circle, rectangle, triangle version. If D is the diameter of the circle and H is the height of the plug, then the rectangle has dimensions D x H and the triangle has base D and side S, so S = ((H2 + D2/4). Making the rectangle a square, i.e. H = D, makes S = D(5/2, while making the triangle equilateral, i.e. S = D, makes H = D(3/2.

van Etten. 1624.

Prob. 22 (misnumbered 15 in 1626) (Prob. 20), pp. 19-20 & figs. opp. p. 16 (pp. 35-36): 2 silhouettes -- one circular, the other triangular, rhomboidal or square. (English ed. omits last case.) The 1630 Examen says the author could have done better and suggests: isosceles triangle, several scalene triangles, oval or circle, which he says can be done with an elliptically cut cone and a scalene cone. I am not sure I believe these. It seems that the authors are allowing the object to fill the hole and to pass through the hole moving at an angle to the board rather than perpendicularly as usually understood. In the English edition the Examination is combined with that of the next problem.

Prob. 23 (21), pp. 20-21 & figs. opp. p. 16 (pp. 37-38): 3 silhouettes -- circle, oval and square or rectangle. The 1630 Examen suggests: square, circle, several parallelograms and several ellipses, which he says can be done with an elliptic cylinder of height equal to the major diameter of the base. The English Examination says "a solid colume ... cut Ecliptick-wise" -- ??

John II Tradescant (1608-1662). Musæum Tradescantianum: Or, A Collection of Rarities Preserved at South-Lambeth neer London By John Tradescant. Nathaniel Brooke, London, 1656. [Facsimile reprint, omitting the Garden List, Old Ashmolean Reprints I, edited by R. T. Gunther, on the occasion of the opening of the Old Ashmolean Museum as what has now become the Museum of the History of Science, Oxford. OUP, 1925.] John I & II Tradescant were gardeners to nobility and then royalty and used their connections to request naval captains to bring back new plants, curiosities and "Any thing that Is strang". These were accumulated at his house and garden in south Lambeth, becoming known as Tradescant's Ark, eventually being acquired by Elias Ashmole and becoming the foundation of the Ashmolean Museum in Oxford. This catalogue was prepared by Elias Ashmole and his friend Thomas Wharton, but they are not named anywhere in the book. It was the world's first museum catalogue.

P. 37, last entry: "A Hollow cut in wood, that will fit a round, square and ovall figure."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. He says square, circle and triangle is in a book in front of him dated 1674. I suspect this must be the 1674 English edition of van Etten, but I don't find the problem in the English editions I have examined. Perhaps Dudeney just meant that the idea was given in the 1674 book, though he is specifically referring to the square, circle, triangle version.

Ozanam. 1725. Vol. II, prob. 58 & 59, pp. 455-458 & plate 25* (53 (note there is a second plate with the same number)). Circle and triangle; circle and rhombus; circle, oval, rectangle; circle, oval, square. Figures are very like van Etten. See Ozanam-Montucla, 1778.

Ozanam. 1725. Vol. IV. No text, but shown as an unnumbered figure on plate 15 (17). 3 silhouettes: circle, rhombus, rectangle.

Simpson. Algebra. 1745. Section XVIII, prob. XXIX, pp. 279-281. (1790: prob. XXXVII, pp. 306-307. Computes the volume of an ungula obtained by cutting a cone with a plane. Cf Riecke, 1867.

Alberti. 1747. No text, but shown as an unnumbered figure on plate XIIII, opp. p. 218 (112), copied from Ozanam, 1725, vol IV. 3 silhouettes: circle, rhombus, rectangle.

Ozanam-Montucla. 1778. Faire passer un même corps par un trou quarré, rond & elliptique. Prob. 46, 1778: 347-348; 1803: 345-346; 1814: 293. Prob. 45, 1840: 149-150. Circle, ellipse, square.

Catel. Kunst-Cabinet. 1790. Die mathematischen Löcher, p. 16 & fig. 42 on plate II. Circle, square, triangle.

E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Op. cit. in 6.P.2. 1799. Vol. 2, Quatrième récréation, p. 45 & figs. 1-4, plate 7, opp. p. 45. 2 silhouettes: circle & triangle; 3 silhouettes: circle, oval, rectangle.

Bestelmeier.

1801. Item 536: Die 3 mathematischen Löcher. (See also the picture of Item 275, but that text is for another item.) Square, triangle and circle.

1807. Item 1126: Tricks includes the square, triangle and circle.

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. P. 14, no. 23: How to make a Peg that will exactly fit three different kinds of Holes. "Let one of the holes be circular, the other square, and the third an oval; ...." Solution is a cylinder whose height equals its diameter.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 16, pp. 26 & 86. Circle, square, triangle, with discussion of the dimensions: "a wedge, except that its base must be a circle".

No. 29, pp. 30 & 89-90. Circle, oval, square.

Rational Recreations. 1824. Feat 19, p. 66. Circle, oval, square.

Endless Amusement II. 1826? P. 62: "To make a Peg that will exactly fit three different kinds of Holes." Circle, oval, square. c= Badcock.

The Boy's Own Book. The triple accommodation. 1828: 419; 1828-2: 424; 1829 (US): 215; 1855: 570; 1868: 677. Circle, square and triangle.

Young Man's Book. 1839. Pp. 294-295. Circle, oval, square. Identical to Badcock.

Crambrook. 1843. P. 5, no. 16: The Mathematical Paradox -- the Circle, Triangle, and Square. Check??

Family Friend 3 (1850) 60 & 91. Practical puzzle -- No. XII. Circle, square, triangle. This is repeated as Puzzle 16 -- Cylinder puzzle in (1855) 339 with solution in (1856) 28.

Magician's Own Book. 1857. Prob. 21: The cylinder puzzle, pp. 273 & 296. Circle, square, triangle. = Book of 500 Puzzles, 1859, prob. 21, pp. 87 & 110. = Boy's Own Conjuring Book, 1860, prob. 20, pp. 235 & 260.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 42, pp. 403 & 442. Identical to Magician's Own Book, with diagram inverted.

F. J. P. Riecke. Op. cit. in 4.A.1, vol. 1, 1867. Art. 33: Die Ungula, pp. 58-61. Take a cylinder with equal height and diameter. A cut from the diameter of one base which just touches the other base cuts off a piece called an ungula (Latin for claw). He computes the volume as 4r3/3. He then makes the symmetric cut to produce the circle, square, triangle shape, which thus has volume (2π - 8/3) r3. Says he has seen such a shape and a board with the three holes as a child's toy. Cf Simpson, 1745.

Magician's Own Book (UK version). 1871. The round peg in the square hole: To pass a cylinder through three different holes, yet to fill them entirely, pp. 49-50. Circle, oval, rectangle; circle & (isosceles) triangle.

Alfred Elliott. Within-Doors. A Book of Games and Pastimes for the Drawing Room. Nelson, 1872. [Toole Stott 251. Toole Stott 1030 is a 1873 ed.] No. 4: The cylinder puzzle, pp. 27-28 & 30-31. Circle, square, triangle.

Mittenzwey. 1880. Prob. 257, pp. 46 & 97; 1895?: 286, pp. 50 & 99-100; 1917: 286, pp. 45 & 94-95. Circle, square, triangle.

Will Shortz has a puzzle trade card with the circle, cross, square problem, c1884.

Tom Tit, vol. 2. 1892. La cheville universelle, pp. 161-162. = K, no. 28: The universal plug, pp. 72-73. = R&A, A versatile peg, p. 106. Circle, square, triangle.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 238-242: Captain S's peg puzzle. Circle, square, triangle.

Hoffmann. 1893. Chap. X, no. 20: One peg to fit three holes, pp. 344 & 381-382 = Hoffmann-Hordern, pp. 238-239, with photo. Circle, square, triangle. Photo on p. 239 shows two examples: one simply a wood board and pieces; the other labelled The Holes and Peg Puzzle, from Clark's Cabinet of Puzzles, 1880-1900, but this seems to be just a card box with the holes.

Williams. Home Entertainments. 1914. The plug puzzle, pp. 103-104. Circle, square, triangle and rectangle with curved ends. This is the only example of this four-fold form that I have seen. Nice drawing of a board with the plug shown in each hole, except the curve on the sloping faces is not always drawn down to the bottom.

E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.

The "cross" plug puzzle, p. 17. Square, circle and cross.

The "wedge" plug puzzle, p. 18. Square, circle and triangle.

Perelman. FMP. c1935? One plug for three holes; Further "plug" puzzles, pp. 339-340 & 346. 6 simple versions; 3 harder versions: square, triangle, circle; circle, square, cross; triangle, square, tee. The three harder versions are also in FFF, 1957: probs. 69-71, pp. 112 & 118-119; 1979: probs. 73-75, pp. 137 & 144 = MCBF: probs. 73-75, pp. 134-135 & 142-143.

Anonymous [Antilog]. An elevation puzzle. Eureka 19 (Mar 1957) 11 & 19. Front and top views are a square with a square inside it. What is the side view? Gives two solutions.

Anonymous. An elevation puzzle. Eureka 21 (Oct 1958) 7 & 29. Front is the lower half of a circle. Plan (= top view) is a circle. What is the side view? Solution is a V shape, but it ought to be the other way up! Nowadays, one can buy potato crisps (= potato chips) in this shape.

Joseph S. Madachy. 3-D in 2-D. RMM 2 (Apr 1961) 51-53 & 3 (Jun 1961) 47. Discusses 3 view and 3 silhouette problems.

3 circular silhouettes, but not a sphere.

Square, circle, triangle.

Ernest R. Ranucci. Non-unique orthographic projections. RMM 14 (Jan-Feb 1964) 50. 3 views such that there are 10 different objects with these views.

Ripley's Puzzles and Games. 1966. Pp. 18-19, item 1. Same problem as Antilog, 1957. Gives one solution.

Cedric A. B. Smith. Simple projections. MG 62 (No. 419) (Mar 1978) 19-25. This is about how different projections affect one's recognition of what an object is. He starts with an example with two views and the isometric projection which is very difficult to interpret. He gives three other views, each of which is easily interpreted.

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 114, with Solution at the back of the book. Front view is a rectangle with an interior rectangle. Side view is a rectangle with a rectangular notch on front side. Solution is a short cylinder with a straight notch in it. This is a fairly classic problem for engineers but I haven't seen it in print elsewhere.

Marek Penszko. Polish your wits -- 3: Loop the loop. Games 11:2 (Feb/Mar 1987) 28 & 58. Draw lines on a glass cube to produce three given projections. Problem asks for all three projections to be the same.

6.W. BURR PUZZLES

When assembled, a burr looks like three sticks crossing orthogonally, forming a 'star' with six points at the vertices of an octahedron. Slocum says Wyatt [Puzzles in Wood, 1928, op. cit. in 5.H.1] is the first to use the word 'burr'. Collins, Book of Puzzles, 1927, p. 135, calls them "Cluster, Parisian or Gordian Knot Puzzles" and states: "it is believed that they were first made in Paris, if, indeed, they were not invented there." Since about 1990, there has been considerable development in new types of burr which use plates or boards rather than sticks, or whose central volume is subdivided more (cf in 6.W.1).

See S&B, pp. 62-85.

See also 6.BJ.

6.W.1. THREE PIECE BURR

Most of these have three pieces which are rectangular in cross-section (1 x 3 x 5) with slots of the same size and some of the pieces have notches from the slot to the outside. When one piece is pushed, it slides, revealing its notch. When placed properly, this allows a second piece to slide off and out.

In the 1990s, a more elaborate type of three piece burr appeared. These have three 3 x 3 x 5 pieces which intersect in a central 3 x 3 x 3 region. Within this region, some of the unit cubes are not present, which allows sliding of the pieces. Some versions of the puzzle permit twisting of pieces though this usually requires a bit of rounding of edges and the actual examples tend to break, so these are not as acceptable.

Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three piece burr, but perhaps is in 6.W.3 -- ?? It is followed by "Maltese Cross 6 pieces".

Edward Hordern's collection has examples in ivory from 1850-1900.

Hoffmann. 1893. Chap. III, no. 35: The cross-keys or three-piece puzzle, pp. 106 & 139 = Hoffmann-Hordern, pp. 104-105, with photo. One piece has an extra small notch which does not appear in other versions where the dimensions are better chosen. I have recently acquired an example which appears identical to the illustrations but does not have the extra notch - this came from a Jaques puzzle box, c1900, and Dalgety has several examples of such boxes with the solution, where the puzzle is named The Cross Keys Puzzle (cf discussion at the beginning of Section 11). The photo on p. 105 is an assembled version, with verbal instructions, by Jaques & Son, 1880-1895 (but Jaques was producing them up to at least c1910). Hordern Collection, p. 67, shows Le Noeud Mystérieux, 1880-1905, with a pictorial solution and this does not have the extra notch.

Benson. 1904. The cross keys puzzle, pp. 205-206.

Pearson. 1907. Part III, no. 56: The cross-keys, pp. 56 & 127-128.

Anon. A puzzle in wood. Hobbies 31 (No. 795) (7 Jan 1911) 345. Three piece burr with small extra notch as in Hoffmann.

Anon. Woodwork Joints. Evans, London, (1918), 2nd ed., 1919. [I have also seen a 4th ed., 1925, which is identical to the 2nd ed., except for advertising pages at the end.] A mortising puzzle, pp. 197-199.

Collins. Book of Puzzles. 1927. Pp. 136-137: The cross-keys puzzle.

E. M. Wyatt. Three piece cross. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 24-25.

Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared in 1908, but versions continued until the 1950s. This looks like 1930s?? 3-Piece Mortise with thin pieces.

A. S. Filipiak. Burr puzzle. Mathematical Puzzles, 1942, op. cit. in 5.H.1, p. 101.

Dic Sonneveld seems to be the first to begin designing three piece burrs of the more elaborate style, perhaps about 1985. Trevor Wood has made several examples for sale.

Bill Cutler. Email announcement to NOBNET on 27 Jan 1999. He has begun analysing the newer style of three piece burr, excluding twist moves. His first stage has examined cases where the centre cube of the central region is occupied and the piece this central cube belongs to has no symmetry. He finds 202 x 109 assemblies (I'm not sure if this is an exact figure) and there are 33 level-8 examples (i.e. where it takes 8 moves to remove the first piece); 6674 level-7 examples; 73362 level-6 examples. He thinks this is about 70% of the total and it is already about six times the number of cases considered for the six piece burr (see 6.W.2).

Bill Cutler. Christmas letter of 4 Dec 1999. Says he has completed the above analysis and found 25 x 1010 possibilities, which took 225 days on a workstation. The most elaborate examples require 8 moves to get a piece out and there are 80 of these. He used one for his IPP19 puzzle. He has a website with many of his results on burrs, etc.: .

6.W.2. SIX PIECE BURR = CHINESE CROSS

The usual form of these has six sticks, 2 x 2 x 6 (or 8), which have various notches in them. In the 1990s, new forms were introduced, using plates or boards. One version makes an open frame shape, something like a 3 x 3 x 3 chessboard. In the other, 1 x 4 x 6 boards are paired side by side and the result looks like a classic six-piece burr with the end rectangle divided lengthwise rather than crosswise. See also 6.W.7.

Jurgis Baltrušaitis. Anamorphoses ou magie artificielle des effets merveilleux. Olivier Perrin Éditeur, Paris, 1969. On pp. 110-116 & 184 is a discussion of a 1698 engraving "L'Académie des Sciences et des Beaux Arts" by Sébastien Leclerc (or Le Clerc). In the right foreground is an object looking like a six piece burr. James Dalgety discusses this in his Latest news on oldest puzzles; Lecture to Second Meeting on the History of Recreational Mathematics, 1 Jun 1996. This image also exists in a large painted version (950 x 480 mm) which is more precise and more legible in many details, so it is supposed that the engraving was done in conjunction with the painting. Though it was normal for a notable painting to be turned into an engraving, the opposite sometimes happened and Leclerc was a famous engraver. The painter is unknown. The divisions between the pairs of pieces of the 'burr' are pretty clear in the engraving, but two of them are not visible in the painting. The 'burr' is also not quite correctly drawn, but all in all, it seems pretty convincing. James Dalgety was the first to discover this picture and he has a copy of the engraving, but has not been able to locate the painting, though it was in the Bernard Monnier Collection exhibited at the Musée des Arts Decoratifs in Paris in 1975/76.

Camille Frémontier. Sébastien Leclerc and the British Encyclopeaedists. Sphæra [Newsletter of the Museum of the History of Science, Oxford] 6 (Aut 1997) 6-7. Discusses the Leclerc engraving which was used as the frontispiece to several encyclopedias, the earliest being Chambers Cyclopaedia of 1728.

Minguet. 1733. Pp. 103-105 (1755: 51-52; 1822: 122-124; 1864: 103-104). Pieces diagrammed. One plain key piece.

Catel. Kunst-Cabinet. 1790. Die kleine Teufelsklaue, p. 10 & fig. 16 on plate I. Figure shows it assembled and fails to draw one of the divisions between pieces. Description says it is 6 pieces, 2 inches long, from plum wood and costs 3 groschen (worth about an English penny of the time). (See also pp. 9-10, fig. 20 on plate I for Die grosse Teufelsklaue -- the 'squirrelcage'.)

Bestelmeier. 1801. Item 147: Die kleine Teufelsklaue. (Note -- there is another item 147 on the next plate.) Only shows it assembled. Brief text may be copying part of Catel. See also the picture for item 1099 which looks like a six-piece burr included in a set of puzzles. (See also Item 142: Die grosse Teufelsklaue.)

Edward Hordern's collection has examples, called The Oak of Old England, from c1840.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these might be here or in 6.W.4 or 6.W.5 -- ??

Magician's Own Book. 1857. Prob. 1: The Chinese cross, pp. 266-267 & 291. One plain key piece. Not the same as in Minguét.

Landells. Boy's Own Toy-Maker. 1858. Pp. 137-139. Identical to Magician's Own Book.

Book of 500 Puzzles. 1859. 1: The Chinese cross, pp. 80-81 & 105. Identical to Magician's Own Book.

A. F. Bogesen (1792-1876). In the Danish Technical Museum, Helsingør (= Elsinore) are a number of wooden puzzles made by him, including a 6 piece burr, a 12 piece burr, an Imperial Scale? and a complex (trick??) joint.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 23: The Chinese Cross, pp. 399 & 439. Identical to Magician's Own Book, except one diagram in the solution omits two labels.

Boy's Own Conjuring Book. 1860. Prob. 1: The Chinese cross, pp. 228 & 254. Identical to Magician's Own Book.

Hoffmann. 1893. Chap. III, no. 36: The nut (or six-piece) puzzle, pp. 106 & 139-140 = Hoffmann-Hordern, pp. 104-106. Different pieces than in Minguét and Magician's Own Book.

Dudeney. Prob. 473 -- Chinese cross. Weekly Dispatch (23 Nov & 7 Dec 1902), both p. 13. "There is considerable variety in the manner of cutting out the pieces, and though the puzzle has been given in some of the old books, I have purposely presented it in a form that has not, I believe, been published."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Chinese Cross," a puzzle of undoubted Oriental origin that was formerly brought from China by travellers as a curiosity, but for a long time has had a steady sale in this country."

Wehman. New Book of 200 Puzzles. 1908. The Chinese cross, pp. 40-41. = Magician's Own Book.

Dudeney. The world's best puzzles. 1908. Op. cit. in 2. P. 779 shows a '"Chinese Cross" which ... is of great antiquity.'

Oscar W. Brown. US Patent 1,225,760 -- Puzzle. Applied: 27 Jun 1916; patented: 15 May 1917. 3pp + 1p diagrams. Coffin says this is the earliest US patent, with several others following soon after.

Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. Eastern joint puzzle, pp. 196-197: Two versions using different pieces. Six-piece joint puzzle, pp. 199-200. Another version.

Western Puzzle Works, 1926 Catalogue. No. 86: 6 piece Wood Block. Several other possible versions -- see 6.W.7.

E. M. Wyatt. Six-piece burr. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 27-28. Describes 17 versions from 13 types of piece.

A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in 5.H.1, pp. 79-87. 73 versions from 38 types of piece.

William H. [Bill] Cutler. The six-piece burr. JRM 10 (1977-78) 241-250. Complete, computer assisted, analysis, with help from T. H. O'Beirne and A. C. Cross. Pieces are considered as 'notchable' if they can be made by a sequence of notches, which are produced by two saw cuts and then chiselling out the space between them. Otherwise viewed, notches are what could be produced by a wide cutter or router. There are 25 of these which can occur in solutions. (In 1994, he states that there are a total of 59 notchable pieces and diagrams all of them.) One can also have more general pieces with 'right-angle notches' which would require four chisel cuts -- e.g. to cut a single 1 x 1 x 1 piece out of a 2 x 2 x 8 rod. Alternatively, one can glue cubes into notches. There are 369 which can occur in solutions. (In 1994, he states that there are 837 pieces which produce 2225 different oriented pieces, and he lists them all.) He only considers solid solutions -- i.e. ones where there are no internal holes. He finds and lists the 314 'notchable' solutions. There are 119,979 general solutions.

C. Arthur Cross. The Chinese Cross. Pentangle, Over Wallop, Hants., UK, 1979. Brief description of the solutions in the general case, as found by Cutler and Cross.

S&B, p. 83, describes holey burrs.

W. H. [Bill] Cutler. Christmas letter, 1987. Sketches results of his (and other's) search for holey burrs with notchable pieces.

Bill Cutler. Holey 6-Piece Burr! Published by the author, Palatine, Illinois. (1986); with addendum, 1988, 48pp. He is now permitting internal holes. Describes holey burrs with notchable pieces, particularly those with multiple moves to release the first piece.

Bill Cutler. A Computer Analysis of All 6-Piece Burrs. Published by the author, ibid., 1994. 86pp. Sketches complete history of the project. (I have included a few details in the description of his 1977/78 article, above.) In 1987, he computed all the notchable holey solutions, using about 2 months of PC AT time, finding 13,354,991 assemblies giving 7.4 million solutions. Two of these were level 10 -- i.e. they require 10 moves to remove the first piece (or pieces), but the highest level occurring for a unique solution was 5. After that he started on the general holey burrs and estimated it would take 400 years of PC AT time -- running at 8 MHz. After some development, the actual time used was about 62.5 PC AT years, but a lot of this was done on by Harry L. Nelson during idle time on the Crays at Lawrence Livermore Laboratories, and faster PCs became available, so the whole project only took about 2½ years, being completed in Aug 1990 and finding 35,657,131,235 assemblies. He hasn't checked if all assemblies come apart fully, but he estimates there are 5.75 billion solutions. He estimates the project used 45 times the computing power used in the proof of the Four Color Theorem and that the project would only take two weeks on the eight RS6000 workstations he now supervises. Some 70,000 high-level solutions were specifically saved and can be obtained on disc from him. The highest level found was 12 and the highest level for a unique solution was 10. See 6.W.1 for a continuation of this work. He has a website with many of his results on burrs, etc.: .

Bill Cutler & Frans de Vreugd. Information leaflet accompanying their separate IPP22 puzzles, 2002. In 2001, they did an analysis of six-board burrs, of the type where the boards are paired side by side. There are 4096 possible such boards, but only 219 usable boards occur. They looked at all combinations of six of these and found 14,563,061,989 assemblies. Of these, the highest level found was 13.

6.W.3. THREE PIECE BURR WITH IDENTICAL PIECES

See S&B, p. 66.

Crambrook. 1843. P. 5, no. 4: Puzzling Cross 3 pieces. This seems likely to be a three piece burr, but perhaps is in 6.W.1 -- ?? It is followed by "Maltese Cross 6 pieces".

Wilhelm Segerblom. Trick wood joining. SA (1 Apr 1899) 196.

6.W.4. DIAGONAL SIX PIECE BURR = TRICK STAR

This version often looks like a stellated rhombic dodecahedron. It has two basic forms, one with a key piece; the other with all pieces identical, which assembles as two groups of three.

See S&B, p. 78.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2 or 6.W.5 -- ??

The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 19. Star Puzzle. The picture does not show which form it is. Slocum's Compendium also shows this.

Samuel P. Chandler. US Patent 393,816 -- Puzzle. Applied: 9 Mar 1888; patented: 23 Apr 1888. 1p + 1p diagrams. Coffin says this is the earliest version, but it is more complex than usual, with 12 pieces, and has a key piece.

John S. Pinnell. US Patent 774,197 -- Puzzle. Applied: 9 Oct 1902; patented: 8 Nov 1904. 2pp + 2pp diagrams. Coffin notes that this extends the idea to 102 pieces!

William E. Hoy. US Patent 766,444 -- Puzzle-Ball. Applied: 16 Oct 1902; patented: 2 Aug 1904. 2pp + 2pp diagrams. Spherical version with a key piece.

George R. Ford. US Patent 779,121 -- Puzzle. Applied: 16 May 1904; patented: 3 Jan 1905. 1p + 1p diagrams. With square rods, all identical. He shows assembly by inserting a last piece rather than joining two groups of three.

Anon. Simple wood puzzle. Hobbies 31 (No. 786) (5 Nov 1910) 127. With key piece.

E. M. Wyatt. Woodwork puzzles. Industrial Arts Magazine 12 (1923) 326-327. Version with a key piece and square rods.

Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139.

Iffland Frères (Lausanne). Swiss Patent 245,402 -- Zusammensetzspiel. Received: 19 Nov 1945; granted: 15 Nov 1946; published: 1 Jul 1947. 2pp + 1p diagrams. Stellated rhombic dodecahedral version with a key piece. (Coffin says this is the first to use this shape, although Slocum has a version c1875.)

6.W.5. SIX PIECE BURR WITH IDENTICAL PIECES

One form has six identical pieces and all move outward or inward together. Another form with flat notched pieces has one piece with an extra notch or an extended notch which allows it to fit in last, either by sliding or twisting, but this is not initially obvious. This form is sometimes made with equal pieces so that it can only be assembled by force, perhaps after steaming, and it then makes an unopenable money box. This might be considered under 11.M.

Edward Hordern's collection has a version with one piece a little smaller than the rest from c1800.

Crambrook. 1843. P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2 or 6.W.4 -- ??

C. Baudenbecher catalogue, c1850s. Op. cit. in 6.W.7. This has an example of the six equal flat pieces making an unopenable(?) money box.

F. Chasemore. Some mechanical puzzles. In: Hutchison; op. cit. in 5.A; 1891, chap. 70, part 1, pp. 571-572. Item 5: The puzzle box, p. 572. Six U pieces make a uniformly expanding cubical box.

Hoffmann. 1893. Chap. III, no.33: The bonbon nut puzzle, pp. 104 & 138 = Hoffmann-Hordern, pp. 102-103, with photo. One piece has an extra notch to simplify the assembly. Photo on p. 103 shows an example, almost certainly by Jaques & Son, 1860-1895.

Burnett Fallow. How to make a puzzle money-box. The Boy's Own Paper 15 (No. 755) (1 Jul 1893) 638. Equal flat notched pieces forced together to make an unopenable box.

Burnett Fallow. How to make a puzzle picture-frame. The Boy's Own Paper 16 (No. 815) (25 Aug 1894) 749. Each corner has the same basic forced construction as used in the puzzle money-box.

Benson. 1904. The bonbon nut puzzle, p. 204.

Bartl. c1920. Several versions on p. 306.

Western Puzzle Works, 1926 Catalogue. Last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are of the present type.

Collins. Book of Puzzles. 1927. The bonbon or nut puzzle, pp. 137-139. As in Hoffmann.

Iona & Robert Opie and Brian Alderson. Treasures of Childhood. Pavilion (Michael Joseph), London, 1989. P. 158 shows a "cluster puzzle which Professor Hoffman [sic] names the 'Nut (or Six-piece) Puzzle', but which is usually called 'The Maltese Puzzle'."

6.W.6. ALTEKRUSE PUZZLE

William Altekruse. US Patent 430,502 -- Block-Puzzle. Applied: 3 Apr 1890; patented: 17 Jun 1890. 1p + 1p diagrams. Described in S&B, p. 72. The standard version has 12 pieces, but variations discovered by Coffin have 14, 36 & 38 pieces.

Western Puzzle Works, 1926 Catalogue. No. 112: 12 piece Wood Block. Possibly Altekruse.

6.W.7. OTHER BURRS

See also 6.BJ for other 3D dissections. I have avoided repeating items, so 6.BJ should also be consulted if you are reading this section.

Catel. Kunst-Cabinet. 1790. Die grosse Teufelsklaue, pp. 9-10 & fig. 20 on plate I. 24 piece 'squirrel cage'. Cost 16 groschen.

Bestelmeier. 1801. Item 142: Die grosse Teufelsklaue. The 'squirrelcage', identical to Catel, with same drawing, but reversed. Text may be copying some of Catel.

C. Baudenbecher, toy manufacturer in Nuremberg. Sample book or catalogue from c1850s. Baudenbecher was taken over by J. W. Spear & Sons in 1919 and the catalogue is now in the Spear's Game Archive, Ware, Hertfordshire. It comprises folio and double folio sheets with finely painted illustrations of the firm's products. One whole folio page shows about 20 types of wooden interlocking puzzles, including most of the types mentioned elsewhere in this section and in 6.W.5 and 6.BJ. Until I get a picture, I can't be more specific.

The Youth's Companion. 1875. [Mail order catalogue.] Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 19. Shows a 'woodchuck' type puzzle, called White Wood Block Puzzle, from The Youth's Companion, 1875. I can't see how many pieces it has: 12 or 18?? Slocum's Compendium also shows this.

Slocum. Compendium. Shows: "Mystery", Magic "Champion Puzzle" and "Puzzle of Puzzles" from Bland's Catalogue, c1890.

The first looks like a 6 piece burr with circular segments added to make it look like a ball. So it may be a 6 piece burr in disguise. See also Hoffmann, Chap. III, no. 38, pp. 107-108 & 141-142 = Hoffmann-Hordern, pp. 106-108 = Benson, p. 205.

The second is a six piece puzzle, but the pieces are flattish and it may be of the type described in 6.W.5.

The third is complex, with perhaps 18 pieces.

Bartl. c1920. Several versions on pp. 306-307, including some that are in 6.W.5 and some 'Chinese block puzzles'.

Western Puzzle Works, 1926 Catalogue. Shows a number of burrs and similar puzzles.

No. 86: 6 piece Wood Block.

No. 112: 12 piece Wood Block. Possibly Altekruse.

No. 212: 11 piece Wood Block

The last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are burrs.

Collins. Book of Puzzles. 1927. Other cluster puzzles, pp. 139-142. Describes and illustrates: The cluster; The cluster of clusters; The gun cluster; The point cluster; The flat cluster; The cluster (or secret) table; The barrel; The Ball; The football. All of these have a key piece.

Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escher-puzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -- ??) (1992) 12-13. Two Penrose tribars made into an impossible 5-piece burr.

6.X. ROTATING RINGS OF POLYHEDRA

Generally, these have edge to edge joints. 'Jacob's ladder' joints are used by Engel -- see 11.L for other forms of this joint.

I am told these may appear in Fedorov (??NYS).

Max Brückner. Vielecke und Vielfläche. Teubner, Leipzig, 1900. Section 162, pp. 215-216 and Tafel VIII, fig. 4. Describes rings of 2n tetrahedra joined edge to edge, called stephanoids of the second order. The figure shows the case n = 5.

Paul Schatz. UK Patent 406,680 -- Improvements in or relating to Boxes or Containers. Convention date (Germany): 10 Dec 1931; application date (in UK): 19 Jul 1932; accepted: 19 Feb 1934. 6pp + 6pp diagrams. Six and four piece rings of prisms which fold into a box.

Paul Schatz. UK Patent 411,125 -- Improvements in Linkwork comprising Jointed Rods or the like. Convention Date (Germany): 31 Aug 1931; application Date (in UK): 31 Aug 1932; accepted: 31 May 1934. 3p + 6pp diagrams. Rotating rings of six tetrahedra and linkwork versions of the same idea, similar to Flowerday's Hexyflex.

Ralph M. Stalker. US Patent 1,997,022 -- Advertising Medium or Toy. Applied: 27 Apr 1933; patented: 9 Apr 1935. 3pp + 2pp diagrams. "... a plurality of tetrahedron members or bodies flexibly connected together." Shows six tetrahedra in a ring and an unfolded pattern for such objects. Shows a linear form with 14 tetrahedra of decreasing sizes.

Sidney Melmore. A single-sided doubly collapsible tessellation. MG 31 (No. 294) (1947) 106. Forms a Möbius strip of three triangles and three rhombi, which is basically a flexagon (cf 6.D). He sees it has two distinct forms, but doesn't see the flexing property!! He describes how to extend these hexagons into a tessellation which has some resemblance to other items in this section.

Alexander M. Shemet. US Patent 2,688,820 -- Changeable Display Amusement Device. Applied: 25 Jul 1950; patented: 14 Sep 1954. 2pp + 2pp diagrams. Basically a rotating ring of six tetrahedra, but says 'at least six'. Gives an unfolded version or net for making it and a mechanism for flexing it continually. Cites Stalker.

Wallace G. Walker invented his "IsoAxis" ® in 1958 while a student at Cranbrook Academy of Art, Michigan. This is approximately a ring of ten tetrahedra. He obtained a US Patent for it in 1967 -- see below. In 1973(?) he sent an example to Doris Schattschneider who soon realised that the basic idea was a ring of tetrahedra and that Escher tessellations could be adapted to it. They developed the idea into "M. C. Escher Kaleidocycles", published by Ballantine in 1977 and reprinted several times since.

Douglas Engel. Flexahedrons. RMM 11 (Oct 1962) 3-5. These have 'Jacob's ladder' hinges, not edge-to-edge hinges. He says he invented these in Fall, 1961. He formed rings of 4, 6, 7, 8 tetrahedra and used a diagonal joining to make rings of 4 and 6 cubes.

Wallace G. Walker. US Patent 3,302,321 -- Foldable Structure. Filed: 16 Aug 1963; issued: 7 Feb 1967. 2pp + 6pp diagrams.

Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Solid Flexagons, pp. 81-84. Based on Engel, but only gives the ring of 6 tetrahedra.

D. Engel. Flexing rings of regular tetrahedra. Pentagon 26 (Spring 1967) 106-108. ??NYS -- cited in Schaaf II 89 -- write Engel.

Paul Bethell. More Mathematical Puzzles. Encyclopædia Britannica International, London, 1967. The magic ring, pp. 12-13. Gives diagram for a ten-tetrahedra ring, all tetrahedra being regular.

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. Presents versions of the flexing cubes and the 'Shinsei Mystery'. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

Jan Slothouber. Flexicubes -- reversible cubic shapes. JRM 6 (1973) 39-46. As above.

Frederick George Flowerday. US Patent 3,916,559 -- Vortex Linkages. Filed: 12 Aug 1974 (23 Aug 1973 in UK); issued: 4 Nov 1975. Abstract + 2pp + 3pp diagrams. Mostly shows his Hexyflex, essentially a six piece ring of tetrahedra, but with just four edges of each tetrahedron present. He also shows his Octyflex which has eight pieces. Text refers to any even number ( 6.

Naoki Yoshimoto. Two stars in a cube (= Shinsei Mystery). Described in Japanese in: Itsuo Sakane; A Museum of Fun; Asahi Shimbun, Tokyo, 1977, pp. 208-210. Shown and pictured as Exhibit V-1 with date 1972 in: The Expanding Visual World -- A Museum of Fun; Exhibition Catalogue, Asahi Shimbun, Tokyo, 1979, pp. 102 & 170-171. (In Japanese). ??get translated??

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 63-66. Describes Walkers IsoAxis and rotating rings of six and eight tetrahedra.

6.Y. ROPE ROUND THE EARTH

The first few examples illustrate what must be the origin of the idea in more straightforward situations.

Lucca 1754. c1330. F. 8r, pp. 31-32. This mentions the fact that a circumference increases by 44/7 times the increase in the radius.

Muscarello. 1478.

Ff. 932-93v, p. 220. A circular garden has outer circumference 150 and the wall is 3½ thick. What is the inner circumference? Takes π as 22/7.

F. 95r, p. 222. The internal circumference of a tower is 20 and its wall is 3 thick. What is the outer circumference? Again takes π as 22/7.

Pacioli. Summa. 1494. Part II, f. 55r, prob. 33. Florence is 5 miles around the inside. The wall is 3½ braccia wide and the ditch is 14 braccia wide -- how far is it around the outside? Several other similar problems.

William Whiston. Edition of Euclid, 1702. Book 3, Prop. 37, Schol. (3.). ??NYS -- cited by "A Lover" and Jackson, below.

"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller, Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty five Surprising PARADOXES, in GORDON's Geography, so the following must have appeared in Gordon. Part I, no. 73, p. 56. "'Tis certainly Matter of Fact, that three certain Travellers went a Journey, in which, Tho' their Heads travelled full twelve Yards more than their Feet, yet they all return'd alive, with their Heads on."

Carlile. Collection. 1793. Prob. XXV, p. 17. Two men travel, one upright, the other standing on his head. Who "sails farthest"? Basically he compares the distance travelled by the head and the feet of the first man. He notes that this argument also applies to a horse working a mill by walking in a circle; the outside of the horse travels about six times the thickness of the horse further than the inside on each turn.

Jackson. Rational Amusement. 1821. Geographical Paradoxes, no. 54, pp. 46 & 115-116. "It is a matter of fact, that three certain travellers went on a journey, in which their heads travelled full twelve yards more than their feet; and yet, they all returned alive with their heads on." Solution says this is discussed in Whiston's Euclid, Book 3, Prop. 37, Schol. (3.). [This first appeared in 1702.]

K. S. Viwanatha Sastri. Reminiscences of my esteemed tutor. In: P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and Reminiscences; 2: Ramanujan -- An Inspiration; Muthialpet High School, Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 1, pp. 89-93. On p. 93, he relates that this was a favourite problem of his tutor, Srinivasan Ramanujan. Though not clearly dated, this seems likely to be c1908-1910, but may have been up to 1914. "Suppose we prepare a belt round the equator of the earth, the belt being 2π feet longer, and if we put the belt round the earth, how high will it stand? The belt will stand 1 foot high, a substantial height."

Dudeney. The paradox party. Strand Mag. 38 (No. 228) (Dec 1909) 673-674 (= AM, p. 139).

Anon. Prob. 58. Hobbies 30 (No. 773) (6 Aug 1910) 405 & (No. 776) (27 Aug 1910) 448. Double track circular railway, five miles long. Move all rails outward one foot. How much more material is needed? Solution notes the answer is independent of the length.

Ludwig Wittgenstein was fascinated by the problem and used to pose it to students. Most students felt that adding a yard to the rope would raise it from the earth by a negligible amount -- which it is, in relation to the size of the earth, but not in relation to the yard. See: John Lenihan; Science in Focus; Blackie, 1975, p. 39.

Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. Prob. 5, p. 87 & Answers p. 7. A yard is added to a band around the earth. Can you raise it 5 inches? Answer notes the size of the earth is immaterial.

Collins. Book of Puzzles. 1927. The globetrotter's puzzle, pp. 68-69. If you walk around the equator, how much farther does your head go?

Abraham. 1933. Prob. 33 -- A ring round the earth, pp. 12 & 24 (9 & 112).

Perelman. FMP. c1935?? Along the equator, pp. 342 & 349. Same as Collins.

Sullivan. Unusual. 1943.

Prob. 20: A global readjustment. Take a wire around the earth and insert an extra 40 ft into it -- how high up will it be?

Prob. 23: Getting ahead. If you walk around the earth, how much further does your head go than your feet?

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Things are seldom what they seem -- No. 42a, 43, 44, pp. 50-51. 42a and 43 ask how much the radius increases for a yard gain of circumference. No. 44 asks if we add a yard to a rope around the earth and then tauten it by pulling outward at one point, how far will that point be above the earth's surface?

Richard I Hess. Puzzles from Around the World. The author, 1997. (This is a collection of 117 puzzles which he published in Logigram, the newsletter of Logicon, in 1984-1994, drawn from many sources. With solutions.) Prob. 28. Consider a building 125 ft wide and a rubber band stretched around the earth. If the rubber band has to stretch an extra 10 cm to fit over the building, how tall is the building? He takes the earth's radius as 20,902,851 ft. He gets three trigonometric equations and uses iteration to obtain 85.763515... ft.

Erwin Brecher & Mike Gerrard. Challenging Science Puzzles. Sterling, 1997. [Reprinted by Goodwill Publishing House, New Delhi, India, nd [bought in early 2000]]. Pp. 38-39 & 77. The M25 is a large ring road around London. A man commutes from the south to the north and finds the distance is the same if goes by the east or the west, so he normally goes to the east in the morning and to the west in the evening. Recalling that the English drive on the left, he realised that his right wheels were on the outside in both journeys and he worried that they were wear out sooner. So he changed and drove both ways by the east. But he then worried whether the wear on the tires was the same since the evening trip was on the outer lanes of the Motorway.

6.Z. LANGLEY'S ADVENTITIOUS ANGLES

Let ABC be an isosceles triangle with ( B = ( C = 80o. Draw BD and CE, making angles 50o and 60o with the base. Then ( CED = 20o.

JRM 15 (1982-83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. ??NYS

Peterhouse and Sidney Entrance Scholarship Examination. Jan 1916. ??NYS.

E. M. Langley. Note 644: A Problem. MG 11 (No. 160) (Oct 1922) 173.

Thirteen solvers, including Langley. Solutions to Note 644. MG 11 (No. 164) (May 1923) 321-323.

Gerrit Bol. Beantwoording van prijsvraag No. 17. Nieuw Archief voor Wiskunde (2) 18 (1936) 14-66. ??NYS. Coxeter (CM 3 (1977) 40) and Rigby (below) describe this. The prize question was to completely determine the concurrent diagonals of regular polygons. The 18-gon is the key to Langley's problem. However Bol's work was not geometrical.

Birtwistle. Math. Puzzles & Perplexities. 1971. Find the angle, pp. 86-87. Short solution using law of sines and other simple trigonometric relations.

Colin Tripp. Adventitious angles. MG 59 (No. 408) (Jun 1975) 98-106. Studies when ( CED can be determined and all angles are an integral number of degrees. Computer search indicates that there are at most 53 cases.

CM 3 (1977) 12 gives 1939 & 1950 reappearances of the problem and a 1974 variation.

D. A. Q. [Douglas A. Quadling]. The adventitious angles problem: a progress report. MG 61 (No. 415) (Mar 1977) 55-58. Reports on a number of contributions resolving the cases which Tripp could not prove. All the work is complicated trigonometry -- no further cases have been demonstrated geometrically.

CM 4 (1978) 52-53 gives more references.

D. A. Q. [Douglas A. Quadling]. Last words on adventitious angles. MG 62 (No. 421) (Oct 1978) 174-183. Reviews the history, reports on geometric proofs for all cases and various generalizations.

J[ohn]. F. Rigby. Adventitious quadrangles: a geometrical approach. MG 62 (No. 421) (Oct 1978) 183-191. Gives geometrical proofs for almost all cases. Cites Bol and a long paper of his own to appear in Geom. Dedicata (??NYS). He drops the condition that ABC be isosceles. His adventitious quadrangles correspond to Bol's triple intersections of diagonals of a regular n-gon.

MS 27:3 (1994/5) 65 has two straightforward letters on the problem, which was mentioned in ibid. 27:1 (1994/5) 7. One letter cites 1938 and 1955 appearances. P. 66 gives another solution of the problem. See next item.

Douglas Quadling. Letter: Langley's adventitious angles. MS 27:3 (1994/5) 65-66. He was editor of MG when Tripp's article appeared. He gives some history of the problem and some life of Langley (d. 1933). Edward Langley was a teacher at Bedford Modern School and the founding editor of the MG in 1894-1895. E. T. Bell was a student of Langley's and contributed an obituary in the MG (Oct 1933) saying that Langley was the finest expositor he ever heard -- ??NYS. Langley also had botanical interests and a blackberry variety is named for him.

6.AA. NETS OF POLYHEDRA

Albrecht Dürer. Underweysung der messung mit dem zirckel uň [NOTE: ň denotes an n with an overbar.] richtscheyt, in Linien ebnen unnd gantzen corporen. Nürnberg, 1525, revised 1538. Facsimile of the 1525 edition by Verlag Dr. Alfons Uhl, Nördlingen, 1983. German facsimile with English translation of the 1525 edition, with notes about the 1538 edition: The Painter's Manual; trans. by Walter L. Strauss; Abaris Books, NY, 1977. Figures 29-43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show nets and pictures of the regular polyhedra, an approximate sphere (16 sectors by 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi-cubo-octahedron, snub cube, great rhombi-cubo-octahedron, truncated cubo-octahedron (having a pattern of four triangles replacing each triangle of the cubo-octahedron -- not an Archimedean solid) and an elongated hexagonal bipyramid (not even regular faced). (See 6.AT.3 for more details.) (Panofsky's biography of Dürer asserts that Dürer invented the concept of a net -- this is excerpted in The World of Mathematics I 618-619.) In the revised version of 1538, figure 43 is replaced by the icosi-dodecahedron and great rhombi-cubo-octahedron (figures 43 & 43a, pp. 414-419 of The Painter's Manual) to make 9 of the Archimedean polyhedra.

Albrecht Dürer. Elementorum Geometricorum (?) -- the copy of this that I saw at the Turner Collection, Keele, has the title page missing, but Elementorum Geometricorum is the heading of the first text page and appears to be the book's title. This is a Latin translation of Unterweysung der Messung .... Christianus Wechelus, Paris, 1532. This has the same figures as the 1525 edition, but also has page numbers. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

Cardan. De Rerum Varietate. 1557, ??NYS = Opera Omnia, vol. III, pp. 246-247. Liber XIII. Corpora, qua regularia diei solent, quomodo in plano formentur. Shows nets of the regular solids, except the two halves of the dodecahedron have been separated to fit into one column of the text.

Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice, (1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.] Pp. 45-104 give nets and drawings of the regular polyhedra and 11 of the 13 Archimedean polyhedra -- he omits the two snub solids.

E. Welper. Elementa geometrica, in usum geometriae studiosorum ex variis Authoribus collecta. J. Reppius, Strassburg, 1620. ??NYS -- cited, with an illustration of the nets of the octahedron, icosahedron and dodecahedron, in Lange & Springer Katalog 163 -- Mathematik & Informatik, Oct 1994, item 1350 & illustration on back cover, but the entry gives Trassburg.

Athanasius Kircher. Ars Magna, Lucis et Umbrae. Rome, 1646. ??NX. Has net of a rhombi-cuboctahedron.

Pike. Arithmetic. 1788. Pp. 458-459. "As the figures of some of these bodies would give but a confused idea of them, I have omitted them; but the following figures, cut out in pasteboard, and the lines cut half through, will fold up into the several bodies." Gives the regular polyhedra.

Dudeney. MP. 1926. Prob. 146: The cardboard box, pp. 58 & 149 (= 536, prob. 316, pp. 109 & 310). All 11 nets of a cube.

Perelman. FMP. c1935? To develop a cube, pp. 179 & 182-183. Asserts there are 10 nets and draws them, but two "can be turned upside down and this will add two more ...." One shape is missing. Of the two marked as reversible, one is symmetric, hence equal to its reverse, but the other isn't.

C. Hope. The nets of the regular star-faced and star-pointed polyhedra. MG 35 (1951) 8-11. Rather technical.

H. Steinhaus. One Hundred Problems in Elementary Mathematics. (As: Sto Zadań, PWN -- Polish Scientific Publishers, Warsaw, 1958.) Pergamon Press, 1963. With a Foreword by M. Gardner; Basic Books, NY, 1964. Problem 34: Diagrams of the cube, pp. 20 & 95-96. (Gives all 11 nets.) Gardner (pp. 5-6) refers to Dudeney and suggests the four dimensional version of the problem should be easy.

M. Gardner. SA (Nov 1966) c= Carnival, pp. 41-54. Discusses the nets of the cube and the Answers show all 11 of them. He asks what shapes these 11 hexominoes will form -- they cannot form any rectangles. He poses the four dimensional problem; the Addendum says he got several answers, no two agreeing.

Charles J. Cooke. Nets of the regular polyhedra. MTg 40 (Aut 1967) 48-52. Erroneously finds 13 nets of the octahedron.

Joyce E. Harris. Nets of the regular polyhedra. MTg 41 (Winter 1967) 29. Corrects Cooke's number to 11.

A. Sanders & D. V. Smith. Nets of the octahedron and the cube. MTg 42 (Spring 1968) 60-63. Finds 11 nets for the octahedron and shows a duality with the cube.

Peter Turney. Unfolding the tesseract. JRM 17 (1984-85) 1-16. Finds 261 nets of the 4-cube. (I don't believe this has ever been confirmed.)

Peter Light & David Singmaster. The nets of the regular polyhedra. Presented at New York Acad. Sci. Graph Theory Day X, 213 Nov 1985. In Notes from New York Graph Theory Day X, 23 Nov 1985; ed. by J. W. Kennedy & L. V. Quintas; New York Acad. Sci., 1986, p. 26. Based on Light's BSc project in 1984-1984 under my supervision. Shows there are 43,380 nets for the dodecahedron and icosahedron. I may organize this into a paper, but several others have since verified the result.

6.AB. SELF-RISING POLYHEDRA

H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938. (= Kalejdoskop Matematyczny. Książnica-Atlas, Lwów and Warsaw, 1938, ??NX.) Pp. 74-75 describes the dodecahedron and says to see the model in the pocket at the end, but makes no special observation of the self-rising property. Described in detail with photographs in OUP, NY, eds: 1950: pp. 161-164; 1960: pp. 209-212; 1969 (1983): pp. 196-198.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, p. 29, section 66: Pop-up dodecahedron.

M. Kac. Hugo Steinhaus -- a reminiscence and a tribute. AMM 81 (1974) 572-581. Material is on pp. 580-581, with picture on p. 581.

A pop-up octahedron was used by Waddington's as an advertising insert in a trade journal at the London Toy Fair about 1981. Pop-up cubes have also been used.

6.AC. CONWAY'S LIFE

There is now a web page devoted to Life run by Bob Wainwright -- address is:

[sic!].

M. Gardner. Solitaire game of "Life". SA (Oct 1970). On cellular automata, self-reproduction, the Garden-of-Eden and the game of "Life". SA (Feb 1971). c= Wheels, chap. 20-22. In the Oct 1970 issue, Conway offered a $50 prize for a configuration which became infinitely large -- Bill Gosper found the glider gun a month later. At G4G2, 1996, Bob Wainwright showed a picture of Gosper's telegram to Gardner on 4 Nov 1970 giving the coordinates of the glider gun. I wasn't clear if Wainwright has this or Gardner still has it.

Robert T. Wainwright, ed. (12 Longue Vue Avenue, New Rochelle, NY, 10804, USA). Lifeline (a newsletter on Life), 11 issues, Mar 1971 -- Sep 1973. ??NYR.

John Barry. The game of Life: is it just a game? Sunday Times (London) (13 Jun 1971). ??NYS -- cited by Gardner.

Anon. The game of Life. Time (21 Jan 1974). ??NYS -- cited by Gardner.

Carter Bays. The Game of Three-dimensional Life. Dept. of Computer Science, Univ. of South Carolina, Columbia, South Carolina, 29208, USA, 1986. 48pp.

A. K. Dewdney. The game Life acquires some successors in three dimensions. SA 256:2 (Feb 1987) 8-13. Describes Bays' work.

Bays has started a quarterly 3-D Life Newsletter, but I have only seen one (or two?) issues. ??get??

Alan Parr. It's Life -- but not as we know it. MiS 21:3 (May 1992) 12-15. Life on a hexagonal lattice.

6.AD. ISOPERIMETRIC PROBLEMS

There is quite a bit of classical history which I have not yet entered. Magician's Own Book notes there is a connection between the Dido version of the problem and Cutting a card so one can pass through it, Section 6.BA. There are several relatively modern surveys of the subject from a mathematical viewpoint -- I will cite a few of them.

Virgil. Aeneid. -19. Book 1, lines 360-370. (p. 38 of the Penguin edition, translated by W. F. Jackson Knight, 1956.) Dido came to a spot in Tunisia and the local chiefs promised her as much land as she could enclose in the hide of a bull. She cut it into a long strip and used it to cut off a peninsula and founded Carthage. This story was later adapted to other city foundations. John Timbs; Curiosities of History; With New Lights; David Bogue, London, 1857, devotes a section to Artifice of the thong in founding cities, pp. 49-50, relating that in 1100, Hengist, the first Saxon King of Kent, similarly purchased a site called Castle of the Thong and gives references to Indian, Persian and American versions of the story as well as several other English versions.

Pappus. c290. Synagoge [Collection]. Book V, Preface, para. 1-3, on the sagacity of bees. Greek and English in SIHGM II 588-593. A different, abridged, English version is in HGM II 389-390.

The Friday Night Book (A Jewish Miscellany). Soncino Press, London, 1933. Mathematical Problems in the Talmud: Arithmetical Problems, no. 2, pp. 135-136. A Roman Emperor demanded the Jews pay him a tax of as much wheat as would cover a space 40 x 40 cubits. Rabbi Huna suggested that they request to pay in two instalments of 20 x 20 and the Emperor granted this. [The Talmud was compiled in the period -300 to 500. This source says he is one of the few mathematicians mentioned in the Talmud, but gives no dates and he is not mentioned in the EB. From the text, the problem would seem to be sometime in the 1-5 C.]

The 5C Saxon mercenary, Hengist or Hengest, is said to have requested from Vortigern: "as much land as can be encircled by a thong". He "then took the hide of a bull and cut it into a single leather thong. With this thong he marked out a certain precipitous site, which he had chosen with the greatest possible cunning." This is reported by Geoffrey of Monmouth in the 12C and this is quoted by the editor in: The Exeter Book Riddles; 8-10C (the book was owned by Leofric, first Bishop of Exeter, who mentioned it in his will of 1072); Translated and edited by Kevin Crossley-Holland; (As: The Exeter Riddle Book, Folio Society, 1978, Penguin, 1979); Revised ed., Penguin, 1993; pp. 101-102.

Lucca 1754. c1330. Ff. 8r-8v, pp. 31-33. Several problems, e.g. a city 1 by 24 has perimeter 50 while a city 8 by 8 has perimeter 32 but is 8/3 as large; stitching two sacks together gives a sack 4 times as big.

Calandri. Arimethrica. 1491. F. 97v. Joining sacks which hold 9 and 16 yields a sack which holds 49!!

Pacioli. Summa. 1494. Part II, ff. 55r-55v. Several problems, e.g. a cord of length 4 encloses 100 ducats worth, how much does a cord of length 10 enclose? Also stitching bags together.

Buteo. Logistica. 1559. Prob. 86, pp. 298-299. If 9 pieces of wood are bundled up by 5½ feet of cord, how much cord is needed to bundle up 4 pieces? 5 pieces?

Pitiscus. Trigonometria. Revised ed., 1600, p. 223. ??NYS -- described in: Nobuo Miura; The applications of trigonometry in Pitiscus: a preliminary essay; Historia Scientarum 30 (1986) 63-78. A square of side 4 and triangle of sides 5, 5, 3 have the same perimeter but different areas. Presumably he was warning people not to be cheated in this way.

J. Kepler. The Six-Cornered Snowflake, op. cit. in 6.AT.3. 1611. Pp. 6-11 (8-19). Discusses hexagons and rhombic interfaces, but only says "the hexagon is the roomiest" (p. 11 (18-19)).

van Etten. 1624. Prob. 90 (87). Pp. 136-138 (214-218). Compares fields 6 x 6 and 9 x 3. Compares 4 sacks of diameter 1 with 1 sack of diameter 4. Compares 2 water pipes of diameter 1 with 1 water pipe of diameter 2.

Ozanam. 1725.

Question 1, 1725: 327. Question 3, 1778: 328; 1803: 325; 1814: 276; 1840: 141. String twice as long contains four times as much asparagus.

Question 2, 1725: 328. If a cord of length 10 encloses 200, how much does a cord of length 8 enclose?

Question 3, 1725: 328. Sack 5 high by 4 across versus 4 sacks 5 high by 1 across. c=  Q. 2, 1778: 328; 1803: 324; 1814: 276; 1840: 140-141, which has sack 4 high by 6 around versus two sacks 4 high by 3 around.

Question 4, 1725: 328-329. How much water does a pipe of twice the diameter deliver?

Les Amusemens. 1749.

Prob. 211, p. 376. String twice as long contains four times as much asparagus.

Prob. 212, p. 377. Determine length of string which contains twice as much asparagus.

Prob. 223-226, pp. 386-389. Various problems involving changing shape with the same perimeter. Notes the area can be infinitely small.

Ozanam-Montucla. 1778.

Question 1, 1778: 327; 1803: 323-324; 1814: 275-276; 1840: 140. Square versus oblong field of the same circumference.

Prob. 35, 1778: 329-333; 1803: 326-330; 1814: 277-280; 1840: 141-143. Les alvéoles des abeilles (On the form in which bees construct their combs).

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 30, pp. 30 & 90. Square field versus oblong (rectangular?) field of the same perimeter.

No. 31, pp. 30 & 90-91. String twice as long contains four times as much asparagus.

Magician's Own Book (UK version). 1871. To cut a card for one to jump through, p. 124, says: "The adventurer of old, who, inducing the aborigines to give him as much land as a bull's hide would cover, and made it into one strip by which acres were enclosed, had probably played at this game in his youth." See 6.BA.

M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften. Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und Minima. Die isoperimetrische Aufgabe. Pp. 1118-1128. General survey, from Zenodorus (-1C) and Pappus onward.

6.AD.1. LARGEST PARCEL ONE CAN POST

New section. I have just added the problem of packing a fishing rod as the diagonal of a box. Are there older examples?

Richard A. Proctor. Greatest content with parcels' post. Knowledge 3 (3 Aug 1883) 76. Height + girth ( 6 ft. States that a cylinder is well known to be the best solution. Either for a cylinder or a box, the optimum has height = 2, girth = 4, with optimum volumes 2 and 8/π = 2.54... ft3.

R. F. Davis. Letter: Girth and the parcel post. Knowledge 3 (17 Aug 1883) 109-110, item 897. Independent discussion of the problem, noting that length ( 3½ ft is specified, though this doesn't affect the maximum volume problem.

H. F. Letter: Parcel post problem. Knowledge 3 (24 Aug 1883) 126, item 905. Suppose 'length' means "the maximum distance in a straight line between any two points on its surface". By this he means the diameter of the solid. Then the optimum shape is the intersection of a right circular cylinder with a sphere, the axis of the cylinder passing through the centre of the sphere, and this has the 'length' being the diameter of the sphere and the maximum volume is then 2⅓ ft3.

Algernon Bray. Letter: Greatest content of a parcel which can be sent by post. Knowledge 3 (7 Sep 1883) 159, item 923. Says the problem is easily solved without calculus. However, for the box, he says "it is plain that the bulk of half the parcel will be greatest when [its] dimensions are equal".

Pearson. 1907. Part II, no. 20: Parcel post limitations, pp. 118 & 195. Length ( 3½ ft; length + girth ( 6 ft. Solution is a cylinder.

M. Adams. Puzzle Book. 1939. Prob. B.86: Packing a parcel, pp. 79 & 107. Same as Pearson, but first asks for the largest box, then the largest parcel.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 18, pp. 27 & 89. Ship a rifle about 1½ yards long when the post office does not permit any dimension to be more than 1 yard.

T. J. Fletcher. Doing without calculus. MG 55 (No. 391) (Feb 1971) 4-17. Example 5, pp. 8-9. He says only that length + girth ( 6 ft. However, the optimal box has length 2, so the maximal length restriction is not critical.

I have looked at the current parcel post regulations and they say length ( 1.5m and length + girth ( 3m, for which the largest box is 1 x ½ x ½, with volume 1/4 m3. The largest cylinder has length 1 and radius 1/π with volume 1/π m3.

I have also considered the simple question of a person posting a fishing rod longer than the maximal length by putting it diagonally in a box. The longest rod occurs at a boundary maximum, at 3/2 x 3/4 x 0 or 3/2 x 0 x 3/4, so one can post a rod of length 3(5/4  =  1.677... m, which is about 12% longer than 1.5m. In this problem, the use of a cylinder actually does worse!

6.AE. 6" HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME

Hamnet Holditch. Geometrical theorem. Quarterly J. of Pure and Applied Math. 2 (1858) ??NYS, described by Broman. If a chord of a closed curve, of constant length a+b, be divided into two parts of lengths a, b respectively, the difference between the areas of the closed curve, and of the locus of the dividing point as the chord moves around the curve, will be πab. When the closed curve is a circle and a = b, then this is the two dimensional version given by Jones, below. A letter from Broman says he has found Holditch's theorem cited in 1888, 1906, 1975 and 1976.

Richard Guy (letter of 27 Feb 1985) recalls this problem from his schooldays, which would be late 1920s-early 1930s, and thought it should occur in calculus texts of that time, but could not find it in Lamb or Caunt.

Samuel I. Jones. Mathematical Nuts. 1932. P. 86. ??NYS. Cited by Gardner, (SA, Nov 1957) = 1st Book, chap. 12, prob. 7. Gardner says Jones, p. 93, also gives the two dimensional version: If the longest line that can be drawn in an annulus is 6" long, what is the area of the annulus?

L. Lines. Solid Geometry. Macmillan, London, 1935; Dover, 1965. P. 101, Example 8W3: "A napkin ring is in the form of a sphere pierced by a cylindrical hole. Prove that its volume is the same as that of a sphere with diameter equal to the length of the hole." Solution is given, but there is no indication that it is new or recent.

L. A. Graham. Ingenious Mathematical Problems and Methods. Dover, 1959. Prob. 34: Hole in a sphere, pp. 23 & 145-147. [The material in this book appeared in Graham's company magazine from about 1940, but no dates are provided in the book. (??can date be found out.)]

M. H. Greenblatt. Mathematical Entertainments, op. cit. in 6.U.2, 1965. Volume of a modified bowling ball, pp. 104-105.

C. W. Trigg. Op. cit. in 5.Q. 1967. Quickie 217: Hole in sphere, pp. 59 & 178-179. Gives an argument based on surface tension to see that the ring surface remains spherical as the hole changes radius. Problem has a 10" hole.

Andrew Jarvis. Note 3235: A boring problem. MG 53 (No. 385) (Oct 1969) 298-299. He calls it "a standard problem" and says it is usually solved with a triple integral (??!!). He gives the standard proof using Cavalieri's principle.

Birtwistle. Math. Puzzles & Perplexities. 1971.

Tangential chord, pp. 71-73. 10" chord in an annulus. What is the area of the annulus? Does traditionally and then by letting inner radius be zero.

The hole in the sphere, pp. 87-88 & 177-178. Bore a hole through a sphere so the remaining piece has half the volume of the sphere. The radius of the hole is approx. .61 of the radius of the sphere.

Another hole, pp. 89, 178 & 192. 6" hole cut out of sphere. What is the volume of the remainder? Refers to the tangential chord problem.

Arne Broman. Holditch's theorem: An introductory problem. Lecture at ICM, Helsinki, Aug 1978. Broman then sent out copies of his lecture notes and a supplementary letter on 30 Aug 1978. He discusses Holditch's proof (see above) and more careful modern versions of it. His letter gives some other citations.

6.AF. WHAT COLOUR WAS THE BEAR?

A hunter goes 100 mi south, 100 mi east and 100 mi north and finds himself where he started. He then shoots a bear -- what colour was the bear?

Square versions: Perelman; Klamkin, Breault & Schwarz; Kakinuma, Barwell & Collins; Singmaster.

I include other polar problems here. See also 10.K for related geographical problems.

"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller, Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty five Surprising PARADOXES, in GORDON's Geography, so the following must have appeared in Gordon. Part I, no. 10, p. 9. "There is a particular Place of the Earth where the Winds (tho' frequently veering round the Compas) do always blow from the North Point."

Philip Breslaw (attrib.). Breslaw's Last Legacy; or the Magical Companion: containing all that is Curious, Pleasing, Entertaining and Comical; selected From the most celebrated Masters of Deception: As well with Slight of Hand, As with Mathematical Inventions. Wherein is displayed The Mode and Manner of deceiving the Eye; as practised by those celebrated Masters of Mirthful Deceptions. Including the various Exhibitions of those wonderful Artists, Breslaw, Sieur, Comus, Jonas, &c. Also the Interpretation of Dreams, Signification of Moles, Palmestry, &c. The whole forming A Book of real Knowledge in the Art of Conjuration. (T. Moore, London, 1784, 120pp.) With an accurate Description of the Method how to make The Air Balloon, and inject the Inflammable Air. (2nd ed., T. Moore, London, 1784, 132pp; 5th ed., W. Lane, London, 1791, 132pp.) A New Edition, with great Additions and Improvements. (W. Lane, London, 1795, 144pp.) Facsimile from the copy in the Byron Walker Collection, with added Introduction, etc., Stevens Magic Emporium, Wichita, Kansas, 1997. [This was first published in 1784, after Breslaw's death, so it is unlikely that he had anything to do with the book. There were versions in 1784, 1791, 1792, 1793, 1794, 1795, 1800, 1806, c1809, c1810, 1811, 1824. Hall, BCB 39-43, 46-51. Toole Stott 120-131, 966-967. Heyl 35-41. This book went through many variations of subtitle and contents -- the above is the largest version.]. I will cite the date as 1784?.

Geographical Paradoxes.

Paradox I, p. 35. Where is it noon every half hour? Answer: At the North Pole in Summer, when the sun is due south all day long, so it is noon every moment!

Paradox II, p. 36. Where can the sun and the full moon rise at the same time in the same direction? Answer: "Under the North Pole, the sun and the full moon, both decreasing in south declination, may rise in the equinoxial points at the same time; and under the North Pole, there is no other point of compass but south." I think this means at the North Pole at the equinox.

Carlile. Collection. 1793. Prob. CXVI, p. 69. Where does the wind always blow from the north?

Jackson. Rational Amusement. 1821. Geographical Paradoxes.

No. 7, pp. 36 & 103. Where do all winds blow from the north?

No. 8, pp. 36 & 110. Two places 100 miles apart, and the travelling directions are to go 50 miles north and 50 miles south.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:3 (Jul 1903) 246-247. A safe catch. Airship starts at the North Pole, goes south for seven days, then west for seven days. Which way must it go to get back to its starting point? No solution given.

Pearson. 1907.

Part II, no. 21: By the compass, pp. 18 & 190. Start at North Pole and go 20 miles southwest. What direction gets back to the Pole the quickest? Answer notes that it is hard to go southwest from the Pole!

Part II, no. 15: Ask "Where's the north?" -- Pope, pp. 117 & 194. Start 1200 miles from the North Pole and go 20 mph due north by the compass. How long will it take to get to the Pole? Answer is that you never get there -- you get to the North Magnetic Pole.

Ackermann. 1925. P. 116. Man at North Pole goes 20 miles south and 30 miles west. How far, and in what direction, is he from the Pole?

Richard Guy (letter of 27 Feb 1985) recalls this problem (I think he is referring to the 'What colour was the bear' version) from his schooldays in the 1920s.

H. Phillips. Week-End. 1932. Prob. 8, pp. 12 & 188. = his Playtime Omnibus, 1933, prob. 10: Popoff, pp. 54 & 237. House with four sides facing south.

H. Phillips. The Playtime Omnibus. Faber & Faber, London, 1933. Section XVI, prob. 11: Polar conundrum, pp. 51 & 234. Start at the North Pole, go 40 miles South, then 30 miles West. How far are you from the Pole. Answer: "Forty miles. (NOT thirty, as one is tempted to suggest.)" Thirty appears to be a slip for fifty??

Perelman. FFF. 1934. 1957: prob. 6, pp. 14-15 & 19-20: A dirigible's flight; 1979: prob. 7, pp. 18-19 & 25-27: A helicopter's flight. MCBF: prob. 7, pp. 18-19 & 25-26: A helicopter's flight. Dirigible/helicopter starts at Leningrad and goes 500km N, 500km  E, 500km S, 500km W. Where does it land? Cf Klamkin et seq., below.

Phillips. Brush. 1936. Prob. A.1: A stroll at the pole, pp. 1 & 73. Eskimo living at North Pole goes 3 mi south and 4 mi east. How far is he from home?

Haldeman-Julius. 1937. No. 51: North Pole problem, pp. 8 & 23. Airplane starts at North Pole, goes 30 miles south, then 40 miles west. How far is he from the Pole?

J. R. Evans. The Junior Week-End Book. Gollancz, London, 1939. Prob. 9, pp. 262 & 268. House with four sides facing south.

Leopold. At Ease! 1943. A helluva question!, pp. 10 & 196. Hunter goes 10 mi south, 10 mi west, shoots a bear and drags it 10 mi back to his starting point. What colour was the bear? Says the only geographic answer is the North Pole.

E. P. Northrop. Riddles in Mathematics. 1944. 1944: 5-6; 1945: 5-6; 1961: 15-16. He starts with the house which faces south on all sides. Then he has a hunter that sees a bear 100 yards east. The hunter runs 100 yards north and shoots south at the bear -- what colour .... He then gives the three-sided walk version, but doesn't specify the solution.

E. J. Moulton. A speed test question; a problem in geography. AMM 51 (1944) 216 & 220. Discusses all solutions of the three-sided walk problem.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 50: A fine outlook, pp. 54-55. House facing south on all sides used by an artist painting bears!

Leeming. 1946. Chap. 3, prob. 32: What color was the bear?, pp. 33 & 160. Man walks 10 miles south, then 10 miles west, where he shoots a bear. He drags it 10 miles north to his base. What color .... He gives only one solution.

Darwin A. Hindman. Handbook of Indoor Games & Contests. (Prentice-Hall, 1955); Nicholas Kaye, London, 1957. Chap. 16, prob. 4: The bear hunter, pp. 256 & 261. Hunter surprises bear. Hunter runs 200 yards north, bear runs 200 yards east, hunter fires south at bear. What colour ....

Murray S. Klamkin, proposer; D. A. Breault & Benjamin L. Schwarz, solvers. Problem 369. MM 32 (1958/59) 220 & 33 (1959/60) 110 & 226-228. Explorer goes 100 miles north, then east, then south, then west, and is back at his starting point. Breault gives only the obvious solution. Schwartz gives all solutions, but not explicitly. Cf Perelman, 1934.

Benjamin L. Schwartz. What color was the bear?. MM 34 (1960) 1-4. ??NYS -- described by Gardner, SA (May 1966) = Carnival, chap. 17. Considers the problem where the hunter looks south and sees a bear 100 yards away. The bear goes 100 yards east and the hunter shoots it by aiming due south. This gives two extra types of solution.

Ripley's Puzzles and Games. 1966. Pp. 18, item 5. 50 mi N, 1000 mi W, 10 mi S to return to your starting point. Answer only gives the South Pole, ignoring the infinitely many cases near the North Pole. Looking at this made me realise that when the sideways distance is larger than the circumference of the parallel at that distance from the pole, then there are other solutions that start near the pole. Here there are three solutions where one starts at distances 109.2, 29.6 or 3.05 miles from the South Pole, circling it 1, 2 or 3 times.

Yasuo Kakinuma, proposer; Brian Barwell and Craig H. Collins, solvers. Problem 1212 -- Variation of the polar bear problem. JRM 15:3 (1982-83) 222 & 16:3 (1983-84) 226-228. Square problem going one mile south, east, north, west. Barwell gets the explicit quadratic equation, but then approximates its solutions. Collins assumes the earth is flat near the pole.

David Singmaster. Bear hunting problems. Submitted to MM, 1986. Finds explicit solutions for the general version of Perelman/Klamkin's problem. [In fact, I was ignorant of (or had long forgotten) the above when I remembered and solved the problem. My thanks to an editor (Paul Bateman ??check) for referring me to Klamkin. The Kakinuma et al then turned up also.] Analysis of the solutions leads to some variations, including the following.

David Singmaster. Home is the hunter. Man heads north, goes ten miles, has lunch, heads north, goes ten miles and finds himself where he started.

Used as: Explorer's problem by Keith Devlin in his Micromaths Column; The Guardian (18 Jun 1987) 16 & (2 Jul 1987) 16.

Used by me as one of: Spring term puzzles; South Bank Polytechnic Computer Services Department Newsletter (Spring 1989) unpaged [p. 15].

Used by Will Shortz in his National Public Radio program 6? Jan 1991.

Used as: A walk on the wild side, Games 15:2 (No. 104) (Aug 1991) 57 & 40.

Used as: The hunting game, Focus 3 (Feb 1993) 77 & 98.

Used in my Puzzle Box column, G&P, No. 11 (Feb 1995) 19 & No. 12 (Mar 1995) 41.

Bob Stanton. The explorers. Games Magazine 17:1 (No. 113) (Feb 1993) 61 & 43. Two explorers set out and go 500 miles in each direction. Madge goes N, W, S, E, while Ellen goes E, S, W, N. At the end, they meet at the same point. However, this is not at their starting point. How come? and how far are they from their starting point, and in what direction? They are not near either pole.

Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 11, prob. 9: What color was that bear? (A lesson in non-Euclidean geometry), pp. 97 & 185-191. Camper walks south 2 km, then west 5 km, then north 2 km; how far is he from his starting point? Solution analyses this and related problems, finding that the distance x satisfies 0 ( x ( 7.183, noting that there are many minimal cases near the south pole and if one is between them, one gets a local maximum, so one has to determine one's position very carefully.

David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics Teacher; Corwin Press, NY, 1996, pp. 273-286. Sketches the explicit solution to Klamkin's problem as an example of the use of symmetric variables to obtain a solution.

Anonymous. Brainteaser B163 -- Shady matters. Quantum 6:3 (Jan/Feb 1996) 15 & 48. Is there anywhere on earth where one's shadow has the same length all day long?

6.AG. MOVING AROUND A CORNER

There are several versions of this. The simplest is moving a ladder or board around a corner -- here the problem is two-dimensional and the ladder is thin enough to be considered as a line. There are slight variations -- the corner can be at a T or + junction; the widths of the corridors may differ; the angle may not be a right angle; etc. If the object being moved is thicker -- e.g. a table -- then the problem gets harder. If one can use the third dimension, it gets even harder.

H. E. Licks. Op. cit. in 5.A, 1917. Art. 110, p. 89. Stick going into a circular shaft in the ceiling. Gets [h2/3 + d2/3)]3/2 for maximum length, where h is the height of the room and d is the diameter of the shaft. "A simple way to solve a problem which has proved a stumbling block to many."

Abraham. 1933. Prob. 82 -- Another ladder, pp. 37 & 45 (23 & 117). Ladder to go from one street to another, of different widths.

E. H. Johnson, proposer; W. B. Carver, solver. Problem E436. AMM 47 (1940) 569 & 48 (1941) 271-273. Table going through a doorway. Obtains 6th order equation.

J. S. Madachy. Turning corners. RMM 5 (Oct 1961) 37, 6 (Dec 1961) 61 & 8 (Apr 1962) 56. In 5, he asks for the greatest length of board which can be moved around a corner, assuming both corridors have the same width, that the board is thick and that vertical movement is allowed. In 6, he gives a numerical answer for his original values and asserts the maximal length for planar movement, with corridors of width w and plank of thickness t, is 2 (w(2 - t). In vol. 8, he says no two solutions have been the same.

L. Moser, proposer; M. Goldberg and J. Sebastian, solvers. Problem 66-11 -- Moving furniture through a hallway. SIAM Review 8 (1966) 381-382 & 11 (1969) 75-78 & 12 (1970) 582-586. "What is the largest area region which can be moved through a "hallway" of width one (see Fig. 1)?" The figure shows that he wants to move around a rectangular corner joining two hallways of width one. Sebastian (1970) studies the problem for moving an arc.

J. M. Hammersley. On the enfeeblement of mathematical skills .... Bull. Inst. Math. Appl. 4 (1968) 66-85. Appendix IV -- Problems, pp. 83-85, prob. 8, p. 84. Two corridors of width 1 at a corner. Show the largest object one can move around it has area < 2 (2 and that there is an object of area ( π/2 + 2/π = 2.2074.

Partial solution by T. A. Westwell, ibid. 5 (1969) 80, with editorial comment thereon on pp. 80-81.

T. J. Fletcher. Easy ways of going round the bend. MG 57 (No. 399) (Feb 1973) 16-22. Gives five methods for the ladder problem with corridors of different widths.

Neal R. Wagner. The sofa problem. AMM 83 (1976) 188-189. "What is the region of largest area which can be moved around a right-angled corner in a corridor of width one?" Survey.

R. K. Guy. Monthly research problems, 1969-77. AMM 84 (1977) 807-815. P. 811 reports improvements on the sofa problem.

J. S. Madachy & R. R. Rowe. Problem 242 -- Turning table. JRM 9 (1976-77) 219-221.

G. P. Henderson, proposer; M. Goldberg, solver; M. S. Klamkin, commentator. Problem 427. CM 5 (1979) 77 & 6 (1979) 31-32 & 49-50. Easily finds maximal area of a rectangle going around a corner.

Research news: Conway's sofa problem. Mathematics Review 1:4 (Mar 1991) 5-8 & 32. Reports on Joseph Gerver's almost complete resolution of the problem in 1990. Says Conway asked the problem in the 1960s and that L. Moser is the first to publish it. Says a group at a convexity conference in Copenhagen improved Hammersley's results to 2.2164. Gerver's analysis gives an object made up of 18 segments with area 2.2195. The analysis depends on some unproven general assumptions which seem reasonable and is certainly the unique optimum solution given those assumptions.

A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5. Does usual problem, getting a quartic. The finds the shortest ladder. [This turns out to be the same as the longest ladder one can get around a corner from corridors of widths w and h, so 6.AG is related to 6.L.]

6.AH. TETHERED GOAT

A goat is grazing in a circular field and is tethered to a post on the edge. He can reach half of the field. How long is the rope? There are numerous variations obtained by modifying the shape of the field or having buildings within it. In recent years, there has been study of the form where the goat is tethered to a point on a circular silo in a large field -- how much area can he graze?

Upnorensis, proposer; Mr. Heath, solver. Ladies Diary, 1748-49 = T. Leybourn, II: 6-7, quest. 302. [I have a reference to p. 41 of the Ladies' Diary.] Circular pond enclosed by a circular railing of circumference 160 yards. Horse is tethered to a post of the railing by a rope 160 yards long. How much area can he graze?

Dudeney. Problem 67: Two rural puzzles -- No. 67: One acre and a cow. Tit-Bits 33 (5 Feb & 5 Mar 1898) 355 & 432. Circular field opening onto a small rectangular paddock with cow tethered to the gate post so that she can graze over one acre. By skilful choice of sizes, he avoids the usual transcendental equation.

Arc. [R. A. Archibald]. Involutes of a circle and a pasturage problem. AMM 28 (1921) 328-329. Cites Ladies Diary and it appears that it deals with a horse outside a circle.

J. Pedoe. Note 1477: An old problem. MG 24 (No. 261) (Oct 1940) 286-287. Finds the relevant area by integrating in polar coordinates centred on the post.

A. J. Booth. Note 1561: On Note 1477. MG 25 (No. 267) (Dec 1941) 309-310. Goat tethered to a point on the perimeter of a circle which can graze over ½, ⅓, ¼ of the area.

Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968.

No. 8: "Don't fence me in", pp. 87. Equilateral triangular field of area 120. Three goats tethered to the corners with ropes of length equal to the altitude. Consider an area where n goats graze as contributing 1/n to each goat. What area does each goat graze over?

No. 53: Around the silo, pp. 71 & 112-113. Goat tethered to the outside of a silo of diameter 20 by a rope of length 10π, i.e. he can just get to the other side of the silo. How big an area can he graze? The curve is a semicircle together with two involutes of a circle, so the solution uses some calculus.

Marshall Fraser. A tale of two goats. MM 55 (1982) 221-227. Gives examples back to 1894.

Marshall Fraser. Letter: More, old goats. MM 56 (1983) 123. Cites Arc[hibald].

Bull, 1998, below, says this problem has been discussed by the Internet newsgroup sci.math some years previously.

Michael E. Hoffman. The bull and the silo: An application of curvature. AMM 105:1 (Jan 1998) ??NYS -- cited by Bull. Bull is tethered by a rope of length L to a circular silo of radius R. If L ( πR, then the grazeable area is L3/3R + πL2/2. This paper considers the problem for general shapes.

John Bull. The bull and the silo. M500 163 (Aug 1998) 1-3. Improves Hoffman's solution for the circular silo by avoiding polar coordinates and using a more appropriate variable, namely the angle between the taut rope and the axis of symmetry.

Keith Drever. Solution 186.5 -- Horse. M550 188 (Oct 2002) 12. A horse is tethered to a point on the perimeter of a circular field of radius 1. He can graze over all but 1/π of the area. How long is the rope? This turns out to make the problem almost trivial -- the rope is (2 long and the angle subtended at the tether is π/2.

6.AI. TRICK JOINTS

S&B, pp. 146-147, show several types.

These are often made in two contrasting woods and appear to be physically impossible. They will come apart if one moves them in the right direction. A few have extra complications. The simplest version is a square cylinder with dovetail joints on each face -- called common square version below. There are also cases where one thinks it should come apart, but the wood has been bent or forced and no longer comes apart -- see also 6.W.5.

See Bogesen in 6.W.2 for a possible early example.

Johannes Cornelus Wilhelmus Pauwels. UK Patent 15,307 -- Improved Means of Joining or Fastening Pieces of Wood or other Material together, Applicable also as a Toy. Applied: 9 Nov 1887; complete specification: 9 Aug 1888; accepted: 26 Oct 1888. 2pp + 1p diagrams. It says Pauwels is a civil engineer of The Hague. Common square version.

Tom Tit, vol. 2. 1892. Assemblage paradoxal, pp. 231-232. = K, no. 155: The paradoxical coupling, pp. 353-354. Common square version with instructions for making it by cutting the corners off a larger square.

Emery Leverett Williams. The double dovetail and blind mortise. SA (25 Apr 1896) 267. The first is a trick T-joint.

T. Moore. A puzzle joint and how to make it. The Woodworker 1:8 (May 1902) 172. S&B, p. 147, say this is the earliest reference to the common square version -- but see Pauwels, above. "... the foregoing joint will doubtless be well-known to our professional readers. There are probably many amateur woodworkers to whom it will be a novelty."

Hasluck, Paul N. The Handyman's Book. Cassell, 1903; facsimile by Senate (Tiger Books), Twickenham, London, 1998. Pp. 220-223 shows various joints. Dovetail halved joint with two bevels, p. 222 & figs. 703-705 of pp. 221-222. "... of but little practical value, but interesting as a puzzle joint."

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Shows the common square version "given to me some ten years ago, but I cannot say who first invented it." He previously published it in a newspaper. ??look in Weekly Dispatch.

Samuel Hicks. Kinks for Handy Men: The dovetail puzzle. Hobbies 31 (No. 790) (3 Dec 1910) 248-249. Usual square dovetail, but he suggests to glue it together!

Dudeney. AM. 1917. Prob. 424: The dovetailed block, pp. 145 & 249. Shows the common square version -- "... given to me some years ago, but I cannot say who first invented it." He previously published it in a newspaper. ??as above

Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. A curious dovetail joint, pp. 193, 195. Common square version. Dovetail puzzle joint, pp. 194-195. A singly mortised T-joint, with an unmortised second piece.

E. M. Wyatt. Woodwork puzzles. Industrial-Arts Magazine 12 (1923) 326-327. Doubly dovetailed tongue and mortise T-joint called 'The double (?) dovetail'.

Sherman M. Turrill. A double dovetail joint. Industrial-Arts Magazine 13 (1924) 282-283. A double dovetail right angle joint, but it leaves sloping gaps on the inside which are filled with blocks.

Collins. Book of Puzzles. 1927. Pp. 134-135: The dovetail puzzle. Common square version.

E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.

The double (?) dovetail, pp. 44-45. Doubly dovetailed tongue and mortise T-joint.

The "impossible" dovetail joint, p. 46. Common square version.

Double-lock dovetail joint, pp. 47-49. Less acceptable tricks for a corner joint.

Two-way fanned half-lap joint, pp. 49-50. Corner joint.

A. B. Cutler. Industrial Arts and Vocational Education (Jan 1930). ??NYS. Wyatt, below, cites this for a triple dovetail, but I could not not find it in vols. 1-40.

R. M. Abraham. Prob. 225 -- Dovetail Puzzle. Winter Nights Entertainments. Constable, London, 1932, p. 131. (= Easy-to-do Entertainments and Diversions with coins, cards, string, paper and matches; Dover, 1961, p. 225.) Common square version.

Abraham. 1933. Prob. 304 -- Hexagon dovetail; Prob. 306 -- The triangular dovetail, pp. 142-143 (100 & 102).

Bernard E. Jones, ed. The Practical Woodworker. Waverley Book Co., London, nd [1940s?]. Vol. 1: Lap and secret dovetail joints, pp. 281-287. This covers various secret joints -- i.e. ones with concealed laps or dovetails. Pp. 286-287 has a subsection: Puzzle dovetail joints. Common square version is shown as fig. 28. A pentagonal analogue is shown as fig. 29, but it uses splitting and regluing to produce a result which cannot be taken apart.

E. M. Wyatt. Wonders in Wood. Bruce Publishing Co., Milwaukee, 1946.

Double-double dovetail joint, pp. 26-27. Requires some bending.

Triple dovetail puzzle, pp. 28-29. Uses curved piece with gravity lock.

S&B, p. 146, reproduces the above Wyatt and shows a 1948 example.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. Dovetail deceptions, p. 64. Common square version and a tapered T joint.

Allan Boardman. Up and Down Double Dovetail. Shown on p. 147 of S&B. Square version with alternate dovetails in opposite directions. This is impossible!

I have a set of examples which belonged to Tom O'Beirne. There is a common square version and a similar hexagonal version. There is an equilateral triangle version which requires a twist. There is a right triangle version which has to be moved along a space diagonal! [One can adapt the twisting method to n-gons!]

Dick Schnacke (Mountain Craft Shop, American Ridge Road, Route 1, New Martinsville, West Virginia, 26155, USA) makes a variant of the common square version which has two dovetails on each face. I bought one in 1994.

6.AJ. GEOMETRIC ILLUSIONS

There are a great many illusions. This will only give some general studies and some specific sources, though the sources of many illusions are unknown.

An exhibition by Al Seckel says there are impossible geometric patterns in a mosaic floor in the Roman villa at Fishbourne, c75, but it is not clear if this was intentional.

Anonymous 15C French illustrator of Giovanni Boccaccio, De Claris Mulieribus, MS Royal 16 Gv in the British Library. F. 54v: Collecting cocoons and weaving silk. ??NYS -- reproduced in: The Medieval Woman An Illuminated Book of Postcards, HarperCollins, 1991. This shows a loom(?) frame with uprights at each corner and the crosspieces joining the tops of the end uprights as though front and rear are reversed compared to the ground.

Seckel, 2002a, below, p. 25 (= 2002b, p. 175), says Leonardo da Vinci created the first anamorphic picture, c1500.

Giuseppe Arcimboldo (1537-1593). One of his paintings shows a bowl of vegetables, but when turned over, it is a portrait. Seckel, 2000, below, fig. 109, pp. 120 & 122 (= 2002b, fig, 107, pp. 118 & 120), noting that this is the first known invertible picture, but see next entry.

Topsy turvy coin, mid 16C. Seckel, 2002a, fig. 65, p. 80 (omitted in 2002b), shows an example which shows the Pope, but turns around to show the Devil. Inscription around edge reads: CORVI MALUM OVUM MALII.

Robert Smith. A Compleat System of Opticks in Four Books. Cambridge, 1738. He includes a picture of a distant windmill for which one cannot tell whether the sails are in front or behind the mill, apparently the first publication of this visual ambiguity. ??NYS -- cited by: Nicholas J. Wade; Visual Allusions Pictures of Perception; Lawrence Erlbaum Associates, Hove, East Sussex, 1990, pp. 17 & 25, with a similar picture.

L. A. Necker. LXI. Observations on some remarkable optical phœnomena seen in Switzerland; and on an optical phœnomenon which occurs on viewing a figure of a crystal or geometrical solid. Phil. Mag. (3) 1:5 (Nov 1832) 329-337. This is a letter from Necker, written on 24 May 1832. On pp. 336-337, Necker describes the visual reversing figure known as the Necker cube which he discovered in drawing rhomboid crystals. This is also quoted in Ernst; The Eye Beguiled, pp. 23-24]. Richard L. Gregory [Mind in Science; Weidenfeld and Nicolson, London, 1981, pp. 385 & 594] and Ernst say that this was the first ambiguous figure to be described.

See Thompson, 1882, in 6.AJ.2, for illusions caused by rotations.

F. C. Müller-Lyer. Optische Urtheilstusehungen. Arch. Physiol. Suppl. 2 (1889) 263-270. Cited by Gregory in The Intelligent Eye. Many versions of the illusion. But cf below.

Wehman. New Book of 200 Puzzles. 1908. The cube puzzle, p. 37. A 'baby blocks' pattern of cubes, which appears to show six cubes piled in a corner one way and seven cubes the other way. I don't recall seeing this kind of puzzle in earlier sources, though this pattern of rhombuses is common on cathedral floors dating back to the Byzantine era or earlier.

James Fraser. British Journal of Psychology (Jan 1908). Introduces his 'The Unit of Direction Illusion' in many forms. ??NYS -- cited in his popular article in Strand Mag., see below. Seckel, 2000, below, has several versions. On p. 44, note to p. 9 (= 2002b, p. 44, note to p. 9), he says Fraser created a series of these illusions in 1906.

H. E. Carter. A clever illusion. Curiosities section, Strand Mag. 378 (No. 219) (Mar 1909) 359. An example of Fraser's illusion with no indication of its source.

James Fraser. A new illusion. What is its scientific explanation? Strand Mag. 38 (No. 224) (Aug 1909) 218-221. Refers to the Mar issue and says he introduced the illusion in the above article and that the editors have asked him for a popular article on it. 16 illustrations of various forms of his illusion.

Lietzmann, Walther & Trier, Viggo. Wo steckt der Fehler? 3rd ed., Teubner, 1923. [The Vorwort says that Trier was coauthor of the 1st ed, 1913, and contributed most of the Schülerfehler (students' mistakes). He died in 1916 and Lietzmann extended the work in a 2nd ed of 1917 and split it into Trugschlüsse and this 3rd ed. There was a 4th ed., 1937. See Lietzmann for a later version combining both parts.] II. Täuschungen der Anschauung, pp. 7-13.

Lietzmann, Walther. Wo steckt der Fehler? 3rd ed., Teubner, Stuttgart, (1950), 1953. (Strens/Guy has 3rd ed., 1963.) (See: Lietzmann & Trier. There are 2nd ed, 1952??; 5th ed, 1969; 6th ed, 1972. Math. Gaz. 54 (1970) 182 says the 5th ed appears to be unchanged from the 3rd ed.) II. Täuschungen der Anschauung, pp. 15-25. A considerable extension of the 1923 ed.

Williams. Home Entertainments. 1914. Colour discs for the gramophone, pp. 207-212. Discusses several effects produced by spirals and eccentric circles on discs when rotated.

Gerald H. Fisher. The Frameworks for Perceptual Localization. Report of MOD Research Project70/GEN/9617, Department of Psychology, University of Newcastle upon Tyne, 1968. Good collection of examples, with perhaps the best set of impossible figures.

Pp. 42-47 -- reversible perspectives.

Pp. 56-65 -- impossible and ambiguous figures.

Appendix 6, p.190 -- 18 reversible figures.

Appendix 7, pp. 191-192 -- 12 reversible silhouettes.

Appendix 8, p. 193 -- 12 impossible figures.

Appendix 14, pp. 202-203 -- 72 geometrical illusions.

Harvey Long. "It's All In How You Look At It". Harvey Long & Associates, Seattle, 1972. 48pp collection of examples with a few references.

Bruno Ernst [pseud. of J. A. F. Rijk]. (Avonturen met Onmogelijke Figuren; Aramith Uitgevers, Holland, 1985.) Translated as: Adventures with Impossible Figures. Tarquin, Norfolk, 1986. Describes tribar and many variations of it, impossible staircase, two-pronged trident. Pp. 76-77 reproduces an Annunciation of 14C in the Grote Kerk, Breda, with an impossible perspective. P. 78 reproduces Print XIV of Giovanni Battista Piranesi's "Carceri de Invenzione", 1745, with an impossible 4-bar.

Diego Uribe. Catalogo de impossibilidades. Cacumen (Madrid) 4 (No. 37) (Feb 1986) 9-13. Good summary of impossible figures. 15 references to recent work.

Bruno Ernst. Escher's impossible figure prints in a new context. In:  H. S. M. Coxeter, et al., eds.; M. C. Escher -- Art and Science; North-Holland (Elsevier), Amsterdam, 1986, pp. 124-134. Pp. 128-129 discusses the Breda Annunciation, saying it is 15C and quoting a 1912 comment by an art historian on it. There is a colour reproduction on p. 394. P. 130 shows and discusses briefly Bruegel's "The Magpie on the Gallows", 1568. Pp. 130-131 discusses and illustrates the Piranesi.

Bruno Ernst. (Het Begoochelde Oog, 1986?.) Translated by Karen Williams as: The Eye Beguiled. Benedikt Taschen Verlag, Köln, 1992. Much expanded version of his previous book, with numerous new pictures and models by new artists in the field. Chapter 6: Origins and history, pp. 68-93, discusses and quotes almost everything known. P. 68 shows a miniature of the Madonna and Child from the Pericope of Henry II, compiled by 1025, now in the Bayersche Staatsbibliothek, Munich, which is similar in form to the Breda Annunciation (stated to be 15C). (However, Seckel, 1997, below, reproduces it as 2( and says it is c1250.) P. 69 notes that Escher invented the impossible cube used in his Belvedere. P. 82 is a colour reproduction of Duchamp's 1916-1917 'Apolinère Enameled' - see 6.AJ.2. Pp. 83-84 shows and discusses Piranesi. Pp. 84-85 show and discuss Hogarth's 'False Perspective' of 1754. Reproduction and brief mention of Brueghel (= Bruegel) on p. 85. Discussion of the Breda Annunciation on pp. 85-86. Pp. 87-88 show and discuss a 14C Byzantine Annunciation in the National Museum, Ochrid. Pp. 88-89 show and discuss Scott Kim's impossible four-dimensional tribar.

J. Richard Block & Harold E. Yuker. Can You Believe Your Eyes? Brunner/Mazel, NY, 1992. Excellent survey of the field of illusions, classified into 17 major types -- e.g. ambiguous figures, unstable figures, ..., two eyes are better than one. They give as much information as they can about the origins. They give detailed sources for the following -- originals ??NYS. These are also available as two decks of playing cards.

W. E. Hill. My wife and my mother-in-law. Puck, (6 Nov 1915) 11. [However, Julian Rothenstein & Mel Gooding; The Paradox Box; Redstone Press, London, 1993; include a reproduction of a German visiting card of 1888 with a version of this illusion. The English caption by James Dalgety is: My Wife and my Mother-in-law. Cf Seckel, 1997, below.] Ernst, just above, cites Hill and says he was a cartoonist, but gives no source. Long, above, asserts it was designed by E. G. Boring, an American psychologist.

G. H. Fisher. Mother, father and daughter. Amer. J. Psychology 81 (1968) 274-277.

G. Kanisza. Subjective contours. SA 234:4 (Apr 1976) 48-52. (Kanisza triangles.)

Al Seckel, 1997. Illusions in Art. Two decks of playing cards in case with notes. Deck 1 -- Classics. Works from Roman times to the middle of the 20th Century. Deck 2 -- Contemporary. Works from the second half of the 20th Century. Y&B Associates, Hempstead, NY, 1997. This gives further details on some of the classic illusions -- some of this is entered above and in 6.AU and some is given below.

10(: Rabbit/Duck. Devised by Joseph (but notes say Robert) Jastrow, c1900. Seckel, 2000, below, p. 159 (= 2002b, p. 156), says Joseph Jastrow, c1900.

10(: My Wife and My Mother-in-Law, anonymous, 1888. However, in an exhibition, Seckel's text implies the 1888 German card doesn't have a title and the title first occurs on an 1890 US card. Seckel, 2000, below, p. 122 (= 2002b, p. 120), says Boring took it from a popular 19C puzzle trading card.

Al Seckel, 2000. The Art of Optical Illusions. Carlton, 2000. 144 well reproduced illusions with brief notes. All figures except 69-70 are included in Seckel, 2002b.

J. Richard Block. Seeing Double Over 200 Mind-Bending Illusions. Routledge, 2002. Update of Block & Yuker, 1992.

Edgar Rubin. Rubin's Vase. 1921. This is the illusion where there appears to be a vase, but the outsides appear to be two face profiles. [Pp. 8-11.] But Seckel, 2000, above, p. 122 (= 2002b, p. 120), says Rubin's inspiration was a 19C puzzle card.

My wife and my mother-in-law. P. 17 says Hill's version may derive from a late 1880s advertising postcard for Phenyo-Caffein (Worcester, Massachusetts), labelled 'My Girl & Her Mother', reproduced on p. 17.

P. 18 has G. H. Fisher's 1968 triple image, labelled 'Mother, Father and Daughter-in-Law'.

P. 44 says that Rabbit/Duck was devised by Joseph Jastrow in 1888.

Al Seckel, 2002a. More Optical Illusions. Carlton, 2002. 137 well reproduced illusions with brief notes, different than in Seckel, 2000, above. All figures except 65-66, 86-87, 95-95, 137 are included in Seckel, 2002b, but with different figure and page numbers.

Al Seckel, 2002b. The Fantastic World of Optical Illusions. Carlton, 2002. This is essentially a combination of Seckel, 2000, and Seckel, 2002a, both listed above. The Introduction is revised. Figures 69-70 of the first book and 65-66, 86-87, 94-95, 137 of the second book are omitted. The remaining figures are then numbered consecutively. The page of Further Reading in the first book is put at the end of this combined book.

Here I make some notes about origins of other illusions, but I have fewer details on these.

The Müller-Lyer Illusion -- vs >---< was proposed by Zollner in 1859 and described by Johannes Peter Müller (1801-1858) & Lyer in 1889. This seems to be a confusion, as the 1889 article is by F. C. Müller-Lyer, cf above. Lietzmann & Trier, p. 7, date it as 1887.

The Bisection Illusion -- with a vertical segment bisecting a horizontal segment, but above it -- was described by Albert Oppel (1831-1865) and Wilhelm Wundt (1832-1920) in 1865.

Zollner's Illusion -- parallel lines crossed by short lines at 45o, alternately in opposite directions -- was noticed by Johann K. F. Zollner (1834-1882) on a piece of fabric with a similar design.

Hering's Illusion -- with parallel lines crossed by numerous lines through a point between the lines -- was invented by Ewals Hering (1834-1918) in 1860.

6.AJ.1 TWO PRONGED TRIDENT

I have invented this name as it is more descriptive than any I have seen. The object or a version of it is variously called: Devil's Fork; Three Stick Clevis; Widgit; Blivit; Impossible Columnade; Trichometric Indicator Support; Triple Encabulator for Tuned Manifold; Hole Location Gage; Poiyut; Triple-Pronged Fork with only Two Branches; Old Roman Pitchfork.

Oscar Reutersvård. Letters quoted in Ernst, 1992, pp. 69-70, says he developed an equivalent type of object, which he calls impossible meanders, in the 1930s.

R. L. Gregory says this is due to a MIT draftsman (= draughtsman) about 1950??

California Technical Industries. Advertisement. Aviation Week and Space Technology 80:12 (23 Mar 1964) 5. Standard form. (I wrote them but my letter was returned 'insufficient address'.)

Hole location gage. Analog Science Fact • Science Fiction 73:4 (Jun 1964) 27. Classic Two pronged trident, with some measurements given. Editorial note says the item was 'sent anonymously for some reason' and offers the contributor $10 or a two year subscription if he identifies himself. (Thanks to Peter McMullen for the Analog items, but he doesn't recall the contributor ever being named.)

Edward G. Robles, Jr. Letter (Brass Tacks column). Analog Science Fact • Science Fiction 74:4 (Dec 1964) 4. Says the Jun 1964 object is a "three-hole two slot BLIVIT" and was developed at JPL (Jet Propulsion Laboratory, Pasadena) and published in their Goddard News. He provides a six-hole five-slot BLIVIT, but as the Editor comments, it 'lacks the classic simple elegance of the Original.' However, a letter of inquiry to JPL resulted in an email revealing that Goddard News is not their publication, but comes from the Goddard Space Flight Center. I have had a response from Goddard, ??NYR.

D. H. Schuster. A new ambiguous figure: a three-stick clevis. Amer. J. Psychol. 77 (1964) 673. Cites Calif. Tech. Ind. ad. [Ernst, 1992, pp. 80-81 reproduces this article.]

Mad Magazine. No. 93 (Mar 1965). (I don't have a copy of this -- has anyone got one for sale?) Cover by Norman Poiyut (?) shows the figure and it is called a poiyut. Miniature reproduction in: Maria Reidelbach; Completely Mad -- A History of the Comic Book and Magazine; Little, Brown & Co., Boston, 1991, p. 82. Shows a standard version. Al Seckel says they thought it was an original idea and they apologised in the next issue -- to all of the following! I now have the relevant issue, No. 95 (Jun 1965) and p. 2 has 15 letters citing earlier appearances in Engineering Digest, The Airman (official journal of the U.S. Airforce), Analog, Astounding Science Fact -- Science Fiction (Jun 1964, see above), The Red Rag (engineering journal at the University of British Columbia), Society of Automotive Engineers Journal (designed by by Gregory Flynn Jr. of General Motors as Triple Encabulator Tuned Manifold), Popular Mechanics, Popular Science (Jul 1964), Road & Track (Jun 1964). Other letters say it was circulating at: the Engineering Graphics Lab of the University of Minnesota at Duluth; the Nevada Test Site; Eastman Kodak (used to check resolution); Industrial Camera Co. of Oakland California (on their letterhead). Two letters give an impossible crate and an impossible rectangular frame (sort of a Penrose rectangle).

Sergio Aragones. A Mad look at winter sports. Mad Magazine (?? 1965); reprinted in: Mad Power; Signet, NY, 1970, pp. 120-129. P. 124 shows a standard version.

Bob Clark, illustrator. A Mad look at signs of the times. Loc. cit. under Aragones, pp. 167-188. P. 186 shows standard version.

Reveille (a UK weekly magazine) (10 Jun 1965). ??NYS -- cited by Briggs, below -- standard version.

Don Mackey. Optical illusion. Skywriter (magazine of North American Aviation) (18 Feb 1966). ??NYS -- cited by Conrad G. Mueller et al.; Light and Vision; Time-Life Books Pocket Edition, Time-Life International, Netherlands, 1969, pp. 171 & 190. Standard version with nuts on the ends.

Heinz Von Foerster. From stimulus to symbol: The economy of biological computation. IN: Sign Image Symbol; ed. Gyorgy Kepes; Studio Vista, London, 1966, pp. 42-60. On p. 55, he shows the "Triple-pronged fork with only two branches" and on p. 54, he notes that although each portion is correct, it is impossible overall, but he gives no indication of its history or that it is at all new.

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Pp. 17-18 shows the unnamed trident in a version from Adcock & Shipley (Sales) Ltd., machine tool makers in Leicester. Cites Reveille, above. Standard versions.

Harold Baldwin. Building better blivets. The Worm Runner's Digest 9:2 (1967) 104-106. Discusses relation between numbers of slots and of prongs. Draws a three slot version and 2 and 4 way versions.

Charlie Rice. Challenge! Op. cit. in 5.C. 1968. P. 10 shows a six prong, four slot version, called the "Old Roman Pitchfork".

Roger Hayward. Blivets; research and development. The Worm Runner's Digest 10 (Dec 1968) 89-92. Several fine developments, including two interlaced frames and his monumental version. Cites Baldwin.

M. Gardner. SA (May 1970) = Circus, pp. 3-15. Says this became known in 1964 and cites Mad & Hayward, but not Schuster.

D. Uribe, op. cit. above, gives several variations.

6.AJ.2. TRIBAR AND IMPOSSIBLE STAIRCASE

Silvanus P. Thompson. Optical illusions of motion. Brain 3 (1882) 289-298. Hexagon of non-overlapping circles.

Thomas Foster. Illusions of motion and strobic circles. Knowledge 1 (17 Mar 1882) 421-423. Says Thompson exhibited these illusions at the British Association meeting in 1877.

Pearson. 1907. Part II, no. 3: Whirling wheels, p. 3. Gives Thompson's form, but the wheels are overlapping, which makes it look a bit like an ancestor of the tribar.

Marcel Duchamp (1887-1968). Apolinère Enameled. A 'rectified readymade' of 1916-1917 which turned a bedframe in an advertisement for Sapolin Enamel into an impossible figure somewhat like a Penrose Triangle and a square version thereof. A version is in the Philadelphia Museum of Art and is reproduced and discussed in Ernst; The Eye Beguiled, p. 82. (Duchamp's 'readymades' were frequently reproduced by himself and others, so there may be other versions of this.)

Oscar Reutersvård. Omöjliga Figure [Impossible Figures -- In Swedish]. Edited by Paul Gabriel. Doxa, Lund, (1982); 2nd ed., 1984. This seems to be the first publication of his work, but he has been exhibiting since about 1960 and some of the exhibitions seem to have had catalogues. P. 9 shows and discusses his Opus 1 from 1934, which is an impossible tribar made from cubes. (Reproduced in Ernst, 1992, p. 69 as a drawing signed and dated 1934. Ernst quotes Reutersvård's correspondence which describes his invention of the form while doodling in Latin class as a schoolboy. A school friend who knew of his work showed him the Penroses' article in 1958 -- at that time he had drawn about 100 impossible objects -- by 1986, he had extended this to some 2500!) He has numerous variations on the tribar and the two-pronged trident. An exhibition by Al Seckel says Reutersvård had produced some impossible staircases, e.g. 'Visualized Impossible Bach Scale', in 1936-1937, but didn't go far with it until returning to the idea in 1953.

Oscar Reutersvård. Swedish postage stamps for 25, 50, 75 kr. 1982, based on his patterns from the 1930s. The 25 kr. has the tribar pattern of cubes which he first drew in 1934. (Also the 60 kr.??)

L. S. & R. Penrose. Impossible objects: A special type of visual illusion. British Journal of Psychology 49 (1958) 31-33. Presents tribar and staircase. Photo of model staircase, which Lionel Penrose had made in 1955. [Ernst, 1992, pp. 71-73, quotes conversation with Penrose about his invention of the Tribar and reproduces this article. Penrose, like the rest of us, only learned about Reutersvård's work in the 1980s.]

Anon.(?) Don't believe it. Daily Telegraph (24 Mar 1958) ?? (clipping found in an old book). "Three pages of the latest issue of the British Journal of Psychology are devoted to "Impossible Objects."" Shows both the tribar and the staircase.

M. C. Escher. Lithograph: Belvedere. 1958.

L. S. & R. Penrose. Christmas Puzzles. New Scientist (25 Dec 1958) 1580-1581 & 1597. Prob. 2: Staircase for lazy people.

M. C. Escher. Lithograph: Ascending and Descending. 1960.

M. C. Escher. Lithograph: Waterfall. 1961.

Oscar Reutersvård, in 1961, produced a triangular version of the impossible staircase, called 'Triangular Fortress without Highest Level'.

Joseph Kuykendall. Letter. Mad Magazine 95 (Jun 1965) 2. An impossible frame, a kind of Penrose rectangle.

S. W. Draper. The Penrose triangle and a family of related figures. Perception 7 (1978) 283-296. ??NYS -- cited and reproduced in Block, 2002, p. 48. A Penrose rectangle.

Uribe, op. cit. above, gives several variations, including a perspective tribar and Draper's rectangle.

Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escher-puzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -- ??) (1992) 12-13. Two Penrose tribars made into an impossible 5-piece burr.

6.AJ.3. CAFÉ WALL ILLUSION

This is the illusion seen in alternatingly coloured staggered brickwork where the lines of bricks distinctly seem tilted. I suspect it must be apparent in brickwork going back to Roman times.

The illusion is apparent in the polychrome brick work on the side wall inside Keble College Chapel, Oxford, by William Butterfield, completed in 1876 [thanks to Deborah Singmaster for observing this].

Lietzmann & Trier, op. cit. at 6.AJ, 1923. Pp. 12-13 has a striking version of this, described as a 'Flechtbogen der Kleinen'. I can't quite translate this -- Flecht is something interwoven but Bogen could be a ribbon or an arch or a bower, etc. They say it is reproduced from an original by Elsner. See Lietzmann, 1953.

Ogden's Optical Illusions. Cigarette card of 1927. No. 5. Original ??NYS -- reproduced in: Julian Rothenstein & Mel Gooding; The Paradox Box; Redstone Press, London, 1993 AND in their: The Playful Eye; Redstone Press, London, 1999, p. 56. Vertical version of this illusion.

B. K. Gentil. Die optische Täuschung von Fraser. Zeitschr. f. math. u. naturw. Unterr. 66 (1935) 170 ff. ??NYS -- cited by Lietzmann.

Nelson F. Beeler & Franklyn M. Branley. Experiments in Optical Illusion. Ill. by Fred H. Lyon. Crowell, 1951, p. 42, fig. 39, is a good example of the illusion.

Lietzmann, op. cit. at 6.AJ, 1953. P. 23 is the same as above, but adds a citation to Gentil, listed above.

Leonard de Vries. The Third Book of Experiments. © 1965, probably for a Dutch edition. Translated by Joost van de Woestijne. John Murray, 1965; Carousel, 1974. Illusion 10, pp. 58-59, has a clear picture and a brief discussion.

Richard L. Gregory & Priscilla Heard. Border locking and the café wall illusion. Perception 8 (1979) 365-380. ??NYS -- described by Walker, below. [I have photos of the actual café wall in Bristol.]

Jearl Walker. The Amateur Scientist: The café-wall illusion, in which rows of tiles tilt that should not tilt at all. SA 259:5 (Nov 1988) 100-103. Good summary and illustrations.

6.AJ.4. STEREOGRAMS

New section, due to reading Glass's assertion as to the inventor, who is different than other names that I have seen.

Don Glass, ed. How Can You Tell if a Spider is Dead? and More Moments of Science. Indiana Univ Press, Bloomington, Indiana, 1996. Now you see it, now you don't, pp. 131-132. Asserts that Christopher Tyler, of the Smith-Kettlewell Eye Research Institute, San Francisco, is the inventor of stereograms.

6.AJ.5. IMPOSSIBLE CRATE.

This is like a Necker Cube where all the edges are drawn as wooden slats in an impossible configuration.

Escher. Man with Cuboid, which is essentially a detail from Belvedere, both 1958, are apparently the first examples of this impossible object.

Chuck Mathias. Letter Mad Magazine 95 (Jun 1965) 2. Gives an impossible crate.

Jerry Andrus developed his actual model in 1981 and it appeared on the cover of Omni in 1981. But Al Seckel's exhibition says the first physical example was The Feemish Crate, due to C. F. Cochran.

Seckel, 2002a, figs. 27 A&B, pp. 36-37 (= 2002b, figs. 169 A&B, pp. 186-187), shows and discusses Andrus' crate from two viewpoints.

6.AK. POLYGONAL PATH COVERING N x N LATTICE OF POINTS,

QUEEN'S TOURS, ETC.

For magic circuits, see 7.N.4.

3x3 problem: Loyd (1907), Pearson, Anon., Bullivant, Goldston, Loyd (1914), Blyth, Abraham, Hedges, Evans, Doubleday - 1, Piggins & Eley

4x4 problem: King, Abraham, M. Adams, Evans, Depew, Meyer, Ripley's,

Queen's tours: Loyd (1867, 1897, 1914), Loyd Jr.

Bishop's tours: Dudeney (1932), Doubleday - 2, Obermair

Rook's tours: Loyd (1878), Proctor, Loyd (1897), Bullivant, Loyd (1914), Filipiak, Hartswick, Barwell, Gardner, Peters, Obermair

Other versions: Prout, Doubleday - 1

Trick solutions: Fixx, Adams, Piggins, Piggins & Eley

Thanks to Heinrich Hemme for pointing out Fixx, which led to adding most of the material on trick solutions.

Loyd. ??Le Sphinx (Mar 1867 -- but the Supplement to Sam Loyd and His Chess Problems corrects this to 15 Nov 1866). = Chess Strategy, Elizabeth, NJ, 1878, no. or p. 336(??). = A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; no. 40, pp. 42-43. Queen's circuit on 8 x 8 in 14 segments. (I.e. closed circuit, not leaving board, using queen's moves.) No. 41 & 42 of White give other solutions. White quotes Loyd from Chess Strategy, which indicates that Loyd invented this problem. Tit-Bits No. 31 & SLAHP: Touring the chessboard, pp. 19 & 89, give No. 41.

Loyd. Chess Strategy, 1878, op. cit. above, no. or p. 337 (??) (= White, 1913, op. cit. above, no. 43, pp. 42-43.) Rook's circuit on 8 x 8 in 16 segments. (I.e. closed circuit, not leaving board, using rook's moves, and without crossings.)

Richard A. Proctor. Gossip column. Knowledge 10 (Dec 1886) 43 & (Feb 1887) 92. 6 x 6 array of cells. Prisoner in one corner can exit from the opposite corner if he passes "once, and once only, through all the 36 cells." "... take the prisoner into either of the cells adjoining his own, and back into his own, .... This puzzle is rather a sell, ...." Letter and response [in Gossip column, Knowledge 10 (Mar 1887) 115-116] about the impossibility of any normal solution.

Loyd. Problem 15: The gaoler's problem. Tit-Bits 31 (23 Jan & 13 Feb 1897) 307 & 363. Rook's circuit on 8 x 8 in 16 segments, but beginning and ending on a central square. Cf The postman's puzzle in the Cyclopedia, 1914.

Loyd. Problem 16: The captive maiden. Tit-Bits 31 (30 Jan & 20 Feb 1897) 325 & 381. Rook's tour in minimal number of moves from a corner to the diagonally opposite corner, entering each cell once. Because of parity, this is technically impossible, so the first two moves are into an adjacent cell and then back to the first cell, so that the first cell has now been entered.

Loyd. Problem 20: Hearts and darts. Tit-Bits 31 (20 Feb, 13 & 20 Mar 1897) 381, 437, 455. Queen's tour on 8 x 8, starting in a corner, permitting crossings, but with no segment going through a square where the path turns. Solution in 14 segments. This is No. 41 in White -- see the first Loyd entry above.

Ball. MRE, 4th ed., 1905, p. 197. At the end of his section on knight's tours, he states that there are many similar problems for other kinds of pieces.

Loyd. In G. G. Bain, op. cit. in 1, 1907. He gives the 3 x 3 lattice in four lines as the Columbus Egg Puzzle.

Pearson. 1907. Part I, no. 36: A charming puzzle, pp. 36 & 152-153. 3 x 3 lattice in 4 lines.

Loyd. Sam Loyd's Puzzle Magazine (Apr 1908) -- ??NYS, reproduced in: A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; no. 56, p. 52. = Problem 26: A brace of puzzles -- No. 26: A study in naval warfare; Tit-Bits 31 (27 Mar 1897) 475 & 32 (24 Apr 1897) 59. = Cyclopedia, 1914, Going into action, pp. 189 & 364. = MPSL1, prob. 46, pp. 44 & 138. = SLAHP: Bombs to drop, pp. 86 & 119. Circuit on 8 x 8 in 14 segments, but with two lines of slope 2. In White, p. 43, Loyd says an ordinary queen's tour can be started "from any of the squares except the twenty which can be represented by d1, d3 and d4." This problem starts at d1. However I think White must have mistakenly put down twenty for twelve??

Anon. Prob. 67. Hobbies 31 (No. 782) (8 Oct 1910) 39 & (No. 785) (29 Oct 1910) 94. 3 x 3 lattice in 4 lines "brought under my notice some time back".

C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV.

No. 1: The travelling draught-man, pp. 515 & 520. Rook's circuit on 8 x 8 in 16 segments, different than Loyd's.

No. 3: Joining the rings. 3 x 3 in 4 segments.

Will Goldston. More Tricks and Puzzles without Mechanical Apparatus. The Magician Ltd., London, nd [1910?]. (BMC lists Routledge & Dutton eds. of 1910.) (There is a 2nd ed., published by Will Goldston, nd [1919].) The nine-dot puzzle, pp. 127-128 (pp. 90-91 in 2nd ed.).

Loyd. Cyclopedia, 1914, pp. 301 & 380. = MPSL2, prob. 133 -- Solve Christopher's egg tricks, pp. 93 & 163 (with comment by Gardner). c= SLAHP: Milkman's route, pp. 34 & 96. 3 x 3 case.

Loyd. Cyclopedia, 1914, pp. 293 & 379. Queen's circuit on 7 x 7 in 12 segments.

Loyd. The postman's puzzle. Cyclopedia, 1914, pp. 298 & 379. Rook's circuit on 8 x 8 array of points, with one point a bit out of line, starting and ending at a central square, in 16 segments. P. 379 also shows another 8 x 8 circuit, but with a slope 2 line. See also pp. 21 & 341 and SLAHP, pp. 85 & 118, for two more examples.

Loyd. Switchboard problem. Cyclopedia, 1914, pp. 255 & 373. (c= MPSL2, prob. 145, pp. 102 & 167.) Rook's tour with minimum turning.

Blyth. Match-Stick Magic. 1921. Four-way game, pp. 77-78. 3 x 3 in 4 segments.

King. Best 100. 1927. No. 16, pp. 12 & 43. 4 x 4 in 6 segments, not closed, but easily can be closed.

Loyd Jr. SLAHP. 1928. Dropping the mail, pp. 67 & 111. 4 x 4 queen's tour in 6 segments.

Collins. Book of Puzzles. 1927. The star group puzzle, pp. 95-96. 3 x 3 in 4 segments.

Dudeney. PCP. 1932. Prob. 264: The fly's tour, pp. 82 & 169. = 536, prob. 422, pp. 159 & 368. Bishop's path, with repeated cells, going from corner to corner in 17 segments.

Abraham. 1933. Probs. 101, 102, 103, pp. 49 & 66 (30 & 118). 3 x 3, 4 x 4 and 6 x 6 cases.

The Bile Beans Puzzle Book. 1933. No. 4: The puzzled milkman. 3 x 3 array in four lines.

Sid G. Hedges. More Indoor and Community Games. Methuen, London, 1937. Nine spot, p. 110. 3 x 3. "Of course it can be done, but it is not easy." No solution given.

M. Adams. Puzzle Book. 1939. Prob. C.64: Six strokes, pp. 140 & 178. 4 x 4 array in 6 segments which form a closed path, though the closure was not asked for.

J. R. Evans. The Junior Week-End Book. Op. cit. in 6.AF. 1939. Probs. 30 & 31, pp. 264 & 270. 3 x 3 & 4 x 4 cases in 4 & 6 segments, neither closed nor staying within the array.

Depew. Cokesbury Game Book. 1939. Drawing, p. 220. 4 x 4 in 6 segments, not closed, not staying within the array.

Meyer. Big Fun Book. 1940. Right on the dot, pp. 99 & 732. 4 x 4 in 6 segments.

A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in 5.H.1, pp. 50-51. Same as Bullivant, but opens the circuit to make a 15 segment path.

M. S. Klamkin, proposer and solver; John L. Selfridge, further solver. Problem E1123 -- Polygonal path covering a square lattice. AMM 61 (1954) 423 & 62 (1955) 124 & 443. Shows N x N can be done in 2N-2 segments. Selfridge shows this is minimal.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Joining the stars, pp. 41 & 129. 5 x 5 array of points. Using a line of four segments, pass through 17 points. This is a bit like the 3 x 3 problem in that one must go outside the array.

R. E. Miller & J. L. Selfridge. Maximal paths on rectangular boards. IBM J. Research and Development 4:5 (Nov 1960) 479-486. They study rook's paths where a cell is deemed visited if the rook changes direction there. They find maximal such paths in all cases.

Ripley's Puzzles and Games. 1966. Pp. 72-73, item 2. 4 x 4 cases with closed solution symmetric both horizontally and vertically.

F. Gregory Hartswick. In: H. A. Ripley & F. Gregory Hartswick, Detectograms and Other Puzzles, Scholastic Book Services, NY, 1969. Prob. 4, pp. 42-43 & 82. Asks for 8 x 8 rook's circuit with minimal turning and having a turn at a central cell. Solution gives two such with 16 segments and asserts there are no others.

Doubleday - 1. 1969. Prob. 60: Test case, pp. 75 & 167. = Doubleday - 4, pp. 83-84. Two 3 x 3 arrays joined at a corner, looking like the Fore and Aft board (cf 5.R.3), to be covered in a minimum number of segments. He does it in seven segments by joining two 3 x 3 solutions.

Brian R. Barwell. Arrows and circuits. JRM 2 (1969) 196-204. Introduces idea of maximal length rook's tours. Shows the maximal length on a 4 x 4 board is 38 and finds there are 3 solutions. Considers also the 1 x n board.

Solomon W. Golomb & John L. Selfridge. Unicursal polygonal paths and other graphs on point lattices. Pi Mu Epsilon J. 5 (1970) 107-117. Surveys problem. Generalizes Selfridge's 1955 proof to M x N for which MIN(2M, M+N-2) segments occur in a minimal circuit.

Doubleday - 2. 1971. Path finder, pp. 95-96. Bishop's corner to corner path, same as Dudeney, 1932.

James F. Fixx. More Games for the Superintelligent. (Doubleday, 1972); Muller, (1977), 1981. 6. Variation on a variation, pp. 31 & 87. Trick solution in three lines, assuming points of finite size.

M. Gardner. SA (May 1973) c= Knotted, chap. 6. Prob. 1: Find rook's tours of maximum length on the 4 x 4 board. Cites Barwell. Knotted also cites Peters, below.

Edward N. Peters. Rooks roaming round regular rectangles. JRM 6 (1973) 169-173. Finds maximum length on 1 x N board is N2/2 for N even; (N-1)2/2 + N-1 for N odd, and believes he has counted such tours. He finds tours on the N x N board whose length is a formula that reduces to 4 BC(N+1, 3) - 2[(N-1)/2]. I am a bit unsure if he has shown that this is maximal.

James L. Adams. Conceptual Blockbusting. Freeman, 1974, pp. 16-22. 3rd ed., (A-W, 1986), Penguin, 1987, pp. 24-33. Trick solution of 3 x 3 case in three lines, assuming points of finite size, which he says was submitted anonymously when he and Bob McKim used the puzzle on an ad for a talk on problem-solving at Stanford. Also describes a version using paperfolding to get all nine points into a line. The material is considerably expanded in the 3rd ed. and adds several new versions. From the references in Piggins and Eley, it seems that these all appeared in the 2nd ed of 1979 -- ??NYS.

Cut out the 3 x 1 parts and tape them into a straight line.

Take the paper and roll it to a cylinder and then draw a slanting line on the cylinder which goes through all nine, largish, points.

Cut out bits with each point on and skewer the lot with a pencil.

Place the paper on the earth and draw a line around the earth to go through all nine points. One has to assume the points have some size.

Wodge the paper, with large dots, into a ball and stick a pencil through it. Open up to see if you have won -- if not, try again!

Use a very fat line, i.e. as thick as the spacing between the edges of the array.

David J. Piggins. Pathological solutions to a popular puzzle. JRM 8:2 (1975-76) 128-129. Gives two trick solutions.

Three parallel lines, since they meet at infinity.

Put the figure on the earth and use a slanting line around the earth. This works in the limit, but otherwise requires points of finite size, a detail that he doesn't mention.

No references for these versions.

David J. Piggins & Arthur D. Eley. Minimal path length for covering polygonal lattices: A review. JRM 14:4 (1981-82) 279-283. Mostly devoted to various trick solutions of the 3 x 3 case. They cite Piggins' solution with three parallel lines. They say that Gardner sent them the trick solution in 1973 and then cite Adams, 1979. They give solutions using points of different sizes, getting both three and two segment solutions and mention a two segment version that depends on the direction of view. They then give the solution on a sphere, citing Adams, 1979, and Piggins. They give several further versions using paper folding, including putting the surface onto a twisted triangular prism joined at the ends to make the surfaces into a Möbius strip -- Zeeman calls this a umbilical bracelet or a Möbius bar.

Obermair. Op. cit. in 5.Z.1. 1984.

Prob. 19, pp. 23 & 50. Bishop's path on 8 x 8 in 17 segments, as in Dudeney, PCP, 1932.

Prob. 41, p. 72. Rook's path with maximal number of segments, which is 57. [For the 2 x 2, 3 x 3, 4 x 4 boards, I get the maximum numbers are 3, 6, 13.]

Nob Yoshigahara. Puzzlart. Tokyo, 1992. Section: The wisdom of Solomon, pp. 40-47, abridged from an article by Solomon W. Golomb in Johns Hopkins Magazine (Oct 1984). Classic 3 x 3 problem. For the 4 x 4 case: 1) find four closed paths; 2), says there are about 30 solutions and gives 19 beyond the previous 4. Find the unique 5-segment closed path on the 3 x 4. Gives 3 solutions on 5 x 5. 10-segment solution on 6 x 6 which stays on the board. Loyd's 1867? Queen's circuit. Queen's circuit on 7 x 7, attributed to Dudeney, though my earliest entry is Loyd, 1914 -- ??CHECK.

6.AL. STEINER-LEHMUS THEOREM

This has such an extensive history that I will give only a few items.

C. L. Lehmus first posed the problem to Jacob Steiner in 1840.

Rougevin published the first proof in 1842. ??NYS.

Jacob Steiner. Elementare Lösung einer Aufgabe über das ebene und sphärische Dreieck. J. reine angew. Math. 28 (1844) 375-379 & Tafel III. Says Lehmus sent it to him in 1840 asking for a purely geometric proof. Here he gives proofs for the plane and the sphere and also considers external bisectors.

Theodor Lange. Nachtrag zu dem Aufsatze in Thl. XIII, Nr. XXXIII. Archiv der Math. und Physik 15 (1850) 221-226. Discusses the problem and gives a solution by Steiner and two by C. L. Lehmus. Steiner also considers the external bisectors.

N. J. Chignell. Note 1031: A difficult converse. MG 16 (No. 219) (Jul 1932) 200-202. [The author's name is omitted in the article but appears on the cover.] 'Three fairly simple proofs', due to: M. J. Newell; J. Travers, improving J. H. Doughty, based on material in Lady's and Gentleman's Diary (1859) 87-88 & (1860) 84-86; Wm. Mason, found by Doughty, in Lady's and Gentleman's Diary (1860) 86.

H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 1.5, ex. 4, p. 16. An easy proof is posed as a problem with adequate hints in four lines.

M. Gardner. SA (Apr 1961) = New MD, chap. 17. Review of Coxeter's book, saying his brief proof came as a pleasant shock.

G. Gilbert & D. MacDonnell. The Steiner-Lehmus theorem. AMM 70 (1963) 79-80. This is the best of the proofs sent to Gardner in response to his review of Coxeter. A later source says this turned out to be identical to Lehmus' original proof!

Léo Sauvé. The Steiner-Lehmus theorem. CM 2:2 (Feb 1976) 19-24. Discusses history and gives 22 references, some of which refer to 60 proofs.

Charles W. Trigg. A bibliography of the Steiner-Lehmus theorem. CM 2:9 (Nov 1976) 191-193. 36 references beyond Sauvé's.

David C. Kay. Nearly the last comment on the Steiner-Lehmus theorem. CM 3:6 (1977) 148-149. Observes that a version of the proof works in all three classical geometries at once and gives its history.

6.AM. MORLEY'S THEOREM

This also has an extensive history and I give only a few items.

T. Delahaye and H. Lez. Problem no. 1655 (Morley's triangle). Mathesis (3) 8 (1908) 138-139. ??NYS.

E. J. Ebden, proposer; M. Satyanarayana, solver. Problem no. 16381 (Morley's theorem). The Educational Times (NS) 61 (1 Feb 1908) 81 & (1 Jul 1908) 307-308 = Math. Quest. and Solutions from "The Educational Times" (NS) 15 (1909) 23. Asks for various related triangles formed using interior and exterior trisectors to be shown equilateral. Solution is essentially trigonometric. No mention of Morley.

Frank Morley. On the intersections of the trisectors of the angles of a triangle. (From a letter directed to Prof. T. Hayashi.) J. Math. Assoc. of Japan for Secondary Education 6 (Dec 1924) 260-262. (= CM 3:10 (Dec 1977) 273-275.

Frank Morley. Letter to Gino Loria. 22 Aug 1934. Reproduced in: Gino Loria; Triangles équilatéraux dérivés d'un triangle quelconque. MG 23 (No. 256) (Oct 1939) 364-372. Morley says he discovered the theorem in c1904 and cites the letter to Hayashi. Loria mentions other early work and gives several generalizations.

H. F. Baker. Note 1476: A theorem due to Professor F. Morley. MG 24 (No. 261) (Oct 1940) 284-286. Easy proof and reference to other proofs. He cites a related result of Steiner.

Anonymous [R. P.] Morley's trisector theorem. Eureka 16 (Oct 1953) 6-7. Short proof, working backward from the equilateral triangle.

Dan Pedoe. Notes on Morley's proof of his theorem on angle trisectors. CM 3:10 (Dec 1977) 276-279. "... very tentative ... first steps towards the elucidation of his work."

C. O. Oakley & Charles W. Trigg. A list of references to the Morley theorem. CM 3:10 (Dec 1977) 281-290 & 4 (1978) 132. 169 items.

André Viricel (with Jacques Bouteloup). Le Théorème de Morley. L'Association pour le Développement de la Culture Scientifique, Amiens, 1993. [This publisher or this book was apparently taken over by Blanchard as Blanchard was selling copies with his label pasted over the previous publisher's name in Dec 1994.] A substantial book (180pp) on all aspects of the theorem. The bibliography is extremely cryptic, but says it is abridged from Mathesis (1949) 175 ??NYS. The most recent item cited is 1970.

6.AN. VOLUME OF THE INTERSECTION OF TWO CYLINDERS

Archimedes. The Method: Preface, 2. In: T. L. Heath; The Works of Archimedes, with a supplement "The Method of Archimedes"; (originally two works, CUP, 1897 & 1912) = Dover, 1953. Supplement, p. 12, states the result. The proof is lost, but pp. 48-51 gives a reconstruction of the proof by Zeuthen.

Liu Hui. Jiu Zhang Suan Chu Zhu (Commentary on the Nine Chapters of the Mathematical Art). 263. ??NYS -- described in Li & Du, pp. 73-74 & 85. He shows that the ratio of the volume of the sphere to the volume of Archimedes' solid, called mou he fang gai (two square umbrellas), is π/4, but he cannot determine either volume.

Zu Geng. c500. Lost, but described in: Li Chunfeng; annotation to Jiu Zhang (= Chiu Chang Suan Ching) made c656. ??NYS. Described on pp. 86-87 of: Wu Wenchun; The out-in complementary principle; IN:  Ancient China's Technology and Science; compiled by the Institute of the History of Natural Sciences, Chinese Academy of Sciences; Foreign Languages Press, Beijing, 1983, pp. 66-89. [This is a revision and translation of parts of: Achievements in Science and Technology in Ancient China [in Chinese]; China Youth Publishing House, Beijing(?), 1978.]

He considers the shape, called fanggai, within the natural circumscribed cube and shows that, in each octant, the part of the cube outside the fanggai has cross section of area h2 at distance h from the centre. This is equivalent to a tetrahedron, whose volume had been determined by Liu, so the excluded volume is ⅓ of the cube.

Li & Du, pp. 85-87, and say the result may have been found c480 by Zu Geng's father, Zu Chongzhi.

Lam Lay-Yong & Shen Kangsheng. The Chinese concept of Cavalieri's Principle and its applications. HM 12 (1985) 219-228. Discusses the work of Liu and Zu.

Shiraishi Chōchū. Shamei Sampu. 1826. ??NYS -- described in Smith & Mikami, pp. 233-236. "Find the volume cut from a cylinder by another cylinder that intersects is orthogonally and touches a point on the surface". I'm not quite sure what the last phrase indicates. The book gives a number of similar problems of finding volumes of intersections.

P. R. Rider, proposer; N. B. Moore, solver. Problem 3587. AMM 40 (1933) 52 (??NX) & 612. Gives the standard proof by cross sections, then considers the case of unequal cylinders where the solution involves complete elliptic integrals of the first and second kinds. References to solution and similar problem in textbooks.

Leo Moser, solver; J. M. Butchart, extender. MM 25 (May 1952) 290 & 26 (Sep 1952) 54. ??NX. Reproduced in Trigg, op. cit. in 5.Q: Quickie 15, pp. 6 & 82-83. Moser gives the classic proof that V = 16r3/3. Butchart points out that this also shows that the shape has surface area 16r2.

6.AO. CONFIGURATION PROBLEMS

NOTATION: (a, b, c) denotes the configuration of a points in b rows of c each. The index below covers articles other than the surveys of Burr et al. and Gardner.

( 5, 2, 3): Sylvester

( 6, 3, 3): Mittenzwey

( 7, 6, 3): Criton

( 9, 8, 3): Sylvester; Carroll; Criton

( 9, 9, 3): Carroll; Bridges; Criton

( 9, 10, 3): Jackson; Family Friend; Parlour Pastime; Magician's Own Book; The Sociable; Book of 500 Puzzles; Charades etc.; Boy's Own Conjuring Book; Hanky Panky; Carroll; Crompton; Berkeley & Rowland; Hoffmann; Dudeney (1908); Wehman; Williams; Loyd Jr; Blyth; Rudin; Young World; Brooke; Putnam; Criton

(10, 5, 4): The Sociable; Book of 500 Puzzles; Carroll; Hoffmann; Dudeney (1908); Wehman; Williams; Dudeney (1917); Blyth; King; Rudin; Young World; Hutchings  & Blake; Putnam

(10, 10, 3): Sylvester

(11, 11, 3): The Sociable; Book of 500 Puzzles; Wehman

(11, 12, 3): Hoffmann; Williams; Young World

(11, 13, 3): Prout

(11, 16, 3): Wilkinson -- in Dudeney (1908 & 1917); Macmillan

(12, 4, 5) -- Trick version of a hollow 3 x 3 square with doubled corners, as in 7.Q: Family Friend (1858); Secret Out; Illustrated Boy's Own Treasury;

(12, 6, 4): Endless Amusement II; The Sociable; Book of 500 Puzzles; Boy's Own Book; Cassell's; Hoffmann; Wehman; Rudin; Criton

(12, 7, 4) -- Trick version of a 3 x 3 square with doubled diagonal: Secret Out; Hoffmann (1876); Mittenzwey; Hoffmann (1893), no. 8

(12, 7, 4): Dudeney (1917); Putnam

(12, 19, 3): Macmillan

(13, 9, 4): Criton

(13, 12, 3): Criton

(13, 18, 3): Sylvester

(13, 22, 3): Criton

(15, 15, 3): Jackson

(15, 16, 3): The Sociable; Book of 500 Puzzles; H. D. Northrop; Wehman

(15, 23, 3): Jackson

(15, 26, 3): Woolhouse

(16, 10, 4): The Sociable; Book of 500 Puzzles; Hoffmann; Wehman

(16, 12, 4): Criton

(16, 15, 4): Dudeney (1899, 1902, 1908); Brooke; Putnam; Criton

(17, 24, 3): Jackson

(17, 28, 3): Endless Amusement II; Pearson

(17, 32, 3): Sylvester

(17, 7, 5): Ripley's

(18, 18, 4): Macmillan

(19, 19, 4): Criton

(19, 9, 5): Endless Amusement II; The Sociable; Book of 500 Puzzles; Proctor; Hoffmann; Clark; Wehman; Ripley; Rudin; Putnam; Criton

(19, 10, 5): Proctor

(20, 12, 5): trick method: Doubleday - 3

(20, 18, 4): Loyd Jr

(20, 21, 4): Criton

(21, 9, 5): Magician's Own Book; Book of 500 Puzzles; Boy's Own Conjuring Book; Blyth; Depew

(21, 10, 5): Mittenzwey

(21, 11, 5): Putnam

(21, 12, 5): Dudeney (1917); Criton

(21, 30, 3): Secret Out; Hoffmann

(21, 50, 3): Sylvester

(22, 15, 5): Macmillan

(22, 20, 4): Dudeney (1899)

(22, 21, 4): Dudeney (1917); Putnam

(24, 28, 3): Jackson; Parlour Pastime

(24, 28, 4): Jackson; Héraud; Benson; Macmillan

(24, 28, 5): Jackson

(25, 12, 5): Endless Amusement II; Young Man's Book; Proctor; Criton

(25, 18, 5): Bridges

(25, 30, 4): Macmillan

(25, 72, 3): Sylvester

(26, 21, 5): Macmillan

(27, 9, 6): The Sociable; Book of 500 Puzzles; Hoffmann; Wehman

(27, 10, 6): The Sociable; Book of 500 Puzzles; Wehman

(27, 15, 5): Jackson

(29, 98, 3): Sylvester

(30, 12, 7): Criton

(30, 22, 5): Criton

(30, 26, 5): Macmillan

(31, 6, 6) -- with 7 circles of 6: The Sociable; Book of 500 Puzzles; Magician's Own Book (UK version); Wehman

(31, 15, 5): Proctor

(36, 55, 4): Macmillan

(37, 18, 5): Proctor

(37, 20, 5): The Sociable; Book of 500 Puzzles; Illustrated Boy's Own Treasury; Hanky Panky; Wehman

(49, 16, 7): Criton

Trick versions -- with doubled counters: Family Friend (1858); Secret Out; Illustrated Boy's Own Treasury; Hoffmann (1876); Mittenzwey; Hoffmann (1893), nos. 8 & 9; Pearson; Home Book ....; Doubleday - 3. These could also be considered as in 7.Q.2 or 7.Q.

A different type of configuration problem is considered by Shepherd, 1947.

Jackson. Rational Amusement. 1821. Trees Planted in Rows, nos. 1-10, pp. 33-34 & 99-100 and plate IV, figs. 1-9. [Brooke and others say this is the earliest statement of such problems.]

1. (9, 10, 3). Quoted in Burr, below.

"Your aid I want, nine trees to plant

In rows just half a score;

And let there be in each row three.

Solve this: I ask no more."

2. (n, n, 3), He does the case n = 15.

3. (15, 23, 3).

4. (17, 24, 3).

5. (24, 24, 3) with a pond in the middle.

6. (24, 28, 4).

7. (27, 15, 5)

8. (25, 28, c) with c = 3, 4, 5.

9. (90, 10, 10) with equal spacing -- decagon with 10 trees on each side.

10. Leads to drawing square lattice in perspective with two vanishing points, so the diagonals of the resulting parallelograms are perpendicular.

Endless Amusement II. 1826?

Prob. 13, p. 197. (19, 9, 5). = New Sphinx, c1840, p. 135.

Prob. 14, p. 197. (12, 6, 4). = New Sphinx, c1840, p. 135.

Prob. 26, p. 202. (25, 12, 5). Answer is a 5 x 5 square array.

Ingenious artists, how may I dispose

Of five-and-twenty trees, in just twelve rows;

That every row five lofty trees may grace,

Explain the scheme -- the trees completely place.

Prob. 35, p. 212. (17, 28, 3). [This is the problem that is replaced in the 1837 ed.]

Young Man's Book. 1839. P. 239. Identical to Endless Amusement II.

Crambrook. 1843. P. 5, no. 15: The Puzzle of the Steward and his Trees. This may be a configuration problem -- ??

Boy's Own Book. 1843 (Paris): 438 & 442, no. 15: "Is it possible to place twelve pieces of money in six rows, so as to have four in each row?" I. e. (12, 6, 5). = Boy's Treasury, 1844, pp. 426 & 429, no. 13. = de Savigny, 1846, pp. 355 & 358, no. 11.

Family Friend 1 (1849) 148 & 177. Family Pastime -- Practical Puzzles -- 1. The puzzle of the stars. (9, 10, 3).

Friends of the Family Friend, pray show

How you nine stars would so bestow

Ten rows to form -- in each row three --

Tell me, ye wits, how this can be?

Robina.

Answer has

Good-tempered Friends! here nine stars see:

Ten rows there are, in each row three!

W. S. B. Woolhouse. Problem 39. The Mathematician 1 (1855) 272. Solution: ibid. 2 (1856) 278-280. ??NYS -- cited in Burr, et al., below, who say he does (15, 26, 3).

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles.

No. 1, p. 176 (1868: 187). (9, 10, 3).

Ingenious artist pray disclose,

How I nine trees can so dispose,

That these ten rows shall formed be,

And every row consist of three?

No. 12, p. 182 (1868: 192-193). (24, 28, 3), but with a central pond breaking 4 rows of 6 into 8 rows of 3.

Magician's Own Book. 1857.

Prob. 33: The puzzle of the stars, pp. 277 & 300. (9, 10, 3),

Friends one and all, I pray you show

How you nine stars would so bestow,

Ten rows to form -- in each row three --

Tell me, ye wits, how this can be?

Prob. 41: The tree puzzle, pp. 279 & 301. (21, 9, 5), unequally spaced on each row. Identical to Book of 500 Puzzles, prob. 41.

The Sociable. 1858. = Book of 500 Puzzles, 1859, with same problem numbers, but page numbers decreased by 282.

Prob. 3: The practicable orchard, pp. 286 & 302. (16, 10, 4).

Prob. 8: The florist's puzzle, pp. 289 & 303-304. (31, 6, 6) with 7 circles of 6.

Prob. 9: The farmer's puzzle, pp. 289 & 304. (11, 11, 3).

Prob. 12: The geometrical orchard, p. 291 & 306. (27, 9, 6).

Prob. 17: The apple-tree puzzle, pp. 292 & 308. (10, 5, 4).

Prob. 22: The peach orchard puzzle, pp. 294 & 309. (27, 10, 6).

Prob. 26: The gardener's puzzle, pp. 295 & 311. (12, 6, 4) two ways.

Prob. 27: The circle puzzle, pp. 295 & 311. (37, 20, 5) equally spaced along each row.

Prob. 29: The tree puzzle, pp. 296 & 312. (15, 16, 3) with some bigger rows. Solution is a 3 x 4 array with three extra trees halfway between the points of the middle line of four.

Prob. 32: The tulip puzzle, pp. 296 & 314. (19, 9, 5).

Prob. 36: The plum tree puzzle, pp. 297 & 315. (9, 10, 3).

Family Friend (Dec 1858) 359. Practical puzzles -- 2. "Make a square with twelve counters, having five on each side." (12, 4, 5). I haven't got the answer, but presumably it is the trick version of a hollow square with doubled corners, as in 7.Q. See Secret Out, 1859 & Illustrated Boy's Own Treasury, 1860.

Book of 500 Puzzles. 1859. Prob. 3, 9, 12, 17, 22, 26, 27, 29, 32, 36 are identical to those in The Sociable, with page numbers decreased by 282.

Prob. 33: The puzzle of the stars, pp. 91 & 114. (9, 10, 3), identical to Magician's Own Book, prob. 33.

Prob. 41: The tree puzzle, pp. 93 & 115. (21, 9, 5), identical to Magician's Own Book, prob. 41. See Illustrated Boy's Own Treasury.

The Secret Out. 1859.

To place twelve Cards in such a manner that you can count Four in every direction, p. 90. (12, 7, 4) trick of a 3 x 3 array with doubling along a diagonal. 'Every direction' must refer to just the rows and columns, but one diagonal also works.

The magical arrangement, pp. 381-382 = The square of counters, (UK) p. 9. (12, 4, 5) -- trick version. Same as Family Friend & Illustrated Boy's Own Treasury, prob. 13.

The Sphynx, pp. 385-386. (21, 30, 3). = Hoffmann, no. 15.

Charades, Enigmas, and Riddles. 1860: prob. 13, pp. 58 & 61; 1862: prob. 13, pp. 133 & 139; 1865: prob. 557, pp. 105 & 152. (9, 10, 3). (The 1862 and 1865 have slightly different typography.)

Sir Isaac Newton's Puzzle (versified).

Ingenious Artist, pray disclose

How I, nine Trees may so dispose,

That just Ten Rows shall planted be,

And every Row contain just Three.

Boy's Own Conjuring Book. 1860.

Prob. 40: The tree puzzle, pp. 242 & 266. (21, 9, 5), identical to Magician's Own Book, prob. 41.

Prob. 42: The puzzle of the stars, pp. 243 & 267. (9, 10, 3), identical to Magician's Own Book, prob. 33, with commas omitted.

Illustrated Boy's Own Treasury. 1860.

Prob. 2, pp. 395 & 436. (37, 20, 5), equally spaced on each row, identical to The Sociable, prob. 27.

Prob. 13, pp. 397 & 438. "Make a square with twelve counters, having five on each side." (12, 4, 5). Trick version of a hollow square with doubled corners. Presumably identical to Family Friend, 1858. Same as Secret Out.

J. J. Sylvester. Problem 2473. Math. Quest. from the Educ. Times 8 (1867) 106-107. ??NYS -- Burr, et al. say he gives (10, 10, 3), (81, 800, 3) and (a, (a-1)2/8, 3).

Magician's Own Book (UK version). 1871. The solution to The florist's puzzle (The Sociable, prob. 8) is given at the bottom of p. 284, apparently to fill out the page as there is no relevant text anywhere.

Hanky Panky. 1872.

To place nine cards in ten rows of three each, p. 291. I.e. (9, 10, 3).

Diagram with no text, p. 128. (37, 20, 5), equally spaced on each line as in The Sociable, prob. 27.

Hoffmann. Modern Magic. (George Routledge, London, 1876); reprinted by Dover, 1978. To place twelve cards in rows, in such a manner that they will count four in every direction, p. 58. Trick version of a 3 x 3 square with extras on a diagonal, giving a form of (12, 7, 4). Same as Secret Out.

Lewis Carroll. MS of 1876. ??NYS -- described in: David Shulman; The Lewis Carroll problem; SM 6 (1939) 238-240.

Given two rows of five dots, move four to make 5 rows of 4. Shulman describes this case, following Dudeney, AM, 1917, then observes that since Dudeney is using coins, there are further solutions by putting a coin on top of another. He refers to Hoffmann and Loyd. The same problem is in Carroll-Wakeling, prob. 1: Cakes in a row, pp. 1-2 & 63, but undated and the answer mentions the possibility of stacking the counters.

(9, 10, 3). Shulman quotes from Robert T. Philip; Family Pastime; London, 1852, p. 30, ??NYS, but this must refer to the item in Family Friend, which was edited by Robert Kemp Philp. BMC indicates Family Pastime which may be another periodical. Shulman then cites Jackson and Dudeney. Carroll-Wakeling, prob. 2: More cakes in a row, pp. 3 & 63, gives the problems (9, 8, 3), (9, 9, 3), (9, 10, 3), undated.

Mittenzwey. 1880.

Prob. 151, pp. 31 & 83; 1895?: 174, pp. 36 & 85; 1917: 174, pp. 33 & 82. (6, 3, 3) in three ways.

Prob. 152, pp. 31 & 83; 1895?: 175, pp. 36 & 85; 1917: 175, pp. 33 & 82. Arrange 16 pennies as a 3 x 3 square so each row and column has four in it. Solution shows a 3 x 3 square with extras on the diagonal -- but this only uses 12 pennies! So this the trick version of (12, 7, 4) as in Secret Out & Hoffmann (1876).

Prob. 153, pp. 31 & 83; 1895?: 176, pp. 36 & 85; 1917: 176, pp. 33 & 82. (21, 10, 5).

Cassell's. 1881. P. 92: The six rows puzzle. = Manson, 1911, p. 146.

J. J. Sylvester. Problem 2572. Math. Quest. from the Educ. Times 45 (1886) 127-128. ??NYS -- cited in Burr, below. Obtains good examples of (a, b, 3) for each a. In most cases, this is still the best known.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. (19, 9, 5). Generalises to (6n+1, 3n, 5).

Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40. Gives several solutions of (19, 9, 5) and asks for (19, 10, 5). Gossip column, (Feb 1887) 92, gives another solution

William Crompton. The odd half-hour. The Boy's Own Paper 13 (No. 657) (15 Aug 1891) 731-732. Sir Isaac Newton's puzzle (versified). (9, 10, 3).

Ingenious artist pray disclose

How I nine trees may so dispose

That just ten rows shall planted be

And every row contain just three.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. IV, p. 3. (9, 10, 3).

Hoffmann. 1893. Chap. VI, pp. 265-268 & 275-281 = Hoffmann-Hordern, pp. 174-182, with photo.

No. 1: (11, 12, 3).

No. 2: (9, 10, 3).

No. 3: (27, 9, 6).

No. 4: (10, 5, 4).

No. 5: (12, 6, 4). Photo on p. 177 shows L'Embarras du Brigadier, by Mauclair-Dacier, 1891-1900, which has a board with a 7 x 6 array of holes and 12 pegs. The horizontal spacing seems closer than the vertical spacing.

No. 6: (19, 9, 5).

No. 7: (16, 10, 4).

No. 8: (12, 7, 4) -- Trick version of a 3 x 3 square with extras on a diagonal as in Secret Out, Hoffmann (1876) & Mittenzwey.

No. 9: 9 red + 9 white, form 10 + 8 lines of 3 each. Puts a red and a white point at the same place, so this is a trick version.

No. 11: (10, 8, 4) -- counts in 8 'directions', so he counts each line twice!

No. 12: (13, 12, 5) -- with double counting as in no. 11.

No. 15: (21, 30, 3) -- but points must lie on a given figure, which is the same as in The Secret Out.

Clark. Mental Nuts. 1897, no. 19: The apple orchard; 1904, no. 91: The lovers' grove. (19, 9, 5). 1897 just has "Place an orchard of nineteen trees so as to have nine rows of five trees each." 1904 gives a poem.

I am required to plant a grove

To please the lady whom I love.

This simple grove to be composed

Of nineteen trees in nine straight rows;

Five trees in each row I must place,

Or I shall never see her face.

Cf Ripley, below.

Dudeney. A batch of puzzles. Royal Magazine 1:3 (Jan 1899) & 1:4 (Feb 1899) 368-372. (22, 20, 4) with trees at lattice points of a 7 x 10 lattice. Compare with AM, prob. 212.

Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320 & 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. (16, 15, 4). Cf 1902.

Dudeney. "The Captain" puzzle corner. The Captain 3:2 (May 1900) 179. This gives a solution of a problem called Joubert's guns, but I haven't seen the proposal. (10, 5, 4) but wants the maximum number of castles to be inside the walls joining the castles. Manages to get two inside. = Dudeney; The puzzle realm; Cassell's Magazine ?? (May 1908) 713-716; no. 6: The king and the castles. = AM, 1917, prob. 206: The king and the castles, pp. 56 & 189.

H. D. Northrop. Popular Pastimes. 1901. No. 11: The tree puzzle, pp. 68 & 73. = The Sociable, no. 29.

Dudeney. The ploughman's puzzle. In: The Canterbury Puzzles, London Magazine 9 (No. 49) (Aug 1902) 88-92 & (No. 50) (Sep 1902) 219. = CP; 1907; no. 21, pp. 43-44 & 175-176. (16, 15, 4). Cf 1899.

A. Héraud. Jeux et Récréations Scientifiques -- Chimie, Histoire Naturelle, Mathématiques. Baillière et Fils, Paris, 1903. P. 307: Un paradoxe mathématique. (24, 28, 4). I haven't checked for this problem in the 1884 ed.

Pearson. 1907.

Part I, no. 77: Lines on an old sampler, pp. 77 & 167. (17, 28, 3).

Part II, no. 83: For the children, pp. 83 & 177. Trick version of (12, 4, 5), as in Family Friend (1858).

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He says (9, 10, 3) "is attributed to Sir Isaac Newton, but the earliest collection of such puzzles is, I believe, in a rare little book that I possess -- published in 1821." [This must refer to Jackson.] Says Rev. Mr. Wilkinson gave (11, 16, 3) "some quarter of a century ago" and that he, Dudeney, published (16, 15, 4) in 1897 (cf under 1902 above). He leaves these as problems but doesn't give their solutions in the next issue.

Wehman. New Book of 200 Puzzles. 1908.

P. 4: The practicable orchard. (16, 10, 4). = The Sociable, prob. 3.

P. 7: The puzzle of the stars. (9, 10, 3). = Magician's Own Book, prob. 33.

P. 8: The apple-tree puzzle. (10, 5, 4). = The Sociable, prob. 17.

P. 8: The peach orchard puzzle. (27, 10, 6). = The Sociable, prob. 22.

P. 8: The plum tree puzzle. (9, 10, 3). = The Sociable, prob. 36.

P. 12: The farmer's puzzle. (11, 11, 3). = The Sociable, prob. 9.

P. 19: The gardener's puzzle. (12, 6, 4) two ways. = The Sociable, prob. 26.

P. 26: The circle puzzle. (37, 20, 5) equally spaced along each row. = The Sociable, prob. 27.

P. 30: The tree puzzle. (15, 16, 3) with some bigger rows. = The Sociable, prob. 29.

P. 31: The geometrical orchard. (27, 9, 6). = The Sociable, prob. 12.

P. 31: The tulip puzzle. (19, 9, 5). = The Sociable, prob. 32.

P. 41: The florist's puzzle. (31, 6, 6) with seven circles of six. = The Sociable, prob. 8.

J. K. Benson, ed. The Pearson Puzzle Book. C. Arthur Pearson, London, nd [c1910, not in BMC or NUC]. [This is almost identical with the puzzle section of Benson, but has 13 pages of different material.] A symmetrical plantation, p. 99. (24, 28, 4).

Williams. Home Entertainments. 1914. Competitions with counters, p. 115. (11, 12, 3); (9, 10, 3); (10, 5, 4).

Dudeney. AM. 1917. Points and lines problems, pp. 56-58 & 189-193.

Prob. 206: The king and the castles. See The Captain, 1900.

Prob. 207: Cherries and plums. Two (10, 5, 4) patterns among 55 of the points of an 8 x 8 array.

Prob. 208: A plantation puzzle. (10, 5, 4) among 45 of the points of a 7 x 7 array.

Prob. 209: The twenty-one trees. (21, 12, 5).

Prob. 210: The ten coins. Two rows of five. Move four to make (10, 5, 4). Cf Carroll, 1876. Shows there are 2400 ways to do this. He shows that there are six basic solutions of the (10, 5, 4) which he calls: star, dart, compasses, funnel, scissors, nail and he describes the smallest arrays on which they can fit.

Prob. 211: The twelve mince-pies. 12 points at the vertices and intersections of a Star of David. Move four to make (12, 7, 4).

Prob. 212: The Burmese plantation. (22, x, 4) among the points of a 7 x 7 array. Finds x = 21. Cf 1899.

Prob. 213: Turks and Russians, pp. 58 & 191-193. Complicated problem leading to (11, 16, 3) -- cites his Afridi problem in Tit-Bits and attributes the pattern to Wilkinson 'some twenty years ago', cf 1908.

Blyth. Match-Stick Magic. 1921.

Four in line, p. 48. (10, 5, 4).

Three in line, p. 77. (9, 10, 3).

Five-line game, pp. 78-79. (21, 9, 5).

King. Best 100. 1927. No. 62, pp. 26 & 54. = Foulsham's no. 21, pp. 9 & 13. (10, 5, 4).

Loyd Jr. SLAHP. 1928. Points and lines puzzle, pp. 20 & 90. Says Newton proposed (9, 10, 3). Asks for (20, 18, 4) on a 7 x 7 array.

R. Ripley. Believe It or Not! Book 2. Op. cit. in 5.E, 1931. The planter's puzzle, p. 197, asks for (19, 9, 5) but no solution is given. See Clark, above, for a better version of the verse.

"I am constrained to plant a grove

For a lady that I love.

This ample grove is too composed;

Nineteen trees in nine straight rows.

Five trees in each row I must place,

Or I shall never see her face."

Rudin. 1936. Nos. 105-108, pp. 39 & 99-100.

No. 105: (9, 10, 3).

No. 106: (10, 5, 4) -- two solutions.

No. 107: (12, 6, 4) -- two solutions.

No. 108: (19, 9, 5).

Depew. Cokesbury Game Book. 1939. The orange grower, p. 221. (21, 9, 5).

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 147, prob. 1 & 2. Place six coins in an L or a cross and make two rows of four, i.e. (6, 2, 4), which is done by the simple trick of putting a coin on the intersection.

R. H. Macmillan. Letter: An old problem. MG 30 (No. 289) (May 1946) 109. Says he believes Newton and Sylvester studied this. Says he has examples of (11, 16, 3), (12, 19, 3), (18, 18, 4), (24, 28, 4), (25, 30, 4), (36, 55, 4), (22, 15, 5), (26, 21, 5), (30, 26, 5).

G. C. Shephard. A problem in orchards. Eureka 9 (Apr 1947) 11-14. Given k points in n-dimensions, the general problem is to draw N(k, n) hyperplanes to produce k regions, each containing one point. The most common example is k = 7, n = 2, N = 3. [See Section 5.Q for determining k as a function of n and N.] The author investigates the question of determining the possible locations of the seventh point given six points. He gives a construction of a set T such that being in T is necessary and sufficient for three such lines to exist.

J. Bridges. Potter's orchard. Eureka 11 (Jan 1949) 30 & 12 (Oct 1949) 17. Start with an orchard (9, 9, 3). Add 16 trees to make (25, 18, 5). The nine trees are three points in a triangle, with the three midpoints of the sides and the three points halfway between these. Six of the new trees are one third of the way along the sides of the original triangle; another six are one third of the way along the lines joining the midpoints of the original triangle; one point is the centre of the original triangle and the last three are easily seen.

W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Thirteen rows of three, pp. 45 & 132. (11, 13, 3).

Young World. c1960. Pp. 10-11.

Three coin lines. (9, 10, 3).

Five coin lines. (10, 5, 4).

Eleven coin trick. (11, 12, 3).

Maxey Brooke. Dots and lines. RMM 6 (Dec 1961) 51-55. Cites Jackson and Dudeney. Says Sylvester showed that n points can be arranged in at least (n-1)(n-2)/6 rows of three. Shows (9, 10, 3) and (16, 15, 4).

R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. F. (10, 5, 4) with points in the centres of cells of a chess board. Actually only needs a 7 x 7 board.

Ripley's Puzzles and Games. 1966. Pp. 18-19, item 4. (17, 7, 5).

Doubleday - 3. 1972. Count down, pp. 125-126. Start with a 4 x 4 array of coins. Add four coins so that each row, column and diagonal has the same number. Solution doubles the coins in the 1, 3, 4, 2 positions in the rows.

S. A. Burr, B. Grünbaum & N. J. A. Sloane. The orchard problem. Geometria Dedicata 2 (1974) 397-424. Establishes good examples of (a, b, 3) slightly improving on Sylvester, and establishes some special better examples. Gives upper bounds for b in (a, b, 3). Sketches history and tabulates best values and upper bounds for b in (a, b, 3), for a = 1 (1) 32.

The following have the maximal possible value of b for given a and c.

(3, 1, 3); (4, 1, 3); (5, 2, 3); (6, 4, 3); (7, 6, 3); (8, 7, 3); (9, 10, 3); (10, 12, 3); (11, 16, 3); (12, 19, 3); (16, 37, 3).

The following have the largest known value of b for the given a and c.

(13, 22, 3); (14, 26, 3); (15, 31, 3); (17, 40, 3); (18, 46, 3); (19, 52, 3); (20, 57, 3); (21, 64, 3); (22, 70, 3); (23, 77, 3); (24, 85, 3); (25, 92, 3); (26, 100, 3); (27, 109, 3); (28, 117, 3); (29, 126, 3); (30, 136, 3); (31, 145, 3); (32, 155, 3).

M. Gardner. SA (Aug 1976). Surveys these problems, based on Burr, Grünbaum & Sloane. He gives results for c = 4.

The following have the maximal possible value of b for the given a and c.

(4, 1, 4); (5, 1, 4); (6, 1, 4); (7, 2, 4); (8, 2, 4); (9, 3, 4); (10, 5, 4); (11, 6, 4); (12, 7, 4).

The following have the largest known value of b for the given a and c.

(13, 9, 4); (14, 10, 4); (15, 12, 4); (16, 15, 4); (17, 15, 4); (18, 18, 4); (19, 19, 4); (20, 20, 4).

Putnam. Puzzle Fun. 1978. Nos. 17-23: Bingo arrangements, pp. 6 & 29-30. (21, 11, 5), (16, 15, 4), (19, 9, 5), (9, 10, 3), (12, 7, 4), (22, 21, 4), (10, 5, 4).

S. A. Burr. Planting trees. In: The Mathematical Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth, 1981. Pp. 90-99. Pleasant survey of the 1974 paper by Burr, et al.

Michel Criton. Des points et des Lignes. Jouer Jeux Mathématiques 3 (Jul/Sep 1991) 6-9. Survey, with a graph showing c at (a, b). Observes that some solutions have points which are not at intersections of lines and proposes a more restrictive kind of arrangement of b lines whose intersections give a points with c points on each line. He denotes these with square brackets which I write as [a, b, c]. Pictures of (7, 6, 3), [9, 8, 3], (9, 9, 3), (12, 6, 4), [13, 9, 4], (13, 12, 3), (13, 22, 3), (16, 12, 4), (19, 19, 4), (19, 19, 5), (20, 21, 4), [21, 12, 5], (25, 12, 5), (30, 12, 7), (30, 22, 5), (49, 16, 7) and mentions of (9, 10, 3), (16, 15, 4),

6.AO.1. PLACE FOUR POINTS EQUIDISTANTLY = MAKE FOUR

TRIANGLES WITH SIX MATCHSTICKS

I am adding the problem of making three squares with nine matchsticks here a it uses the same thought process -- see Mittenzwey and see the extended discussion at Anon., 1910.

Pacioli. De Viribus. c1500. Ff. 191r - 192r. LXXX. Do(cumento). commo non e possibile piu ch' tre ponti o ver tondi spere tocarse in un piano tutti (how it is not possible for more than three points or discs or spheres to all touch in a plane). = Peirani 252-253. Says you can only get three discs touching in the plane, but you can get a fourth so they are all touching by making a pyramid.

Endless Amusement II. 1826? Prob. 21, p. 200. "To place 4 poles in the ground, precisely at an equal distance from each other." Uses a pyramidal mound of earth.

Young Man's Book. 1839. P. 235. Identical to Endless Amusement II.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 6, p. 178 (1868: 189). Plant four trees at equal distances from each other.

Frank Bellew. The Art of Amusing. 1866. Op. cit. in 5.E. 1866: pp. 97-98 & 105-106; 1870: pp. 93-94 & 101-102.

Mittenzwey. 1880.

Prob. 161, pp. 32 & 84; 1895?: 184, pp. 37 & 86; 1917: 184, pp. 34 & 83. Use six sticks to make four congruent triangles. Solution is a rectangle (should be a square) with its diagonals, but then two of the sticks have to be longer than the others.

Prob. 163, pp. 32 & 84; 1895?: 186 & 194, pp. 37 & 86-87; 1917: 186 & 194, pp. 34 & 83-84. Use six equally long sticks to make four congruent triangles -- solution is a tetrahedron. The two problems in the 1895? are differently phrased, but identical in content, while the first solution is a picture and the second is a description.

Prob. 171, pp. 33 & 85; 1895?: 195, pp. 38 & 87; 1917: 195, pp. 34 & 84. Use nine equal sticks to make three squares. Solution is three faces of a cube.

F. Chasemore. Loc. cit. in 6.W.5. 1891. Item 3: The triangle puzzle, p. 572.

Hoffmann. 1893.

Chap. VII, no. 15, pp. 290 & 298 = Hoffmann-Hordern, pp. 195. Four matches.

Chap. X, no. 19: The four wine glasses, pp. 344 & 381 = Hoffmann-Hordern, pp. 238-239, with photo on p. 239 of a version by Jaques & Son, 1870-1900. I usually solve the second version by setting one glass on top of the other three, but here he wants the centre of the feet of the glasses to be equally spaced and he turns one glass over and places it in the centre of the other three, appropriately spaced.

Loyd. Problem 34: War-ships at anchor. Tit-Bits 32 (22 May & 12 Jun 1897) 135 & 193. Place four warships equidistantly so that if one is attacked, the others can come to assist it. Solution is a tetrahedron of points on the earth's oceans.

Parlour Games for Everybody. John Leng, Dundee & London, nd [1903 -- BLC], p. 30. "With 6 matches form 4 triangles of equal size."

Pearson. 1907. Part III, no. 77: Three squares, p. 77. Make three squares with nine matches. Solution is a triangular prism!

Anon. Prob. 66. Hobbies 31 (No. 781) (1 Oct 1910) 2 & (No. 784) (22 Oct 1910) 68. Use nine matches to make three squares. "... the only possible solution" is to make two adjacent squares with seven matches, then bisect each square to produce a third square which overlaps the other two.

I re-invented this problem in Apr 1999 and posted it on NOBNET on 19 Apr 1999. Solution (1) is the idea I had when I made up the puzzle, but various friends gave more examples and then I found solution (3).

(1). Arrange the nine matches to form the following.

_

|_| |_|

| |

Then 4 is a square, 9 is a square and 49 is a square.

(2). Use the matches to form a triangular prism. One may object that this also makes two triangles.

(3). Make three squares forming three faces of a cube, all meeting at one corner. Cf Mittenzwey 171.

(4). Make two adjacent squares with seven of the matches. Now bisect each of the squares with a match parallel to the common edge of the squares. This produces a row of four adjacent half-squares as below. The middle two form a new square. Here one may object that the squares are overlapping.

─── ───

│ │ │ │ │

─── ───

(5). Use the matches to make the figures 0, 1 and 4.

One can use the matches to make squares whose edge is half the match length, but one only needs eight matches to make three squares.

There are other solutions which use the fact that matches have squared off ends and have square cross-section, but these properties do not hold for paper matches torn from a matchbook or for other equivalent objects like toothpicks and hence I don't consider them quite reasonable.

Anon. Prob. 76. Hobbies 31 (No. 791) (10 Dec 1910) 256 & (No. 794) (31 Dec 1910) 318. Make as many triangles as possible with six matches. From the solution, it seems that the tetrahedron was expected with four triangles, but many submitted the figure of a triangle with its altitudes drawn, but only one solver noted that this figure contains 16 triangles! However, if the altitudes are displaced to give an interior triangle, I find 17 triangles!!

Williams. Home Entertainments. 1914. Tricks with matches: To form four triangles with six matches, p. 106.

Blyth. Match-Stick Magic. 1921. Four triangle puzzle, p. 23. Make four triangles with six matchsticks.

King. Best 100. 1927. No. 59, pp. 24 & 53. = Foulsham's no. 20, pp. 8 & 12. Use six matches to make four triangles.

6.AO.2. PLACE AN EVEN NUMBER ON EACH LINE

See also section 6.T.

Sometimes the diagonals are considered, but it is not always clear what is intended.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 564-31, pp. 254 & 396. From a 6 x 6 array, remove 6 to leave an even number in each row. (The German 'Reihe' can be interpreted as row or column or both.) If we consider this in the first quadrant with coordinates going from 1 to 6, the removed points are: (1,2), (1,3), (2,1), (2,2), (6,1), (6,3). The use of the sixth column is peculiar and has the effect of making both diagonals odd, while the more usual use of the third column would make both diagonals even.

Prob. 583-5, pp. 285 & 403: Von folgenden 36 Punkten sechs zu streichen. As above, but each file ('Zeile') in 'all four directions' has four or six points. Deletes: (1,1), (1,2), (2,2), (2,3), (6,1), (6,3) which makes one diagonal even and one odd.

Mittenzwey. 1880. Prob. 154, pp. 31 & 83; 1895?: 177, pp. 36 & 85; 1917: 177, pp. 33 & 82. Given a 4 x 4 array, remove 6 to leave an even number in each row and column. Solution removes a 2 x 3 rectangle from a corner. [This fails -- it leaves two rows and a diagonal with an odd number. One can use the idea mentioned for Leske 564-31 to get a solution with both diagonals also being even.]

Hoffmann. 1893. Chap. VI, pp. 271-272 & 285 = Hoffmann-Hordern, pp. 186-187.

No. 22: The thirty-six puzzle. Place 30 counters on a 6 x 6 board so each horizontal and each vertical line has an even number. Solution places the six blanks in a 3 x 3 corner in the obvious way. This also makes the diagonals have even numbers.

No. 23: The "Five to Four" puzzle. Place 20 counters on a 5 x 5 board subject to the above conditions. Solution puts blanks on the diagonal. This also makes the diagonals have even number.

Dudeney. The puzzle realm. Cassell's Magazine ?? (May 1908) 713-716. The crack shots. 10 pieces in a 4 x 4 array making the maximal number of even lines -- counting diagonals and short diagonals -- with an additional complication that pieces are hanging on vertical strings. The picture is used in AM, prob. 270.

Loyd. Cyclopedia. 1914. The jolly friar's puzzle, pp. 307 & 380. (= MPSL2, no. 155, pp. 109 & 172. = SLAHP: A shifty little problem, pp. 64 & 110.) 10 men on a 4 x 4 board -- make a maximal number of even rows, including diagonals and short diagonals. This is a simplification of Dudeney, 1908.

King. Best 100. 1927. No. 72, pp. 29 & 56. As in Hoffmann's No. 22, but specifically asks for even diagonals as well.

The Bile Beans Puzzle Book. 1933. No. 19: Thirty-six coins. As in Hoffmann's No. 22, but specifically asks for even diagonals as well.

Rudin. 1936. No. 151, pp. 53-54 & 111. Place 12 counters on a 6 x 6 board with two in each 'row, column and diagonal'. Reading the positions in each row, the solution is: 16, 34, 25, 25, 34, 16. Some of the short diagonals and some of the broken diagonals are empty, so he presumably isn't including these, or he meant to ask for each of these to have an even number of at most two.

M. Adams. Puzzle Book. 1939. Prob. C.179: Even stars, pp. 169 & 193. Same as Loyd.

Doubleday - 1. 1969. Prob. 61: Milky Way, pp. 76 & 167. = Doubleday - 5, pp. 85-86. 6 x 6 array with two opposite corners already filled. Add ten more counters so that no row, column or diagonal has more than two counters in it. Reading the positions in each row, the solution is: 13, 35, 12, 67, 24, 46. Some short diagonals are empty or have one counter and some broken diagonals have one or four counters, so he seems to be ignoring them. Hence this is the same problem as Rudin, but with a less satisfactory solution.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 37, pp. 38 & 68. 52 men on an 8 x 8 board with all rows, columns and diagonals (both long and short) having an even number.

6.AP . DISSECTIONS OF A TETRAHEDRON

6.AP.1. TWO PIECES

Richard A. Proctor. Our puzzles; Knowledge 10 (Feb 1887) 83 & Solutions of puzzles; Knowledge 10 (Mar 1887) 108-109. "Puzzle XIX. Show how to cut a regular tetrahedron (equilateral triangular pyramid) so that the face cut shall be a square: also show how to plug a square hole with a tetrahedron." Solution shows the cut clearly.

Edward T. Johnson. US Patent 2,216,915 -- Puzzle. Applied: 26 Apr 1939; patented: 8 Oct 1940. 2pp + 1p diagrams. Described in S&B, p. 46.

E. M. Wyatt. Wonders in Wood. Op. cit. in 6.AI. 1946. Pp. 9 & 11: the tetrahedron or triangular pyramid. P. 9 is reproduced in S&B, p. 46.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, p. 26, section 62: Pyramid puzzle. Gives instructions for making the pieces from paper.

Claude Birtwistle. Editor's footnote. MTg 21 (Winter 1962) 32. "The following interesting puzzle was given to us recently."

Birtwistle. Math. Puzzles & Perplexities. 1971. Bisected tetrahedron, pp. 157-158. Gives the net so one can make a drawing, cut it out and fold it up to make one piece.

6.AP.2. FOUR PIECES

These dissections usually also work with a tetrahedron of spheres and hence these are related to ball pyramid puzzles, 6.AZ.

The first version I had in mind dissects each of the two pieces of 6.AP.1 giving four congruent rhombic pyramids. Alternatively, imagine a tetrahedron bisected by two of its midplanes, where a midplane goes halfway between a pair of opposite edges. This puzzle has been available in various versions since at least the 1970s, including one from Stokes Publishing Co., 1292 Reamwood Avenue, Sunnyvale, California, 94089, USA., but I have no idea of the original source. The same pieces are part of a more complex dissection of a cube, PolyPackPuzzle, which was produced by Stokes in 1996. (I bought mine from Key Curriculum Press.)

In 1997, Bill Ritchie, of Binary Arts, sent a quadrisection of the tetrahedron that they are producing. Each piece is a hexahedron. The easiest way to describe it is to consider the tetrahedron as a pile of spheres with four on an edge and hence 20 altogether. Consider a planar triangle of six of these spheres with three on an edge and remove one vertex sphere to produce a trapezium (or trapezoid) shape. Four of these assemble to make the tetrahedron. Writing this has made me realise that Ray Bathke has made and sold these 5-sphere pieces as Pyramid 4 for a few years. However, the solid pieces used by Binary Arts are distinctly more deceptive.

Len Gordon produced another quadrisection of the 20 sphere tetrahedron 0 0

using the planar shape at the right. This was c1980?? 0 0 0  

David Singmaster. Sums of squares and pyramidal numbers. MG 66 (No. 436) (Jun 1982) 100-104. Consider a tetrahedron of spheres with 2n on an edge. The quadrisection described above gives four pyramids whose layers are the squares 1, 4, ..., n2. Hence four times the sum of the first n squares is the tetrahedral number for 2n, i.e. 4 [1 + 4 + ... + n2]  =  BC(n+2, 3).

6.AQ. DISSECTIONS OF A CROSS, T OR H

The usual dissection of a cross has two diagonal cuts at 45o to the sides and passing through two of the reflex corners of the cross and yielding five pieces. The central piece is six-sided, looking like a rectangle with its ends pushed in and being symmetric. Depending on the relative lengths of the arms, head and upright of the cross, the other pieces may be isosceles right triangles or right trapeziums. Removing the head of the cross gives the usual dissection of the T into four pieces -- then the central piece is five-sided. Sometimes the central piece is split in halves. Occasionally the angle of the cuts is different than 45o. Dissections of an H have the same basic idea of using cuts at 45o -- the result can be a bit like two Ts with overlapping stems and the number of pieces depends on the relative size and positioning of the crossbar of the H -- see: Rohrbough.

S&B, pp. 20-21, show several versions. They say that crosses date from early 19C. They show a 6-piece Druid's Cross, by Edwards & Sons, London, c1855. They show several T-puzzles -- they say the first is an 1903 advertisement for White Rose Ceylon Tea, NY -- but see 1898 below. They also show some H-puzzles.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has a cross cut into 5 pieces in the usual way.

Endless Amusement II. 1826? Prob. 30, p. 207. Usual five piece cross. The three small pieces are equal. = New Sphinx, c1840, pp. 139-140.

Crambrook. 1843. P. 4.

No. 10: Five pieces to form a Cross.

No. 11: The new dissected Cross.

Without pictures, I cannot tell what dissections are used??

Boy's Own Book. 1843 (Paris): 435 & 440, no. 2. Usual five piece cross, very similar to Endless Amusement. One has to make three pieces of fig. 2. = Boy's Treasury, 1844, pp. 424 & 428. = de Savigny, 1846, pp. 353 & 357, no. 1.

Family Friend 2 (1850) 58 & 89. Practical Puzzle -- No. II. = Illustrated Boy's Own Treasury, 1860, No. 32, pp. 401 & 440. Usual five piece cross to "form that which, viewed mentally, comforts the afflicted." Three pieces of fig. 1.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 7, p. 178-179 (1868: 189). Five piece dissection of a cross, but the statement of the problem doesn't say which piece to make multiple copies of.

Magician's Own Book. 1857. Prob. 17: The cross puzzle, pp. 272 & 295. Usual 5 piece cross, essentially identical to Family Friend, except this says to "form a cross." = Book of 500 Puzzles, 1859, prob. 17, pp. 86 & 109. = Boy's Own Conjuring Book, 1860, prob. 16, pp. 234 & 258.

Charades, Enigmas, and Riddles. 1860: prob. 33, pp. 60 & 66; 1862: prob. 33, pp. 136 & 143; 1865: prob. 577, pp. 108 & 156. Usual five piece cross, showing all five pieces.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 584-12, pp. 288 & 406: Ein Kreuz. Begins as the usual five piece cross, but the central piece is then bisected into two mitres and the base has two bits cut off to give an eight piece puzzle.

Frank Bellew. The Art of Amusing. 1866. Op. cit. in 5.E. 1866: pp. 239-240; 1870: pp. 236-238. Usual five piece cross.

Elliott. Within-Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 1: The cross puzzle, pp. 27 & 30. Usual five piece cross, but instructions say to cut three copies of the wrong piece.

Mittenzwey. 1880. Prob. 188, pp. 35 & 88; 1895?: 213, pp. 40 & 91; 1917: 213, pp. 37 & 87. This is supposed to be a 10 piece dissection of a cross obtained by further dissecting the usual five pieces. However, pieces 3 & 4 are drawn as trapezoids in the problem and triangles in the solution and piece 2 in the solution is half the size given in the problem. Further, pieces 1 & 2 appear equilateral in the problem, but are isosceles right triangles in the solution. One could modify this to get a 9 piece version where eight of the pieces are right trapezoids -- four having edges 1, 1, 2, (2 and four having edges (2, (2, 2(2, 2, but the arms would be twice as long as they are wide. Or one can make the second four pieces be (2, (2, 2 isosceles right triangles. In either case, the ninth piece would be a rectangle.

Lemon. 1890. A card board puzzle, no. 33, pp. 8 & 98. Usual five piece cross.

Hoffmann. 1893. Chap. III, no. 12: The Latin cross puzzle, pp. 93 & 126 = Hoffmann-Hordern, pp. 82-83, with photo. As in Indoor & Outdoor. Photo on p. 83 shows two versions: one in metal by Jaques & Sons, 1870-1895; the other in ivory, 1850-1900. Hordern Collection, p. 59, shows a Druid's Cross Puzzle.

Lash, Inc. -- Clifton, N.J. -- Chicago, Ill. -- Anaheim, Calif. T Puzzle. Copyright Sept. 1898. 4-piece T puzzle to be cut out from a paper card, but the angle of the cuts is about 35o instead of 45o which makes it less symmetric and less confusing than the more common version. The resulting T is somewhat wider than usual, being about 16% wider than it is tall. It advertises: Lash's Bitters The Original Tonic Laxative. Photocopy sent by Slocum.

Benson. 1904. The cross puzzle, pp. 191-192. Usual 5 piece version.

Wehman. New Book of 200 Puzzles. 1908. The cross puzzle, p. 17. Usual 5 piece version.

A. Neely Hall. Op. cit. in 6.F.5. 1918. The T-puzzle, pp. 19-20. "A famous old puzzle ...." Usual 4-piece version, but with long arms.

Western Puzzle Works, 1926 Catalogue. No. 1394: Four pieces to form Letter T. The notched piece is less symmetric than usual.

Collins. Book of Puzzles. 1927. The crusader's cross puzzle, pp. 1-2. The three small pieces are equal.

Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared in 1908, but versions continued until the 1950s. This looks like 1930s?? Usual 5 piece cross.

A. F. Starkey. The T puzzle. Industrial Arts and Vocational Education 37 (1938) 442. "An interesting novelty ...."

Rohrbough. Puzzle Craft. 1932. The "H" Puzzle, p. 23. Very square H -- consider a 3 x 3 board with the top and bottom middle cells removed. Make a cut along the main diagonal and two shorter cuts parallel to this to produce four congruent isosceles right triangles and two odd pentagons.

See Rohrbough in 6.AS.1 for a very different T puzzle.

6.AR. QUADRISECTED SQUARE PUZZLE

This is usually done by two perpendicular cuts through the centre. A dissection proof of the Theorem of Pythagoras described by Henry Perigal (Messenger of Mathematics 2 (1873) 104) uses the same shapes -- cf 6.AS.2.

The pieces make a number of other different shapes.

Crambrook. 1843. P. 4, no. 17: Four pieces to form a Square. This might be the dissection being considered here??

A. Héraud. Jeux et Récréations Scientifiques -- Chimie, Histoire Naturelle, Mathématiques. (1884); Baillière, Paris, 1903. Pp. 303-304: Casse-tête. Uses two cuts which are perpendicular but are not through the centre. He claims there are 120 ways to try to assemble it, but his mathematics is shaky -- he adds the numbers of ways at each stage rather than multiplying! Also, as Strens notes in the margin of his copy (now at Calgary), if the crossing is off-centre, then many of the edges have different lengths and the number of ways to try is really only one. Actually, I'm not at all sure what the number of ways to try is -- Héraud seems to assume one tries each orientation of each piece, but some intelligence sees that a piece can only fit one way beside another.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. P. 14: The divided square puzzle. Crossing is off-centre.

Tom Tit, vol 3. 1893. Carré casse-tête, pp. 179-180. = K, no. 26: Puzzle squares, pp. 68-69. = R&A, Puzzling squares, p. 99. Not illustrated, but described: cut a square into four parts by two perpendicular cuts, not necessarily through the centre.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 77: Pattern making, pp. 69-70 & 109. Make five other shapes.

M. Adams. Puzzle Book. 1939. Prob. C.12: The broken square, pp. 125 & 173. As above, but notes that the pieces also make a square with a square hole.

6.AS. DISSECTION OF SQUARES INTO A SQUARE

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 102 asserts that dissections of squares to various hexagons and heptagons were known c1800 while square to rectangle dissections were known to Montucla -- though she illustrates the latter with examples like 6.Y, she must mean 6.AS.5.

6.AS.1. TWENTY 1, 2, (5 TRIANGLES MAKE A SQUARE

OR FIVE EQUAL SQUARES TO A SQUARE

The basic puzzle has been varied in many ways by joining up the 20 triangles into various shapes, but I haven't attempted to consider all the modern variants. A common form is a square with a skew # in it, with each line joining a corner to the midpoint of an opposite side, giving the 9 piece version. This has four of the squares having a triangle cut off. For symmetry, it is common to cut off a triangle from the fifth square, giving 10 pieces, though the assembly into one square doesn't need this. See Les Amusemens for details.

Cf Mason in 6.S.2 for a similar puzzle with twenty pieces.

If the dividing lines are moved a bit toward the middle and the central square is bisected, we get a 10 piece puzzle, having two groups of four equal pieces and a group of two equal pieces, called the Japan square puzzle. I have recently noted the connection of this puzzle with this section, so there may be other examples which I have not previously paid attention to -- see: Magician's Own Book, Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Landells, Hanky Panky, Wehman.

Les Amusemens. 1749. P. xxxii. Consider five 2 x 2 squares. Make a cut from a corner to the midpoint of an opposite side on each square. This yields five 1, 2, (5 triangles and five pieces comprising three such triangles. The problem says to make a square from five equal squares. So this is the 10 piece version.

Minguet. 1755. Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Not in 1733 ed. 10 piece version. Also a 15 piece version where triangles are cut off diagonally opposite corners of each small square leaving parallelogram pieces as in Guyot.

Vyse. Tutor's Guide. 1771? Prob. 6, 1793: p. 304, 1799: p. 317 & Key p. 357. 2 x 10 board to be cut into five pieces to make into a square. Cut into a 2 x 2 square and four 2, 4, 2(5 triangles.

Ozanam-Montucla. 1778. Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840: 127. 9 piece version. Remarks that any number of squares can be made into a square -- see 6.AS.5.

Catel. Kunst-Cabinet. 1790.

Das mathematische Viereck, pp. 10-11 & fig. 15 on plate I. 10 piece version with solution shown. Notes these make five squares.

Das grosse mathematische Viereck, p. 11 & fig. 14 on plate I. Cut the larger pieces to give five more 1, 2, (5 triangles and five (5, (5, 2 triangles. Again notes these make five squares.

Guyot. Op. cit. in 6.P.2. 1799. Vol. 2: première récréation: Cinq quarrés éqaux étant sonnés, en former un seul quarré, pp. 40-41 & plate 6, opp. p. 37. 10 piece version. Suggests cutting another triangle off each square to give 10 triangles and 5 parallelograms.

Bestelmeier. 1801. Item 629: Die 5 geometrisch zerschnittenen Quadrate, um aus 5 ein einziges Quadrat zu machen. As in Les Amusemens. S&B say this is the first appearance of the puzzle. Only shown in a box with one small square visible.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 8, pp. 25 & 84 & plate I, fig. 5, no. 1. = Vyse.

No. 10, pp. 25 & 84-85 & plate I, fig. 7, no. 1. Five squares to one. Nine piece version.

Rational Recreations. 1824. Feat 35, pp. 164-166. Usual 20 piece form.

Manuel des Sorciers. 1825. Pp. 201-202, art. 18. ??NX Five squares to one -- usual 10 piece form and 15 piece form as in Guyot.

Endless Amusement II. 1826?

[1837 only] Prob. 35, p. 212. 20 triangles to form a square. = New Sphinx, c1840, p. 141, with problem title: Dissected square.

Prob. 37, p. 215. 10 piece version. = New Sphinx, c1840, p. 141.

Boy's Own Book. The square of triangles. 1828: 426; 1828-2: 430; 1829 (US): 222; 1855: 576; 1868: 676. Uses 20 triangles cut from a square of wood. Cf 1843 (Paris) edition, below. c= de Savigny, 1846, p. 272: Division d'un carré en vingt triangles.

Nuts to Crack IV (1835), no. 195. 20 triangles -- part of a long section: Tricks upon Travellers. The problem is used as a wager and the smart-alec gets it wrong.

The Riddler. 1835. The square of triangles, p. 8. Identical to Boy's Own Book, but without illustration, some consequent changing of the text, and omitting the last comment.

Crambrook. 1843. P. 4.

No. 7: Egyptian Puzzle. Probably the 10 piece version as in Les Amusemens. See S&B below, late 19C. Check??

No. 23: Twenty Triangles to form a Square. Check??

Boy's Own Book. 1843 (Paris): 436 & 441, no. 5: "Cut twenty triangles out of ten square pieces of wood; mix them together, and request a person to make an exact square with them." As stated, this is impossible; it should be as in Boy's Own Book, 1828 etc., qv. = Boy's Treasury, 1844, pp. 425 & 429. = de Savigny, 1846, pp. 353 & 357, no. 4. Also copied, with the error, in: Magician's Own Book, 1857, prob. 29: The triangle puzzle; Book of 500 Puzzles, 1859, prob. 29: The triangle puzzle; Boy's Own Conjuring Book, 1860, prob. 28: The triangle puzzle. c= Hanky Panky, 1872, p. 122.

Magician's Own Book. 1857.

How to make five squares into a large one without any waste of stuff, p. 258. 9 piece version.

Prob. 29: The triangle puzzle, pp. 276 & 298. Identical to Boy's Own Book, 1843 (Paris).

Prob. 35: The Japan square puzzle, pp. 277 & 300. Make two parallel cuts and then two perpendicular to the first two so that a square is formed in the centre. This gives a 9 piece puzzle, but here the central square is cut by a vertical through its centre to give a 10 piece puzzle. = Landells, Boy's Own Toy-Maker, 1858, pp. 145-146.

Charles Bailey (manufacturer in Manchester, Massachusetts). 1858. An Ingenious Puzzle for the Amusement of Children .... The 10 pieces of Les Amusemens, with 19 shapes to make, a la tangrams. Sent by Jerry Slocum -- it is not clear if there were actual pieces with the printed material.

The Sociable. 1858.

Prob. 10: The protean puzzle, pp. 289 & 305-306. Cut a 5 x 1 into 11 pieces to form eight shapes, e.g. a Greek cross. It is easier to describe the pieces if we start with a 10 x 2. Then three squares are cut off. One is halved into two 1 x 2 rectangles. Two squares have two 1, 2, (5 triangles cut off leaving triangles of sides 2, (5, (5. The remaining double square is almost divided into halves each with a 1, 2, (5 triangle cut off, but these two triangles remain connected along their sides of size 1, thus giving a 4, (5, (5 triangle and two trapeziums of sides 2, 2, 1, (5. = Book of 500 Puzzles, 1859, prob. 10, pp. 7 & 23-24.

Prob. 42: The mechanic's puzzle, pp. 298 & 317. Cut a 10 x 2 in five pieces to make a square, as in Vyse. = Book of 500 Puzzles, 1859, prob. 16, pp. 16 & 35.

Book of 500 Puzzles. 1859.

Prob. 10: The protean puzzle, pp. 7 & 23-24. As in The Sociable.

Prob. 42: The mechanic's puzzle, pp. 16 & 35. As in The Sociable.

How to make five squares into a large one without any waste of stuff, p. 72. Identical to Magician's Own Book.

Prob. 29: The triangle puzzle, pp. 90 & 113. Identical to Boy's Own Book, 1843 (Paris).

Prob. 35: The Japan square puzzle, pp. 91 & 114.

Indoor & Outdoor. c1859. Part II, prob. 11: The mechanic's puzzle, pp. 130-131. Identical to The Sociable.

Boy's Own Conjuring Book. 1860.

Prob. 28: The triangle puzzle, pp. 238 & 262. Identical to Boy's Own Book, 1843 (Paris) and Magician's Own Book.

Prob. 34: The Japan square puzzle, pp. 240 & 264. Identical to Magician's Own Book.

Illustrated Boy's Own Treasury. 1860.

Prob. 9, pp. 396 & 437. [The Japan square puzzle.] Almost identical to Magician's Own Book.

Optics: How to make five squares into a large one without any waste of stuff, p. 445. Identical to Book of 500 Puzzles, p. 72.

Vinot. 1860. Art. LXXV: Avec cinq carrés égaux, en faire un seul, p. 90. Nine piece version.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 174, pp. 87-88. Nine piece version.

Prob. 584-6, pp. 287 & 405. Ten piece version of five squares to one.

Hanky Panky. 1872.

The puzzle of five pieces, p. 118. 9 piece version.

Another [square] of four triangles and a square, p. 120. 10 x 2 into five pieces to make a square.

[Another square] of ten pieces, pp. 121-122. Same as the Japan square puzzles in Magician's Own Book.

[Another square] of twenty triangles, p. 122. Similar to Boy's Own Book, 1843 (Paris), but with no diagram and less text, making it quite cryptic.

Mittenzwey. 1880. Prob. 175, pp. 33-34 & 85; 1895?: 200, pp. 38 & 87; 1917: 200, pp. 35 & 84. 10 pieces as in Les Amusemens. See in 6.AS.2 and 6.S.2 for the use of these pieces to make other shapes.

See Mason, 1880, in 6.S.2 for a similar, but different, 20 piece puzzle.

S&B, pp. 11 & 19, show a 10 piece version called 'Egyptian Puzzle', late 19C?

Lucas. RM2. 1883. Les vingt triangles, pp. 128-129. Notes that they also make five squares in the form of a cross.

Tom Tit, vol. 2. 1892. Diviser un carré en cinq carrés égaux, pp. 147-148. = K, no. 2: To divide a square into five equal squares, pp. 12-14. = R&A, Five easy pieces, p. 105. Uses 9 pieces, but mentions use of 10 pieces.

Hoffmann. 1893.

Chap. III, no. 21: The five squares, pp. 100 & 132-133 = Hoffmann-Hordern, p. 94, with photo. 9 piece version, as in Magician's Own Book. Photo on p. 94 shows: an ivory version, 1850-1900;, and a wood version, named Egyptian Puzzle, by C. N. Mackie, 1860-1890; both with boxes.

Chap. III, no. 24: The twenty triangles, pp. 101 & 134 = Hoffmann-Hordern, pp. 96-97, with photo. As in Boy's Own Book. Photo on p. 97 shows The Twenty Triangle Puzzle, with box, by Jaques & Son, 1870-1895. Hordern Collection, p. 64, shows Apollonius, with box, by W. X., Paris, 1880-1900, in a solution very different to the usual one.

Chap. III, no. 30: The carpenter's puzzle -- no. 1, pp. 103 & 136-137 = Hoffmann-Hordern, p. 101. Cut a 5 x 1 board into five pieces to make a square.

Chap. X, no. 25: The divided square, pp. 346 & 384 = Hoffmann-Hordern, p. 242. 9 piece puzzle as a dissection of a square which forms 5 equal squares. He places the five squares together as a 2 x 2 with an adjacent 1 x 1, but he doesn't see the connection with 6.AS.2.

Montgomery Ward & Co. Catalog No 57, Spring & Summer, 1895. Facsimile by Dover, 1969, ??NX. P. 237 shows the Mystic Square, as item 25463, which is the standard 10 piece version.

Benson. 1904.

The carpenter's puzzle (No. 2), p. 191. = Hoffmann, p. 103.

The five-square puzzle, pp. 196-197. = Hoffmann, p. 100.

The triangle puzzle, p. 198. = Hoffmann, p. 101.

Wehman. New Book of 200 Puzzles. 1908.

P. 3: The triangle puzzle. 20 pieces. = Boy's Own Book, omitting the adjuration to use wood and smooth the edges

P. 12: The protean puzzle. c= The Sociable, prob. 10, with the instructions somewhat clarified.

P. 14: The Japan square puzzle. c= Magician's Own Book.

P. 19: To make five squares into a large one. 10 piece version.

P. 27: The mechanic's puzzle. = The Sociable, prob. 42.

J. K. Benson, ed. The Pearson Puzzle Book. C. Arthur Pearson, London, nd [c1910, not in BMC or NUC]. [This is almost identical with the puzzle section of Benson, but has 13 pages of different material.] Juggling geometry, pp. 97-98. Five triangles, which should be viewed as 2, 4, 2(5. Cut one from the midpoint of the hypotenuse to the midpoint of the long leg and assemble into a square, so this becomes a six-piece or five-piece version as in Vyse, etc.

I have seen a 10 piece French example, called Jeu du Carré, dated 1900-1920.

I have seen a 9(?) piece English example, dated early 20C, called The Euclid Puzzle.

Dudeney. Perplexities column, no. 109: A cutting-out puzzle. Strand Magazine 45 (No. 265) (Jan 1913) 113 & (No. 266) (Feb 1913) 238. c= AM, prob. 153 -- A cutting-out puzzle, pp. 37 & 172. Cut a 5 x 1 to make a square. He shows a solution in five pieces and asks for a solution in four pieces. AM states the generalized form given in 6.AS.5.

Rohrbough. Puzzle Craft. 1932. Square "T", p. 23 (= The "T" Puzzle, p. 23 of 1940s?). 1 x 1 square and two 1 x 2 rectangles cut diagonally can be formed into a square or into a T.

Gibson. Op. cit. in 4.A.1.a. 1963. Pp. 71 & 76: Square away. A five piece puzzle, approximately that formed by drawing parallel lines from two diagonally opposite corners to the midpoints of opposite sides and then cutting a square from the middle of the central strip. As drawn, the lines meet the opposite sides a bit further along than the midpoints.

Ripley's Puzzles and Games. 1966.

Pp. 58-59, item 6. Five right triangles to a square. Though not specified, the triangles have sides proportional to 1, 2, (5. Solution is as in Benson.

Pp. 60-61, item 5. Start with the large square, which is 2(5 on a side. Imagine the 9 piece puzzle where one line goes from the upper left corner to the midpoint of the right side. Number the outer pieces clockwise from the upper left, so that pieces 1, 3, 5, 7 are the small triangles and 2, 4, 6, 8 are the trapezia. The pieces of this puzzle are as follows: combine pieces 1, 8, 7 into a (5, 2(5, 5 triangle; combine pieces 2, 3, 4 into an irregular hexagon; take separate pieces 5 and 6 and the central square. These five pieces form: a square; a Greek cross; a 5 x 4 rectangle; a triangle of sides 2(5, 4(5, 10; etc. I think there are earlier versions of this, e.g. in Loyd, but I have just observed the connection with this section.

6.AS.1.a. GREEK CROSS TO A SQUARE

Note that a proper Greek cross is formed from five equal squares.

Lucas. RM2. 1883. Loc. cit. in 6.AS.1. Uses 20 triangles.

Lemon. 1890. The Maltese cross squared, no. 369, pp. 51 & 111. Cut a Maltese cross (really a Greek cross) by two cuts into four pieces that make a square.

Hoffmann. 1893. Chap. III, no. 13: The Greek cross puzzle, pp. 94 & 126 = Hoffmann-Hordern, pp. 82 & 84. Has four pieces made by two cuts.

Loyd. Tit-Bits 31 (10, 17 & 31 Oct 1896) 25, 39 & 75. = Cyclopedia, 1914, p. 14. Four pieces as in Hoffmann.

Loyd. Problem 23: A new "square and cross" puzzle. Tit-Bits 31 (13 Mar 1897) 437 & 32 (3 Apr 1897) 3. = Cyclopedia, 1914, pp. 58, 270 & 376. Four congruent pieces.

Loyd. Problem 27: The swastika problem. Tit-Bits 32 (3 & 24 Apr 1897) 3 & 59. = Cyclopedia, 1914, p. 58. Quadrisect square to make two equal Greek crosses.

Loyd. Problem 30: The Easter problem. Tit-Bits 32 (24 Apr & 15 May 1897) 59 & 117. Dissect square into five pieces to make two unequal Greek crosses.

Dudeney. Problem 56: Two new cross puzzles. Tit-Bits 33 (23 Oct & 13 Nov 1897) 59 & 119. Dissect a half square (formed by cutting a square either vertically or diagonally) to a Greek cross. Solutions in 3 and 4 pieces. [The first case = Loyd, Cyclopedia, 1914, Easter 1903, pp. 46 & 345.]

Benson. 1904. The Greek cross puzzle, p. 197. = Hoffmann, p. 94.

Dudeney. Cutting-out paper puzzles. Cassell's Magazine ?? (Dec 1909) 187-191 & 233-235.

States that the dissection with four pieces in two cuts is relatively 'recent'. c= AM, 1917, p. 29, which dates this to 'the middle of the nineteenth century'.

Fold a Greek cross so that one cut gives four congruent pieces which form a square. = AM, 1917, prob. 145, pp. 35 & 169.

M. Adams. Indoor Games. 1912. The Greek cross, p. 349 with figs. on p. 347.

6.AS.1.b. OTHER GREEK CROSS DISSECTIONS

See also 6.F.3 and 6.F.5.

Dudeney. A batch of puzzles. Royal Magazine 1:3 (Jan 1899) & 1:4 (Feb 1899) 368-372. Squares and cross puzzle. = AM, 1917, p. 34. Dissect a Greek cross into five pieces which make two squares, one three times the edge of the other. If the squares in the Greek cross have edge (2, then the cross has area 10 and the two squares have areas 1 and 9. The dissection arise by joining the midpoints of the edges of the central square of the cross and extending these lines in one direction symmetrically.

Dudeney. AM. 1917. Greek cross puzzles, pp. 28-35. This discusses a number of examples and gives a few problems.

Collins. Book of Puzzles. 1927. The Greek cross puzzle, pp. 98-100. Take a Greek cross whose squares have side 2, so the cross has area 10. Take another cross of area 5 and place it inside the large cross. If this is done centrally and the small one turned to meet the edges of the large one, there are four congruent heptagonal pieces surrounding the small one which make another Greek cross of area 5.

Eric Kenneway. More Magic Toys, Tricks and Illusions. Beaver Books (Arrow (Hutchinson)), London, 1985. On pp. 56-58, he considers a Greek cross cut by two pairs of parallel lines into nine pieces which would make five squares. The lines join an outer corner to the midpoint of an opposite segment. This produces a tilted square in the centre. By pairing the other pieces, he gets four identical pieces which make a square and a Greek cross in a square.

6.AS.2. TWO (ADJACENT) SQUARES TO A SQUARE

The smaller square often has half the edge of the larger, which connects this with 6.AS.1, but this is not essential. The two squares are usually viewed as one piece, i.e. a P-pentomino. These items are dissection proofs of the Theorem of Pythagoras -- see Yates (op. cit. in 6.B, pp. 38-39) for some other examples of this point.

See Ripley's for a similar example, but the 2 x 2 square has a (2, (2, 2 triangle attached to an edge.

Another version has squares of area 1 and 8. The area 8 square is cut into four pieces which combine with the area 1 square to make an area 9 square. I call this the 4 - 5 piece square.

Walther Karl Julius Lietzmann (1880-1959). Der Pythagoreische Lehrsatz. Teubner, (1911, 2nd ed., 1917), 6th ed., 1951. [There was a 7th ed, 1953.] Pp. 23-24 gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. He says it was known to Indian mathematicians at the end of the 9C as the Bride's Chair (Stuhl der Braut). (I always thought this name referred to the figure of the Euclid I, 47 -- ??)

Thabit ibn Qurra (= Thābit ibn Qurra). c875. Gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. [Q. Mushtaq & A. L. Tan; Mathematics: The Islamic Legacy; Noor Publishing House, Farashkhan, Delhi, 1993, pp. 71-72] give this and cite Lietzmann. Greg Frederickson [email of 18 Oct 1996] cites Aydin Sayili; Thabit ibn Qurra's generalization of the Pythagorean theorem; Isis 51 (1960) 35-37.

Abu'l 'Abbas al-Fadhl ibn Hatim al-Narizi (or Annairizi). (d. c922.) Ed. by Maximilian Curtze, from a translation by Gherardo of Cremona, as: Anaritii In decem libros priores Elementorum Euclidis Commentarii, IN: Euclidis Opera Omnia; Supplementum; Teubner, Leipzig, 1899. ??NYS -- information supplied by Greg Frederickson.

Johann Christophorus Sturm. Mathesis Enumerata, 1695, ??NYS. Translated by J. Rogers? as: Mathesis Enumerata: or, the Elements of the Mathematicks; Robert Knaplock, London, 1700, ??NYS -- information provided by Greg Frederickson, email of 14 Jul 1995. Fig. 29 shows it clearly and he attributes it to Frans van Schooten (the Younger, who was the more important one), but this source hasn't been traced yet.

Les Amusemens. 1749. Prob. 216, p. 381 & fig. 97 on plate 8: Réduire les deux quarrés en un seul. Usual dissection of two adjacent squares, attributed to 'Sturmius', a German mathematician, i.e the previous entry.

Ozanam-Montucla. 1778. Diverses démonstrations de la quarante-septieme du premier livre d'Euclide, ..., version 2. Fig. 27, plate 4. 1778: 288; 1803: 284; 1814: 241-243; 1840: 123-124. This is a version of the proof that (a + b)2 = c2 + 4(ab/2), but the diagram includes extra lines which produce the standard dissection of two adjacent triangles.

Crambrook. 1843. P. 4, no. 19: One Square to form two Squares -- ??

E. S. Loomis. The Pythagorean Proposition. 2nd ed., 1940; reprinted by NCTM, 1968. On pp. 194-195, he describes the usual dissection by two cuts as Geometric Proof 165 and gives examples back to 1849, Schlömilch.

Family Friend 2 (1850) 298 & 353. Practical Puzzle -- No. X. = Illustrated Boy's Own Treasury, 1860, Prob. 11, pp. 397 & 437. The larger square has twice the edge of the smaller and is shown divided into four, so this is clearly related to 6.AS.1, though the shape is considered as one piece, i.e. a P-pentomino, to be cut into three parts to make a square.

Magician's Own Book. 1857. To form a square, p. 261. = Book of 500 Puzzles, 1859, p. 75. An abbreviated version of Family Friend. Refers to dotted lines in the figure which are drawn solid.

Charades, Enigmas, and Riddles. 1860: prob. 31, pp. 60 & 65; 1862: prob. 32, pp. 136 & 142; 1865: prob. 576, pp. 108 & 155. Dissect a P-pentomino into three parts which make a square. Usual solution.

Peter Parley, the Younger. Amusements of Science. Peter Parley's Annual for 1866, pp. 139-155.

Pp. 143-144: "To form two squares of unequal size into one square, equal to both the original squares." Usual method, with five pieces. On pp. 146-148, he discusses the Theorem of Pythagoras and shows the dissection gives a proof of it.

P. 144: "To make two smaller squares out of one larger." Cuts the larger square along both diagonals and assembles the pieces into two squares.

Hanky Panky. 1872. To form a square, pp. 116-117. Very similar to Magician's Own Book.

Henry Perigal. Messenger of Mathematics 2 (1873) 104. ??NYS -- described in Loomis, op. cit. above, pp. 104-105 & 214, where some earlier possible occurrences are mentioned. He gives a dissection proof of the theorem of Pythagoras using the shapes that occur in the quadrisection of the square -- Section 6.AR. For sides a < b, perpendicular cuts through the centre are made in the square of side b so they meet the sides at distance (b-a)/2 from a corner. These pieces then fit around the square of side a to make a square of side c.

I invented a hinged version of this, in the 1980s?, which is described in: Greg N. Frederickson; Hinged Dissections: Swinging & Twisting; CUP, 2002, pp. 33-34. I am shown demonstrating this on Frederickson's website: cs.purdue.edu/homes/gnf/book2/Booknews2/singm.html .

I have seen the assembly of these four pieces and the square of edge a into the square on the hypotenuse in a photo of the Tomb of Ezekiel in the village of Al-Kifil, near Hillah, Iraq.

Mittenzwey. 1880.

Prob. 176, pp. 34 & 85; 1895?: 201, pp. 38 & 88; 1917: 201, pp. 35 & 84. Use the 10 pieces of 6.AS.1, as in Les Amusemens, to make squares of edge 1 and edge 2.

Prob. 180, pp. 34 & 86; 1895?: 205, pp. 39 & 89; 1917: 205, pp. 35-36 & 85. Cut a 2 x 2 and a 4 x 4 into five pieces which make a square. Both the problem and the solutions are inaccurately drawn. The smaller square has a 1, 2, (5 cut off, as for 6.AS.1. The larger square has the same cut off at the lower left and a 2, 4, 2(5 cut off at the lower right -- these two touch at the midpoint of the bottom edge -- leaving a quadrilateral with edges 4, 2(5, (5, 3 and two right angles. This is a variant of the standard five piece method.

Alf. A. Langley. Letter: Three-square puzzle. Knowledge 1 (9 Dec 1881) 116, item 97. Cuts two squares into five pieces which form a single square.

Alexander J. Ellis. Letter: The three-square puzzle. Knowledge 1 (23 Dec 1881) 166, item 146. Usual dissection of two adjacent squares, considered as one piece, into three parts by two cuts, which gives Langley's five pieces if the two squares are divided. Suppose the two squares are on a single piece of paper and are ABCD and DEFG, with E on side CD of the larger square ABCD. He notes that if one folds the paper so that B and F coincide, then the fold line meets the line ADG at the point H such that the desired cuts are BH and HF.

R. A. Proctor. Letter or editorial reply: Three square puzzle. Knowledge 1 (30 Dec 1881) 184, item 152. Says there have been many replies, cites Todhunter's Euclid, p. 266 and notes the pieces can be obtained by flipping the large square over and seeing how it cuts the two smaller ones.

R. A. Proctor. Our mathematical column: Notes on Euclid's first book. Knowledge 5 (2 May 1884) 318. "The following problem, forming a well-known "puzzle" exhibits an interesting proof of the 47th proposition." Gives the usual three piece form, as in Ellis.

B. Brodie. Letter: Superposition. Knowledge 5 (30 May 1884) 399, item 1273. Response to the above, giving the five piece version, as in Langley.

Hoffmann. 1893. Chap. III, no. 11: The two squares, pp. 93 & 125-126 = Hoffmann-Hordern, pp. 82-83, with photo. Smaller square has half the edge. The squares are viewed as a single piece. Photo on p. 83 shows The Five Squares Puzzle in paper with box, by Jaques & Son, 1870-1895, and an ivory version, with box, 1850-1900.

Loyd. Tit-Bits 31 (3, 10 & 31 Oct 1896) 3, 25 & 75. General two cut version.

Herr Meyer. Puzzles. The Boy's Own Paper 19 (No. 937) (26 Dec 1896) 206 & (No. 948) (13 Mar 1897) 383. As in Hoffmann.

Benson. 1904. The two-square puzzle, pp. 192-193.

Pearson. 1907. Part II, no. 108: Still a square, pp. 108 & 182. Smaller square has half the edge.

Loyd. Cyclopedia. 1914. Pythagoras' classical problem, pp. 101 & 352. c= SLAHP, pp. 15-16 & 88. The adjacent squares are viewed as one piece of wood to be cut. Uses two cuts, three pieces.

Williams. Home Entertainments. 1914. Square puzzle, p. 118. P-pentomino to be cut into three pieces to make a square. No solution given.

A. W. Siddons. Note 1020: Perigal's dissection for the Theorem of Pythagoras. MG 16 (No. 217) (Feb 1932) 44. Here he notes that the two cutting lines of Perigal's 1873 dissection do not have to go through the centre, but this gives dissections with more pieces. He shows examples with six and seven pieces. These cannot be hinged.

The Bile Beans Puzzle Book. 1933.

No. 37: No waste. Consider a square of side 2 extended by an isosceles triangle of hypotenuse 2. Convert to a square using two cuts.

No. 39: Square building. P-pentomino to square in two cuts.

Slocum. Compendium. Shows 4 - 5 piece square from Johnson Smith catalogue, 1935.

M. Adams. Puzzle Book. 1939. Prob. C.120: One table from two, pp. 154 & 185. 3 x 3 and 4 x 4 tiled squares to be made into a 5 x 5 but only cutting along the grid lines. Solves with each table cut into two pieces. (I think there are earlier examples of this -- I have just added this variant.)

Ripley's Puzzles and Games. 1966. Pp. 58-59.

Item 5. Two joined adjacent squares to a square, using two cuts and three pieces.

Item 6. Consider a 2 x 2 square with a (2, (2, 2 triangle attached to an edge. Two cuts and three pieces to make a square.

6.AS.2.a. TWO EQUAL SQUARES TO A SQUARE

Further subdivision of the pieces gives us 6.AS.4.

Jackson. Rational Amusement. 1821. Geometrical Puzzles, no. 13, pp. 25 & 85-86. Cut two equal squares each into two pieces to make a square.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 172, p. 87. Cut one square into pieces to make two equal squares. Cuts along the diagonals.

Mittenzwey. 1880. Prob. 241, pp. 44 & 94; 1895?: 270, pp. 48 & 96; 1917: 270, pp. 44 & 92. As is Leske.

6.AS.3. THREE EQUAL SQUARES TO A SQUARE

Crambrook. 1843. P. 4, no. 21: One [Square to form] three [Squares] -- ??

"Student". Proposal [A pretty geometrical problem]. Knowledge 1 (13 Jan 1882) 229, item 184. Dissect an L-tromino into a square. Says there are 25 solutions -- editor says there are many more.

Editor. A pretty geometrical problem. Knowledge 1 (3 Mar 1882) 380. Says only the proposer has given a correct solution, which cuts off one square, then cuts the remaining double square into three parts, so the solution has four pieces. Says there are several other ways with four pieces and infinitely many with five pieces.

Hoffmann. 1893. Chap. III, no. 23: The dissected square, pp. 101 & 134 = Hoffmann-Hordern, pp. 96-97, with photo. Cuts three squares identically into three pieces to form one square. Photo on p. 97 shows The Dissected Square, with box, by Jaques & Son, 1870-1895. Hordern Collection, p. 63, shows Arabian Puzzle, with box and some problem shapes to make, 1870-1890.

Loyd. Problem 3: The three squares puzzle. Tit-Bits 31 (17 Oct, 7 & 14 Nov 1896) 39, 97 & 112. Quadrisect 3 x 1 rectangle to a square. Sphinx (i.e. Dudeney) notes it also can be done with three pieces.

M. Adams. Indoor Games. 1912. The divided square, p. 349 with figs. on pp. 346-347. 3 squares, 4 cuts, 7 pieces.

Loyd. Cyclopedia. 1914. Pp. 14 & 341. = SLAHP: Three in one, pp. 44 & 100. Viewed as a 3 x 1 rectangle, solution uses 2 cuts, 3 pieces. Viewed as 3 squares, there are 3 cuts, 6 pieces.

Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 6, pp. 61 & 160. Claims the problem is posed by Abu'l-Wefa, late 10C, though other problems in this section are not strictly as posed by the historic figures cited. Two of the squares to be divided into 8 parts so all nine parts make a square. The solution has the general form of the quadrisection of the square of side (2 folded around to surround a square of side 1 (as in Perigal's(?) dissection proof of the Theorem of Pythagoras), thus forming the square of side (3. The four quadrisection pieces are cut into two triangles of sides: 1, (3/2, (1+(2)/2 and 1, (3/2, ((2-1)/2. Two of each shape assemble into a square of side 1 which can be viewed as having a diagonal cut and then cuts from the other corners to the diagonal, cutting off ((2-1)/2 on the diagonal.

6.AS.3.a. THREE EQUAL 'SQUARES' TO A HEXAGON

Catel. Kunst-Cabinet. Vol. 2, 1793. Das Parallelogramm, pp. 14-15 & fig. 249 a,b,c,d on plate XII. This shows three squares, each dissected the same way into 4 pieces which will make a hexagon or two equal equilateral triangles. Consider a hexagon and connect three alternate vertices to the centre. Join up the same vertices and drop perpendiculars from the centre to three of the sides of the hexagon. However, close examination shows that the squares have dimensions 3/2 by (3. The figure of the three adjacent squares has the divisions between them hard to make out.

Bestelmeier. 1801. Item 292/293 -- Das Parallelogram. Almost identical to Catel, except the diagrams are reversed, and worse, several of the lines are missing. Mathematical part of the text is identical.

6.AS.4. EIGHT EQUAL SQUARES TO A SQUARE

Divide four of the squares in half diagonally.

Magician's Own Book. 1857. Prob. 8: The accommodating square, pp. 269 & 293. c= Landells, Boy's Own Toy-Maker, 1858, p. 144. = Book of 500 Puzzles, 1859, prob. 8, pp. 83 & 107. = Boy's Own Conjuring Book, 1860, prob. 7, pp. 230 & 256. = Illustrated Boy's Own Treasury, 1860, no. 24, pp. 399 & 439.

Hanky Panky. 1872. [Another square] of four squares and eight triangles, p. 120.

Cassell's. 1881. Pp. 92-93: The accommodating square. = Manson, 1911, p. 131.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 321-322: Square puzzle.

Hoffmann. 1893. Chap. III, no. 20: Eight squares in one, pp. 100 & 132 = Hoffmann-Hordern, p. 94.

Wehman. New Book of 200 Puzzles. 1908. The accommodating square, p. 13. c= Magician's Own Book.

6.AS.5. RECTANGLE TO A SQUARE OR OTHER RECTANGLE

New section. See comment at 6.AS. The dissection of a 5 x 1 into five pieces which make a square is explicitly covered in 6.AS.1, and the other cases above can be viewed as dissections of 2 x 1, 3 x 1 and 8 x 1. There must be older examples of the general case??

Ozanam-Montucla. 1778.

Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840: 127. 9 pieces. Remarks that any number of squares can be made into a square.

Prob. 19 & fig. 124-126, plate 15 & 16, 1778: 297-301; 1803: 293-296; 1814: 250-253; 1840: 127-129. Dissect a rectangle to a square.

Prob. 20 & fig. 125-126, plate 15 & 16, 1778: 301-302; 1803: 297; 1814: 253; 1840: 129. Dissect a square into 4, 5, 6, etc. parts which form a rectangle.

"Mogul". Proposal [A pretty geometrical problem]. Knowledge 1 (13 Jan 1882) 229, item 184. Dissect a rectangle into a square. Editor's comment in (3 Mar 1882) 380 says only the proposer has given a correct solution but it will be held over.

"Mogul". Mogul's Problem. Knowledge 1 (31 Mar 1882) 483. Gives a general construction, noting that if the ratio of length to width is ( 2, then it takes two cuts; if the ratio is in the interval (2, 5], it takes three cuts; if the ratio is in (5, 10], it takes four cuts; if the ratio is in (10, 17], it takes five cuts. In general if the ratio is in (n2+1, (n+1)2+1], it takes n+2 cuts.

Richard A. Proctor. Our puzzles; Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40 & Solution of puzzles; Knowledge 10 (Jan 1887) 60-61. "Puzzle XII. Given a rectangular carpet of any shape and size to divide it with the fewest possible cuts so as to fit a rectangular floor of equal size but of any shape." He says this was previously given and solved by "Mogul". Solution notes that this is not the problem posed by "Mogul" and that the shape of the second rectangle is assumed as given. He distinguishes between the cases where the actual second rectangular area is given and where only its shape is given. Gives some solutions, remarking that more cuts may be needed if either rectangle is very long. Poses similar problems for a parallelogram.

Tom Tit, vol. 3. 1893. Rectangle changé en carré. en deux coups de ciseaux, pp. 175-176. = K, no. 24: By two cuts to change a rectangle into a square, pp. 64-65. Consider a square ABCD of side one. If you draw AA' at angle α to AB and then drop BE perpendicular to AA', the resulting three pieces make a rectangle of size sin α by csc α, where α must be ( 450, so the rectangle cannot be more than twice as long as it is wide. If one starts with such a rectangle ABCD, where AB is the length, then one draws AA' so that DA' is the geometric mean of AB and AB - AD. Dropping CE perpendicular to AA' gives the second cut.

Dudeney. Perplexities column, no. 109: A cutting-out puzzle. Strand Magazine 45 (No. 265) (Jan 1913) 113 & (No. 266) (Feb 1913) 238. c= AM, prob. 153 -- A cutting-out puzzle, pp. 37 & 172. Cut a 5 x 1 into four pieces to make a square. AM states the generalized form: if length/breadth is in [(n+1)2, n2), then it can be done with n+2 pieces, of which n-1 are rectangles of the same breadth but having the desired length. The cases 1 x (n+1)2 are exceptional in that one of pieces vanishes, so only n+1 pieces are needed. He doesn't describe this fully and I think one can change the interval above to ((n+1)2, n2].

Anonymous. Two dissection problems, no. 1. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951) 23. An n-step is formed by n lines of unit squares of lengths 1, 2, ..., n, with all lines aligned at one end. Hence a 1-step is a unit square, a 2-step is an L-tromino and an n-step is what is left when an (n-1)-step is removed from a corner of an n x n square. Show any n-step can be cut into four pieces to make a square, with three pieces in one case. Cut parallel to a long side at distance (n+1)/2 from it. The small piece can be rotated 180o about a corner to make an n x (n+1)/2 rectangle. Dudeney's method cuts this into three pieces which make a square, and the cuts do not cut the small part, so we can do this with a total of four pieces. When n = 8, the rectangles is 8 x 9/2, which is similar to 16 x 9 which can be cut into two pieces by a staircase cut, so the problem can be done with a total of three pieces. A little calculation shows this is the only case where n x (n+1)/2 is similar to k2 x (k-1)2.

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 105. Dissect a 2 x 5 rectangle into four pieces that make a square.

6.AT. POLYHEDRA AND TESSELLATIONS

These have been extensively studied, so I give only the major works. See 6.AA for nets of polyhedra.

6.AT.1. REGULAR POLYHEDRA

Gwen White. Antique Toys and Their Background. Batsford, 1971. (Reprinted by Chancellor Press, London, nd [c1989].) P. 9 has a sketch of "Ball of stone, Scotland", which seems to be tetrahedral and she says: "... one of the earliest toys known is a stone ball. Perhaps it is not a plaything, no one knows why it was made, but it is a convenient size to hold in the hand."

Dorothy N. Marshall. Carved stone balls. Proc. Soc. of Antiquaries of Scotland 108 (1976-7) 40-72. Survey of the Scottish neolithic carved stone balls. Lists 387 examples in 36 museums and private collections, mostly of 70mm diameter and mostly from eastern Scotland. Unfortunately Marshall is not interested in the geometry and doesn't clearly describe the patterns -- she describes balls with 3, 4, 5, 6, 7, 8, 9, 10 - 55 and 70 - 160 knobs, but emphasises the decorative styles. From the figures, there are clearly tetrahedral, cubical, dodecahedral(?) and cubo-octahedral shapes. Many are in the National Museum of Antiquities of Scotland (= Royal Museum, see below), but the catalogue uses a number of unexplained abbreviations of collections.

Royal Museum of Scotland, Queen Street, Edinburgh, has several dozen balls on display, showing cubical, tetrahedral, octahedral and dodecahedral symmetry, and one in the form of the dual of the pentagonal prism. [This museum has now moved to a new building beside its other site in Chambers Street and has been renamed the Museum of Scotland. When I visited in 1999, I was dismayed to find that only three of the carved stone balls were on display, in a dimly lit case and some distance behind the glass so that it was difficult to see them. Admittedly, the most famous example, the tetrahedral example with elaborate celtic decorative spirals, NMA AS10 from Glasshill, Towie, Aberdeenshire, is on display -- photo in [Jenni Calder; Museum of Scotland; NMS Publishing, 1998, p. 21]. They are on Level 0 in the section called In Touch with the Gods.]

Ashmolean Museum, Oxford, has six balls on display in case 13a of the John Evans Room. One is tetrahedral, three are cubical, one is dodecahedral and one is unclear.

Keith Critchlow. Time Stands Still -- New Light on Megalithic Science. Gordon Fraser, London, 1979. Chap. 7: Platonic spheres -- a millennium before Plato, pp. 131-149. He discusses and depicts Neolithic Scottish stones carved into rounded polyhedral shapes. All the regular polyhedra and the cubo-octahedron occur. He is a bit vague on locations -- a map shows about 50 discovery sites and he indicates that some of these stones are in the Ashmolean Museum, Dundee City Museum and 'in Edinburgh'. Likewise, the dating is not clear -- he only says 'Neolithic' -- and there seem to be no references.

D. V. Clark. Symbols of Power at the Time of Stonehenge. National Museum of Antiquities, Edinburgh, 1985. Pp. 56-62 & 171. ??NYS -- cited by the Christie's Catalogue, below.

Robert Dixon. Mathographics. Blackwell, 1987, fig. 5.1B, p. 130, is a good photo of the Towie example.

Anna Ritchie. Scotland BC. HMSO, Edinburgh, for Scottish Development Department -- Historic Buildings and Monuments, 1988.

P. 8 has a colour photo of a neolithic cubical ball from the Dark Age fort of Dunadd, Argyll.

P. 14 has a colour photo of a cubical and a tetrahedral ball from Skara Brae, Orkney Islands, c-2800.

Simant Bostock of Glastonbury has made a facsimile of the Towie example, casts of which are available from Glastonbury Film Office, 3 Market Place, Glastonbury, Somerset, BA6 9HD; tel: 01458-830228. You can also contact him at 24 Northload Street, Glastonbury, Somerset, BA6 9JJ; tel: 01458-833267 and he has a mail order catalogue. Since he worked from photographs, there are some slight differences from the original, and the facsimile is slightly larger.

Three examples of tetrahedral stone balls were in Christie's South Kensington antiquities sale of 12 Apr 2000, lots 124 and 125 (2 balls), p. 62, with colour photo of item 124 and the better example in lot 124 on p. 63. (Thanks to Christine Insley Green for a copy of the catalogue.) The descriptive text says 'their exact use is unclear'. Cites Clark, above.

The British Museum has icosahedral dice from Egypt, dated -700/-500.

Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Vol. I, 4th ed., 1906, pp. 174-176. He feels all the regular solids were known to Pythagoras, with the tetrahedron, cube and octahedron having been known long before. Says to see various notices by Count Leopold Hugo in CR 77 for a bronze dodecahedron, a work by Conze on a Celtic bronze example and the paper of Lindemann, below, for a north Italian example. However, he says the dates of these are not determined and I think these are now all dated to later Roman times -- see below. He also notes that moderately regular dodecahedra and icosahedra occur in mineral deposits on Elba and in the Alps and wonders if Pythagoras could have known of these.

Thomas L. Heath. Note about Scholium 1 of Book XIII of Euclid. The Thirteen Books of Euclid's Elements; trans. & ed. by Thomas L. Heath; (1908?); 2nd ed., (1926); Dover, 3 vols., 1956, vol. 3, p. 438. "And it appears that dodecahedra have been found, of bronze or other material, which may belong to periods earlier than Pythagoras' time by some centuries (for references see Cantor's Geschichte der Mathematik I3, pp. 175-6)."

HGM I 160 cites Hugo and Lindemann, dating the Monte Loffa example as -1000/-500.

HGM I 162 discusses the Scholium, giving it as: "the five so-called Platonic figures, which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and icosahedron are due to Theaetetus". He cites Heiberg's Euclid, vol. v., p. 654.

Thomas, SIHGM I 223 says "A number of objects of dodecahedral form have survived from pre-Pythagorean days." But he gives no details or references. Cf Heath's note to Euclid, above.

Plato (-427/-347). Timaeus. c-350. Page references are based on the 1578 edition of Plato which has been used for all later references: pp. 54-56. I use the version in: Edith Hamilton and Huntington Cairns, eds; The Collected Dialogues of Plato including the Letters; Bollingen Series LXXI, Pantheon Books (Random House), (1961), corrected 2nd ptg, 1963, which is the translation of Benjamin Jowett in his The Dialogues of Plato, OUP, (1872), 4th revised ed, 1953. Discusses the regular polyhedra, describing the construction of the tetrahedron, octahedron, icosahedron and cube from triangles since he views the equilateral triangle as made from six 30-60-90 triangles and the square made from eight 45-45-90 triangles. "There was yet a fifth combination which God used in the delineation of the universe with figures of animals." He then relates the first four to the elements: tetrahedron -- fire; octahedron -- air; icosahedron -- water; cube -- earth. [These associations are believed to derive from the Pythagoreans.] However, these associations contribute very little to the rest of the dialogue. The incidental appearance of the dodecahedron lends support to the belief that it was discovered or became known after the initial relation between regular polyhedra and the elements had been established and had to be added in some ad hoc manner. [It is believed that the later Pythagoreans related it to the universe as a whole.]

Scholium 1 of Book XIII of Euclid. Discussed in: The Thirteen Books of Euclid's Elements; op. cit. above, vol. 3, p. 438. Heath's discussion of the Scholia in vol. 1, pp. 64-74, indicates this may be c600. The Scholium asserts that only the tetrahedron, cube and dodecahedron were known to the Pythagoreans and that the other two were due to Theaetetus. Heath thinks the Pythagoreans had all five solids (cf his note to IV.10, vol. 2, pp. 97-100) and the Scholium is taken from Geminus, who may have been influenced by the fact that Theaetetus was the first to write about all five solids and hence the first to write much about the latter two polyhedra.

Stefano de'Stefani. Intorno un dodecaedro quasi regolare di pietra a facce pentagonali scolpite con cifre scoperto nelle antichissime capanne di pietra del Monte Loffa. Atti del Reale Istituto Veneto di Scienze e Lettere, (Ser. 6) 4 (1885) 1437-1459 + plate 18. Separately reprinted by G. Antonelli, Venezia, 1886, which has pp. 1-25 and Tavola 18. This describes perhaps the oldest known reasonably regular dodecahedron, in the Museo Civico di Storia Naturale, Palazzo Pompei, Largo Porta Vittoria 9, Verona, Italy, in the central case of Sala XIX. This is discussed by Herz-Fischler [op. cit. below, p. 61], I have been to see it and the Director, Dr. Alessandra Aspes, has kindly sent me a slide and a photocopy of this article.

The dodecahedron was discovered in 1886 at Monte Loffa, NE of Verona, and has been dated as far back as -10C, but is currently considered to be -3C or -2C [Herz-Fischler, p. 61]. Dr. Aspes said the site was inhabited by tribes who had retreated into the mountains when the Romans came to the area, c-3C. These tribes were friendly with the Romans and were assimilated over a few centuries, so it is not possible to know if this object belongs to the pre-Roman culture or was due to Roman influence. She dates it as -4C/-1C. The stone apparently was cut with a bronze saw and these existed before the Roman incursion (stated in Lindemann, below). It is clearly not perfectly regular -- some of the face angles appear to be 90o and some edges are clearly much shorter than others. But it also seems clear that it is an attempt at a regular dodecahedron -- the faces are quite flat. Its faces are marked with holes and lines, but their meaning and the function of the object are unknown. de'Stefani conjectures it is a kind of die. Lindemann notes that the symbols are not Etruscan nor Greek, but eventually gets to an interpretation of them, which seems not too fanciful, using the values: 3, 6, 9, 10, 12, 15, 16, 20, 21, 24, 60, 300. (Are there any ancient Greek models of the regular polyhedra?) But see also the carved stone balls above.

About 90 examples of a Roman dodecahedron have been found at Roman sites, north of the Alps, from Britain to the Balkans, dating about 200-400. These are bronze and hollow, but also each face has a hole in it, almost always circular, and each corner has a knob at it, making it look like it could be used for Hamilton's Icosian game! The shape is quite precise.

The Society of Antiquaries, London, has the largest extant specimen, dug up on the north side of the Church of St. Mary, Carmarthen, in 1768 and presented to the Society about 1780. The edge length is 2 1/5 in (= 56mm) and, unusually, has plain faces -- almost all examples have some incised decoration on the faces. [Rupert Bruce-Mitford; The Society of Antiquaries of London Notes on Its History and Possessions; The Society, 1951, p. 75, with photograph as pl. XXIV (b) on p. 74.

The Gallo-Romeins Museum (Kielenstraat 15, B-3700 Tongeren, Belgium; Tel: 12-233914) has an example which is the subject of an exhibition and they have produced facsimiles for sale. "The precise significance and exact use of this object have never been explained and remain a great mystery." Luc de Smet says the bronze facsimile is slightly smaller than the original and that the Museum also sells a tin and a bronzed version of the original size.

Other examples are in the Newcastle University Museum (only about half present) and the Hunt Museum, Limerick.

C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp. 24-25. In the first edition, he illustrated this as an example of the mysterious objects which archaeologists turn up and said that it had been described as a toy, a die, a model for teaching measurement of cylinders, a candleholder. His picture shows one opening as being like a key-hole. In the second edition, he added that he had over a hundred suggestions as to what it was for and thinks the most probable answer is that it was a musical instrument.

Jacques Haubrich has recently sent an example of a hollow cubical stone object with different size holes in the faces, apparently currently made in India, sold as a candleholder.

See Thomas, SIHGM I 216-225, for brief references by Philolaus, Aëtius, Plato, Iamblichus.

Euclid. Elements. c-300. Book 13, props. 13-18 and following text. (The Thirteen Books of Euclid's Elements, edited by Sir Thomas L. Heath. 2nd ed., (CUP, 1925??), Dover, vol. 3, pp. 467-511.) Constructs the 5 regular polyhedra in a sphere, compares them. In Prop. 18, he continues "I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another."

Leonardo Pisano, called Fibonacci (c1170->1240). La Practica Geometriae. 1221. As: La Practica Geometriae di Leonardo Pisano secondo la lezione del Codice Urbinate no. 202 della Biblioteca Vaticana. In: Scritti di Leonardo Pisano; vol. II, ed. and pub. by B. Boncompagni; Tipografia delle Scienze Matematiche e Fisiche, Rome, 1862, pp. 1-224. On p. 159, he says there are many polyhedra and mentions there are ones with 8, 12 and 20 faces which Euclid constructs in a sphere in his book XIIII. On pp. 161-162, he describes division in mean and extreme ratio and the construction of the regular pentagon in a circle, then says you can construct, in a sphere, a solid with 20 equilateral triangular faces or with 12 pentagonal faces. After some discussion, he says you can also construct solids with 4, 6, 8, 12, 20 faces, in a sphere. Division in mean and extreme ratio and the construction of the icosahedron are later covered in detail on pp. 196-202. His only drawings of solids are of cubes and pyramids.

Drawings of all the regular polyhedra are included in works, cited in 6.AA, 6.AT.2 and 6.AT.3, by della Francesca (c1480 & c1487), Pacioli (1494), Pacioli & da Vinci (1498), Dürer (1525), Jamnitzer (1568), and Kepler (1619).

F. Lindemann. Zur Geschichte der Polyeder und der Zahlzeichen. Sitzungsber. der math.-phys. Classe k. b. Akademie der Wissenschaften zu München 26 (1896) 625-758 & plates I-IX. Discusses and illustrates many ancient polyhedra. Unfortunately, most of these are undated and/or without provenance. He generally dates them as -7C/5C.

A bronze rhombic triacontahedron, which he dates as first half of the first millennium AD.

Roman knobbed dodecahedra, which he describes as Celtic, going back to the La Tène period (Bronze Age) -- these are now dated to late Roman times. He lists 26 examples listed from the works of Conze and Hugo (cf Cantor, above).

A dodecahedral die; an irregular rhombi-cubo-octahedral die; a bronze dodecahedral die (having two 1s, three 2s, two 3s, one 4 and four 5s).

The Verona dodecahedron (from de'Stefani), which he dates as -1000/-500.

An enamelled icosahedron in Turin with Greek letters on the faces.

An octagonal bipyramid (elongated) from Meclo, South Tyrol, marked with a form of Roman numerals in a somewhat irregular order. It is dated to before the Barbarian migrations.

Three bronze cubo-octahedra.

He then does a long analysis of north-Italian culture and its relations to other cultures and of their number symbols, eventually obtaining an interpretation of the symbols on the Monte Loffa dodecahedron, which he then justifies further with Pythagorean number relations.

Roger Herz-Fischler. A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier University Press, Waterloo, Ontario, 1987. Corrected and extended as: A Mathematical History of the Golden Number; Dover, 1998. P. 61 discusses the history of the dodecahedron and refers to the best articles on the history of polyhedra. Discusses the Verona dodecahedron, see above.

Judith V. Field. Kepler's Geometrical Cosmology. Athlone Press, London, 1988. This gives a good survey of the work of Kepler and his predecessors. In particular, Appendix 4: Kepler and the rhombic solids, pp. 201-219, is most informative. Kepler described most of his ideas several times and this book describes all of them and the relationships among the various versions.

The regular polyhedra in four dimensions were described by Ludwig Schläfli, c1850, but this was not recognised and in the 1880s, several authors rediscovered them.

H. S. M. Coxeter. Regular skew polyhedra. Proc. London Math. Soc. (2) 43 (1937) 33-62. ??NYS -- cited and discussed by Gott, qv.

J. R. Gott III. Pseudopolyhedrons. AMM 74:5 (May 1967) 497-504. Regular polyhedra have their sum of face angles at a vertex being less than 360o and approximate to surfaces of constant positive curvature, while tessellations, with angle sum equal to 360o, correspond to surfaces of zero curvature. The pseudopolyhedra have angle sum greater than 360o and approximate to surfaces of negative curvature. There are seven regular pseudopolyhedra. Each is a periodic structure. He subsequently discovered that J. F. Petrie and Coxeter had discovered three of these in 1926 and had shown that they were the only examples satisfying an additional condition that arrangement of polygons at any vertex have rotational symmetry, and hence that the dihedral angles between adjacent faces are all equal. Coxeter later refers to these structures as regular honeycombs. Some of Gott's examples have some dihedral angles of 180o. Two of these consist of two planes, with a regular replacement of pieces in the planes by pieces joining the two planes. The other five examples go to infinity in all directions and divide space into two congruent parts. He makes some remarks about extending this to general and Archimedean pseudopolyhedra.

6.AT.2 STAR AND STELLATED POLYHEDRA

I have heard it stated that Kircher was the first to draw star polygons.

Paolo Uccello (1397-1475). Mosaic square on the floor at the door of San Pietro in San Marco, Venice. 1425-1430. (This doorway is not labelled on the maps that I have seen -- it is the inner doorway corresponding to the outer doorway second from the left, i.e. between Porta di Sant'Alipio and Porta di San Clemente, which are often labelled.) This seems to show the small stellated dodecahedron {5/2, 5}. This mosaic has only recently (1955 & 1957) been attributed to Uccello, so it can only be found in more recent books on him. See, e.g., Ennio Flaiano & Lucia Tongiorgi Tomase; L'Opera Completa di Paolo Uccello; Rizzoli, Milan, 1971 (and several translations). The mosaic is item 5.A: Rombo con elementi geometrici in the Catalogo delle Opere, with description and a small B&W picture on p. 85. [Bokowski & Wills, below, give the date 1420.]

Coxeter [Elem. der Math. 44 (1989) 25-36] says it "is evidently intended to be a picture of this star polyhedron."

However, J. V. Field tells me that the shape is not truly the small stellated dodecahedron, but just a 'spiky' dodecahedron. She has examined the mosaic and the 'lines' of the pentagrams are not straight. [The above cited photo is too small to confirm this.] She says it appears to be a direct copy of a drawing in Daniele Barbaro; La Practica della Perspettiva; Venice, 1568, 1569, see below, and is most unlikely to be by Uccello. See Field, Appendix 4, for a discussion of early stellations.

In 1998, I examined the mosaic and my photos of it and decided that the 'lines' are pretty straight, to the degree of error that a mason could work, and some are dead straight, so I agree with Coxeter that it is intended to be the small stellated dodecahedron. I now have a postcard of this. However, I have recently seen a poster of a different mosaic of the same shape which is distinctly irregular, so the different opinions may be based on seeing different mosaics!

Both mosaics are viewed directly onto a pentagonal pyramid, but the pyramids are distinctly too short in the poster version. The only spiky dodecahedron in Barbaro is on p. 111, fig. 52, and this is viewed looking at a common edge of two of the pyramids and the pyramids are distinctly too tall, so this is unlikely to be the source of the mosaics. The 'elevated dodecahedron' in Pacioli & da Vinci, plate XXXI, f. CVI-v, has short pyramids and looks quite like the second mosaic, but it is viewed slightly at an angle so the image does not have rotational symmetry. If anyone is in Venice, perhaps they could check whether there are two (or more?) mosaics and get pictures of them.

Luca Pacioli & Leonardo da Vinci. Untitled MS of 1498, beginning: Tavola dela presente opera e utilissimo compendio detto dela divina proportione dele mathematici discipline e lecto -- generally called De divina proportione. Ill. by Leonardo da Vinci. See the entry in 6.AT.3 for fuller details of the facsimiles and details about which plates are in which of the editions.

Discussed by Mackinnon (see in 6.AT.3 below) and Field, pp. 214-215. Clearly shows the stella octangula in one of the superb illustrations of Leonardo, described as a raised or elevated octahedron (plates XVIIII & XX). Field, p. 214, gives the illustration. None of the other raised shapes is a star, but the raised icosahedron is close to a star shape.

Barbaro, Daniele (1514-1570). La Practica della Perspettiva. Camillo & Rutilio Borgominieri, Venice, 1568, 2nd ptg, 1569. (Facsimile from a 1569 copy, Arnaldo Forni, Milan, 1980. The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.) P. 111 has a dodecahedron with pyramids on each face, close to, but clearly not the stellated dodecahedron. P. 112 has an icosahedron with pyramids on each face, again close to, but clearly not the stellated icosahedron. I would have expected a reasonably accurate drawing, but in both drawings, several of the triples of segments which should lie on a single straight line clearly do not. P. 113 shows an icosi-dodecahedron with pyramids on the triangular faces. If the pyramids extended the edges of the pentagons, this would produce the dodecahedron! But here the pyramids distinctly point much further out and the overall perspective seems wrong. [Honeyman, no. 207, observing that some blocks come from the 1566 edition of Serlio which was dedicated to Barbaro.]

Wentzel Jamnitzer (or Jamitzer). Perspectiva Corporum Regularum. With 50 copper plates by Jost Amman. (Nürnberg, 1568.) Facsimile by Akademische Druck- u. Verlagsanstalt, Graz, 1973. [Facsimiles or reprints have also been issued by Alain Brieux, Paris, 1964 and Verlag Biermann und Boukes, Frankfurt, 1972.]

This includes 164 drawings of polyhedra in various elaborations, ranging from the 5 regular solids through various stellations and truncations, various skeletal versions, pseudo-spherical shapes and even rings. Some polyhedra are shown in different views on different pages. Nameable objects, sometimes part of larger drawings, include: tetrahedron, cubo-octahedron, truncated tetrahedron, stella octangula, octahedron, cube, truncated octahedron, rhombi-cuboctahedron, compound of a cube and an octahedron (not quite correct), great rhombi-cuboctahedron, icosahedron, great dodecahedron, dodecahedron, icosi-dodecahedron, rhombi-icosi-dodecahedron, truncated cube, and skeletal versions of: stella octangula, octahedron, cube, icosahedron, dodecahedron, icosi-dodecahedron. There are probably some uniform polyhedra, but I haven't tried to identify them, and some of the truncated and stellated objects might be nameable with some effort.

J. Kepler. Letter to Herwart von Hohenberg. 6 Aug 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 21, letter 130, line 457. ??NYS. Cited by Field, op. cit. below. Refers to (small??) stellated dodecahedron.

J. Kepler. Letter to Maestlin (= Mästlin). 29 Aug 1599. Ibid. Vol. 14, p. 43, letter 132, lines 142-145. ??NYS. Cited by Field, below, and in [Kepler's Geometrical Cosmology; Athlone Press, London, 1988, p. 202]. Refers to (small??) stellated dodecahedron.

J. Kepler. Harmonices Mundi. Godfrey Tampach, Linz, Austria, 1619; facsimile: (Editions) Culture et Civilization, Brussels, 1968 (but my copy is missing three plates!) [Editions probably should have É, but my only text which uses the word Editions is a leaflet in English.] = Joannis Kepleri Astronomi Opera Omnia; ed. Ch. Frisch, Heyder & Zimmer, Frankfurt & Erlangen, 1864, vol. 5. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 6, ??NYS. Book II. Translated by J. V. Field; Kepler's star polyhedra; Vistas in Astronomy 23 (1979) 109-141.

Prop. XXVI, p. 60 & figs. Ss & Tt on p. 53. Describes both stellated dodecahedra, {5/2, 5} and {5/2, 3}. This is often cited as the source of the stella octangula, but the translation is referring to an 'eared cube' with six octagram faces and the stella octangula is clearly shown by Pacioli & da Vinci and by Jamnitzer.

Louis Poinsot. Mémoire sur les polygones et les polyèdres. J. de L'École Polytechnique 4 (1810) 16-48 & plate opp. p. 48. Art. 33-40, pp. 39-42, describe all the regular star polyhedra. He doesn't mention Kepler here, but does a few pages later when discussing Archimedean polyhedra.

A. L. Cauchy. Recherches sur les polyèdres. J. de L'École Polytechnique 16 (1813) 68-86. ??NYS. Shows there are no more regular star polyhedra and this also shows there are no more stellations of the dodecahedron.

H. S. M. Coxeter, P. Du Val, H. T. Flather & J. F. Petrie. The Fifty-Nine Icosahedra. Univ. of Toronto Press, 1938; with new Preface by Du Val, Springer, 1982. Shows that there are just 59 stellations of the icosahedron. They cite earlier workers: M. Brückner (1900) found 12; A. H. Wheeler (1924) found 22.

Dorman Luke. Stellations of the rhombic dodecahedron. MG 41 (No. 337) (Oct 1957) 189-194. With a note by H. M. Cundy which says that the first stellation is well known (see 6.W.4) and that the second and third are in Brückner's Vielecke und Vielfläche, but that the new combinations shown here complete the stellations in the sense of Coxeter et al.

J. D. Ede. Rhombic triacontahedra. MG 42 (No. 340) (May 1958) 98-100. Discusses Coxeter et al. and says the main process generates 8 solids for the icosahedron. He finds that the main process gives 13 for the rhombic triacontahedron, but makes no attempt to find the analogues of Coxeter et al.'s 59.

6.AT.3. ARCHIMEDEAN POLYHEDRA

Archimedes discovered the Archimedean solids, -3C, but his work is lost. Heron quotes some of it and Pappus summarises it. See HGM II 98-101.

Hero of Alexandria (c150). Definitiones. IN: Heronis Alexandrini Opera quae supersunt omnia; Vol. IV, Heronis Definitiones Cum Variis Collectionibus Heronis Quae Feruntur Geometrica, ed. by J. L. Heiberg, Teubner, 1912, pp. 64-67. Heath, HGM I 294-295 has a translation, but it doesn't give the complete text which seems open to two interpretations. The German goes: Archimedes aber sagt, es gebe in ganzen dreizehn Körper, die in einer Kugel eingeschreiben werden können, indem er ausser den genannten fünf noch acht hinzufügt; von diesen habe auch Platon das Tessareskaidekaeder gekannt, dies aber sei ein zweifaches, das eine aus acht Dreiecken und sechs Quadraten zusammengesetzt, aus Erde und Luft, welches auch einige von den Alten gekannt hätten, das andere umgekehrt aus acht Quadraten und sechs Dreiecken, welches schwieriger zu sein scheint. My translation: But Archimedes said, there are in total 13 bodies, which can be inscribed in a sphere, as he added eight beyond the named five [regular solids, which he had just defined]; but of these Plato knew the 14-hedron, however this is a double, one is composed of eight triangles and six squares, from Earth and Air, which some of the ancient also knew, the other conversely [is composed] of eight squares and six triangles, which seems to be more difficult.

Note that Hero has got the numbers wrong - Archimedes found 13 more than the 5 regular solids. Secondly, the 'more difficult' solid does not exist! Heath notes this and suggests that either the truncated cube or the truncated octahedron was intended. The question of interpretation arises at the first semicolon -- is this continuing the statement of Archimedes or is Hero commenting on Archimedes' results? Heath seems to say Archimedes is making the attribution to Plato, but see below. MacKinnon, below, seems to be say this is being made by Hero. Heath's discussion on HGM II 100 says "We have seen that, according to Heron, two of the semi-regular [i.e. Archimedean] solids had already been discovered by Plato" undoubtedly using the method of truncation. However, I don't see that Heron is saying that Plato discovered the cubo-octahedron and the other solid, only that he knew it. Mackinnon says "Plato is said by Heron to have discovered the cuboctahedron by making a model of it from a net." But I don't see that Heron says this.

Pappus. Collection. c290. Vol. 19. In: SIHGM II 194-199. Describes the 13 Archimedean solids. "..., but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons." He then describes each one. Pappus' work has survived in a single MS (Vat. gr. 218) of the 10C in the Vatican and was not copied until 1550, but see Mackinnon, pp. 175-177, on whether it had been seen by Piero. For the history of this MS, see also: Noel M. Swerdlow; The recovery of the exact sciences of antiquity: mathematics, astronomy, geography; IN: Anthony Grafton, ed.; Rome Reborn The Vatican Library and Renaissance Culture Catalog of an exhibition at the Library of Congress, Washington, D.C., Jan. 6 - Apr. 30, 1993; Library of Congress, Washington & Yale University Press, New Haven & London; in association with the Biblioteca Apostolica Vaticana; 1993; pp. 137-139. [This exhibition is on-line at expo/vatican.exhibit/vatican.exhibit.html.]

R. Ripley. Believe It Or Not. 18th series, Pocket Books, NY, 1971. P. 116 asserts the Romans used dice in the shape of cubo-octahedra.

The British Museum, Room 72, Case 9, has two Roman cubo-octahedral dice on display.

F. Lindemann, op. cit. in 6.AT.1, 1896, describes and illustrates an antique rhombic triacontahedron, possibly a die, possibly from the middle of the Byzantine era.

Nick Mackinnon. The portrait of Fra Luca Pacioli. MG 77 (No. 479) (Jul 1993) plates 1-4 & pp. 129-219. Discusses the various early authors, but has mistakes.

della Francesca. Trattato. c1480. Ff. 105r - 117v (224-250) treats solid bodies, discussing all the regular polyhedra, with figures, though Arrighi gives only a projection of the octahedron. Discusses and gives good diagrams of the truncated tetrahedron and cubo-octahedron, apparently the first drawings of any Archimedean polyhedra. Jayawardene refers to the cubo-octahedron as a truncated cube.

Davis notes that Pacioli's Summa, Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Piero della Francesca. Libellus de Quinque Corporibus Regularibus. c1487 [Davis, p. 44, dates it to 1482-1492]. Piero would have written this in Italian and it is believed to have been translated into Latin by Matteo da Borgo [Davis, p. 54], who improved the style. First post-classical discussion of the Archimedean polyhedra, but it was not published until an Italian translation (probably by Pacioli) was printed in Pacioli & da Vinci, qv, in 1509, as: Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., ff. 1-27. A Latin version was discovered by J. Dennistoun, c1850, and rediscovered by Max Jordan, 1880, in the Urbino manuscripts in the Vatican -- MS Vat.Urb.lat. 632; the Duke of Urbino was a patron of Piero and in the MS, Piero asks that it be placed by his De Prospectiva Pingendi in the Duke's library. This was published by Girolamo Mancini in: L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, Memorie della Reale Accademia dei Lincei, Classe di Scienze Morali, Storiche e Filologiche (5) 14:7B (1915) 441-580 & 8 plates, also separately published by Tipografia della Reale Accademia dei Lincei, Rome, 1916. Davis identifies 139 problems in this, of which 85 (= 61%) are taken from the Trattato. There is debate as to how much of this work is due to Piero and how much to Pacioli. The Latin text differs a bit from the Italian. See the works of Taylor and Davis in Section 1 under Pacioli and the discussion on della Francesca's Trattato and Pacioli's Summa in the common references.

He describes a sphere divided into 6 zones and 12 sectors. Mackinnon says Piero describes seven of the Archimedean polyhedra, but without pictures, namely: cuboctahedron, truncated tetrahedron, truncated cube, truncated octahedron, truncated dodecahedron, truncated icosahedron, rhombi-cuboctahedron. Field, op. cit. in 6.AT.1, p. 107, says Piero gives six of the Archimedean polyhedra. In recent lectures Field has given a table showing which Archimedean polyhedra appear in Piero, Pacioli, Dürer and Barbaro and this lists just the first six of the above as being in Piero. I find just the five truncated regular polyhedra -- see above for the cubo-octahedron -- and there is an excellent picture of the truncated tetrahedron on f. 22v of the printed version. Mancini gives different diagrams than in the 1509 printed version, including clear pictures of the truncated icosahedron and the truncated dodecahedron. della Francesca clearly has the general idea of truncation. An internet biographical piece, apparently by, or taken from, J. V. Field, (), shows that the counting is confused by the presence of the cubo-octahedron in the Trattato but not in the Libellus. So della Francesca rediscovered six Archimedean polyhedra, but only five appear in the Libellus. The work of Pappus was not known at this time.

Pacioli. Summa. 1494.

f. 4. Brief descriptions of the cubo-octahedron, truncated tetrahedron, icosidodecahedron, truncated icosahedron. No drawings.

Part II, ff. 68v - 72r, sections 2 (unlabelled) - 35. Discussion and some crude drawings of the regular polyhedra, the truncated tetrahedron and the cubo-octahedron. Mackinnon says these are the first printed illustrations of any Archimedean polyhedra. Davis notes that Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Jacopo de'Barbari or Leonardo da Vinci. Portrait of Fra Luca Pacioli. 1495. In the Museo Nazionale di Capodimonte, Naples. The upper left shows a glass rhombi-cuboctahedron half filled with water. Discussed by Mackinnon, with colour reproduction on the cover. Colour reproduction in Pacioli, Summa, 1994 reprint supplement.

Luca Pacioli & Leonardo da Vinci. Untitled MS of 1498, beginning: Tavola dela presente opera e utilissimo compendio detto dela divina proportione dele mathematici discipline e lecto -- generally called De divina proportione. Three copies of this MS were made. One is in the Civic Library of Geneva, one is the Biblioteca Ambrosiana in Milan and the third is lost. Three modern versions of this exist.

Transcription published as Fontes Ambrosiani XXXI, Bibliothecae Ambrosianae, Milan, 1956. This was sponsored by Mediobanca as a private edition. There is a copy at University College London.

Colour facsimile of the Milan copy, Silvana Editoriale, Milano, (1982), 2nd ptg, 1986. With a separate booklet giving bibliographical details and an Introduzione di Augusto Marinoni, 20pp + covers. The booklet indicates this is Fontes Ambrosiani LXXII.

Printed version: [De] Divina proportione Opera a tutti glingegni perspicaci e curiosi necessaria Ove ciascun studioso di Philosophia: Prospectiva Pictura Sculptura: Architectura: Musica: e altre Mathematice: suavissima: sottile: e admirabile doctrina consequira: e delectarassi: cōvarie questione de secretissima scientia. Ill. by Leonardo da Vinci. Paganino de Paganini, Brescia, 1509. Facsimile in series Fontes Ambrosiana, no. XXXI, Milan, 1956; also by Editrice Dominioni, Maslianico (Como), 1967. (On f. 23r, the date of completion of the original part is printed as 1497, but both MSS have 1498.)

The printed version was assembled from three codices dating from 1497-1498 and contains the above MS with several additional items. However, the diagrams in the text are simplified and the plates are in a different order. The MS has 60 coloured plates, double sided; the printed version has 59 B&W plates, single sided. There are errors of pagination and plate numbering in both versions. On f. 3 of the printed version is a list of plates and one sees that plate LXI should be numbered LVIIII and that plates LX, LXI were omitted and were to have been a hexagonal pyramid in solid and framework views (the framework view is in the MS, but the solid pyramid is not).

NOTE. Simon Finch's Catalogue 48, item 4, describes the copy that was in the Honeyman Collection and says it has 59 printed plates of geometric figures and is unique in having two contemporary additional MS plates showing the hexagonal pyramid (numbered LX and LXI), which are given in the list of plates, but which do not appear in any other known copy. It seems that these figures were overlooked in printing and that the owner of the Honeyman copy decided to make his own versions, or, more likely, got someone to make versions in the original style. There is a framework hexagonal pyramid in the MS, and this makes it seem likely that these figures had been prepared and were omitted in printing -- indeed the Honeyman leaves could be the overlooked drawings. That leaves the question of whether there was a solid hexagonal pyramid in the MS?

Most pictures come in pairs -- a solid figure and then a framework figure. There are the five regular polyhedra, the following six Archimedean polyhedra: truncated tetrahedron, cubo-octahedron, truncated octahedron, truncated icosahedron, icosi-dodecahedron, rhombi-cubo-octahedron and also the stella octangula. There are raised or elevated versions of the tetrahedron, cube, cubo-octahedron, icosahedron, dodecahedron, icosi-dodecahedron, rhombi-cubo-octahedron. Also triangular, square, pentagonal and hexagonal prisms and tall triangular, square and pentagonal pyramids. Also a triangular pyramid not quite regular and a sphere divided into 12 sectors and 6 zones. There are also a solid sphere, a solid cylinder, a solid cone and a framework hexagonal pyramid (the last is not in the printed version). Mackinnon says they give the same seven Archimedean polyhedra as Piero, but Piero gives five or six and Pacioli & da Vinci gives six, with only four common polyhedra. Pacioli & da Vinci assert that the rhombi-cuboctahedron arises by truncating a cuboctahedron, but this is not exactly correct.

Part of the printed version is Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., which is an Italian translation (probably by Luca Pacioli) of Piero della Francesca's Libellus de quinque corporibus regularibus. There is debate as to whether this was actually written by Pacioli or whether Pacioli plagiarized it and whether it actually appeared in the 1509 printing or was added to a later reprinting, etc.

Davis [p. 65] says the drawings were made from models prepared by Da Vinci. Davis [p. 74] cites Summa, Part II, f. 68v, and she quotes part of it on pp. 100-101. This is also referred to by MacKinnon [p. 170] and Taylor [p. 344], neither giving details and no two of the three agreeing on what the passage means. I have not been able to make complete sense of the passage, but it seems clearly to say that in Apr 1489, Pacioli presented models of at least the regular solids to the Duke of Urbino at the palace of Pacioli's protector [Cardinal Giuliano della Rovere, later Pope Julius II] in Rome. He then says many other dependants [= variations] of the regular solids can be made, and models were made for Pietro Valletari, Bishop of Carpentras. There is no reference to the number of models, nor their material, nor to a set being given to the Cardinal, nor whether the Cardinal was present when the models were given to the Duke. Due to a missing right parenthesis, ), the sense of one statement involving 'his own hands' could mean either that Pacioli presented the models to the Duke's own hands or that Pacioli had decorated the models himself. I doubt whether there were many other solids at this time, otherwise he would have mentioned them in the Summa -- the Summa only describes four of the Archimedean solids and only two of them are in Part II. I suggest that he didn't start developing the other shapes until about 1494, or later, in 1496 when he went to Milan and met Leonardo.

However, on f. 28v of De Divina Proportione, Pacioli is clearer and says that he arranged, coloured and decorated with his own hands 60 models in Milan and two other sets for Galeazzo Sanseverino in Milan and for Piero Soderini in Florence. This refers to his time in Milan, which was 1496-1499, though the Soderini set might have been made after Pacioli and da Vinci moved to Florence.

Albrecht Dürer. Underweysung der messung .... 1525 & 1538. Op. cit. in 6.AA. Figures 29-43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show a net of each of the regular polyhedra, an approximate sphere (16 sectors and 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi-cubo-octahedron, snub cube, great rhombi-cubo-octahedron, polyhedron of six dodecagons and thirty-two triangles (having a pattern of four triangles replacing each triangle of the cubo-octahedron, so a sort of truncated cubo-octahedron -- not an Archimedean solid and not correctly drawn) and an elongated hexagonal bipyramid (not even regular faced). This gives 7 of the 13 non-regular Archimedean polyhedra. Mackinnon says Figures 29-41 show a net of each of the regular polyhedra and the same seven Archimedean ones as given by Pacioli & da Vinci, but they give six and there are only four common ones. In the revised version of 1538, figure 43 is replaced by the truncated icosahedron and icosi-dodecahedron (figures 43 & 43a, pp. 414-419 in The Painter's Manual), giving 9 of the 13 non-regular Archimedean solids. P. 457 shows the remaining four Archimedean cases from an 1892 edition.

Albrecht Dürer. Elementorum Geometricorum (?). 1534. Op. cit. in 6.AA. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

See Barbaro, 1568, in 6.AT.2, pp. 45-104 for drawings and nets of 11 of the 13 Archimedean solids - he omits the two snub solids.

See Jamnitzer, 1568, in 6.AT.2 for drawings of eight of the 13 Archimedean solids.

J. Kepler. Letter to Maestlin (= Mästlin). 22 Nov 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 87, letter 142, lines 21-22. ??NYS. Described by Field, p. 202. Describes both rhombic solids.

J. Kepler. Strena seu De Nive Sexangula [A New Year's Gift or The Six-Cornered Snowflake]. Godfrey Tampach, Frankfurt am Main, 1611. (Reprinted in: Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 4, ??NYS.) Reprinted, with translation by C. Hardie and discussion by B. J. Mason & L. L. Whyte, OUP, 1966. I will cite the pages from Kepler (and then the OUP pages). P. 7 (10-11). Mentions 'the fourteen Archimedean solids' [sic!]. Describes the rhombic dodecahedron and mentions the rhombic triacontahedron. The translator erroneously adds that the angles of the rhombi of the dodecahedron are 6Oo and 120o. Kepler adds that the rhombic dodecahedron fills space. Kepler's discussion is thorough and gives no references, so he seems to feel it was his own discovery.

J. Kepler. Harmonices Mundi, 1619. Book II, opp. cit. above. Prop. XXVII, p. 61. Proves that there are just two rhombic 'semi-regular' solids, the rhombic dodecahedron and the rhombic triacontahedron, though the cube and the 'baby blocks' tessellation can also be considered as limiting cases. He illustrates both polyhedra. Def. XIII, p. 50 & plate (missing in facsimile). Mentions prisms and antiprisms. Prop. XXVIII, pp. 61-65. Finds the 13 Archimedean solids and illustrates them -- the first complete set -- but he does not formally show existence.

J. Kepler. Epitome Astronomiae Copernicanae. Linz, 1618-1621. Book IV, 1620. P. 464. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 7, p. 272. Shows both rhombic solids.

The following table shows which Archimedean polyhedra appear in the various early books from della Francesca to Kepler.

dF = della Francesca, c1487, folio

T indicates the object appears in the Trattato of c1480.

P = Pacioli, Summa, Part II, 1494, folio.

P&dV = Pacioli & da Vinci, 1498, plate number of the solid version; the framework version is

the next plate

D = Dürer, 1525, plate

+ indicates the object is added in the 1538 edition.

B = Barbaro, 1568, page

J = Jamnitzer, 1568, plate

K = Kepler, 1619, figure

DF P P&dV D B J K

Truncated tetrahedron T,22v 4v,II-69v 3 35 56 A3 2

Truncated cube 22r 36 61 G2 1

Truncated dodecahedron 21r 76 3

Truncated octahedron 21v 17 38 68 B2,B4 5

Great rhombi-cubo-octahedron 41 88 B6 6

Great rhombi-icosidodecahedron 100 7

Truncated icosahedron (football!) 20v 4v 23 + 81 4

Cubo-octahedron T 4v,II-69v 9 37 58 B6,F1 8

Icosidodecahedron 4v 29 71 D4,F6 9

Rhombi-cubo-octahedron 35 39 64 B6 10

Rhombi-icosidodecahedron 94 D4 11

Snub cube 40 12

Snub dodecahedron 13

NUMBER 5(6) 4 6 7(9) 11 8 13

Richard Buckminster Fuller. Centre spread card version of his Dymaxion World map on the cubo-octahedron. Life (15 Mar 1943). Reproduced in colour, with extended discussion, in: Joachim Krausse & Claude Lichtenstein, eds; Your Private Sky R. Buckminster Fuller The Art of Design Science [book accompanying a travelling exhibition in 2000]; Lars Müller Publishers, Bade, Switzerland, 1999, pp. 250-275. (This quotes a Life article on 1 Mar 1943 and a Fuller article, Fluid Geography, of 1944 -- ??NYS. It also reproduces a 1952 colour example of the icosahedral version.)

Richard Buckminster Fuller. US Patent 2,393,676 -- Cartography. Filed: 25 Feb 1944; granted: 29 Jan 1946. 3pp + 5pp diagrams. His world map on the cubo-octahedron. It was later put on the icosahedron. One page is reproduced in: William Blackwell; Geometry in Architecture; Key Curriculum Press, Berkeley, 1984, p. 157.

J. H. Conway. Four-dimensional Archimedean polytopes. Proc. Colloq. Convexity, Copenhagen, 1965 (1967) 38-39. ??NYS -- cited by Guy, CMJ 13:5 (1982) 290-299.

6.AT.4. UNIFORM POLYHEDRA

H. S. M. Coxeter, M. S. Longuet-Higgins & J. C. P. Miller. Uniform polyhedra. Philos. Trans. Roy. Soc. 246A (1954) 401-450. They sketch earlier work and present 53 uniform polyhedra, beyond the 5 Platonic, 13 Archimedean and 4 Kepler-Poinsot polyhedra and the prisms and anti-prisms. Three of these uniform polyhedra are actually infinite families. "... it is the authors' belief that the enumeration is complete, although a rigorous proof has still to be given."

S. P. Sopov. Proof of completeness of the list of uniform polyhedra. Ukrain. Geometr. Sb. 8 (1970) 139-156. ??NYS -- cited in Skilling, 1976.

J. S. Skilling. The complete set of uniform polyhedra. Philos. Trans. Roy. Soc. London Ser. A 278 (1975) 111-135. Demonstrates that the 1954 list of Coxeter, et al., is complete. If one permits more than two faces to meet at an edge, there is one further polyhedron -- the great disnub dirhombidodecahedron.

J. S. Skilling. Uniform compounds of uniform polyhedra. Math. Proc. Camb. Philos. Soc. 79 (1976) 447-468. ??NYS -- I am told it determines that there are 75 uniform compounds and also cites Sopov.

6.AT.5. REGULAR-FACED POLYHEDRA

O. Rausenberger. Konvexe pseudoreguläre Polyeder. Zeitschr. für math. und naturwiss. Unterricht 46 (1915) 135-142. Finds the eight convex deltahedra.

H. Freudenthal & B. L. van der Waerden. Over een bewering van Euclides [On an assertion of Euclid] [in Dutch]. Simon Stevin 25 (1946/47) 115-121. ??NYS. Finds the eight convex deltahedra -- ignorant of Rausenberger's work.

H. Martyn Cundy. "Deltahedra". MG 36 (No. 318) (Dec 1952) 263-266. Suggests the name "deltahedra". Exposits the work of Freudenthal and van der Waerden, but is ignorant of Rausenberger. Considers non-convex cases with two types of vertex and finds only 17 of them. Considers the duals of Brückner's trigonal polyhedra.

Norman W. Johnson. Convex polyhedra with regular faces. Canad. J. Math. 18 (1966) 169-200. (Possibly identical with an identically titled set of lecture notes at Carleton College, 1961, ??NYS.) Lists 92 such polyhedra beyond the 5 regular and 13 Archimedean polyhedra and the prisms and antiprisms.

Viktor A. Zalgaller. Convex polyhedra with regular faces [in Russian]. Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, vol. 2 (1967). ??NYS. English translation: Consultants Bureau, NY, 1969, 95pp. Gives details of computer calculations which show that Johnson's list is complete. Defines a notion of simplicity and shows that the simple regular-faced polyhedra are the prisms, the antiprisms (excepting the octahedron) and 28 others. Names all the polyhedra and gives drawings of the simple ones.

6.AT.6. TESSELLATIONS

Albrecht Dürer. Underweysung der messung .... 1525 & 1538. Op. cit. in 6.AA. Figures 22-27 (pp. 156-169 in The Painter's Manual, Dürer's 1525 ff. E-vi-v - F-iii-v) show: the three regular tessellations; the quasi-regular one, 3636, and some of its dual; several irregular ones, including some partial tessellations with pentagons; and the truncated square lattice, 482. In the revised version of 1538, he adds some tilings by rhombuses (figures 23a & 24, pp. 410-411 in The Painter's Manual).

Albrecht Dürer. Elementorum Geometricorum (?). Op. cit. in 6.AA, 1532. Book II, fig. 22-27, pp. 62-67, is the material from the 1525 version.

J. Kepler. Letter to Herwart von Hohenberg. 6 Aug 1599. Op. cit. in 6.AT.2. Field, p. 105, says Kepler discusses tessellations here and this is the earliest of his writings to do so.

J. Kepler. Harmonices Mundi. 1619. Book II. Opp. cit. above. Prop. XVIII, p. 51 & plate (missing in my facsimile). Shows there are only three regular plane tessellations and mentions the dual of 3636, which is the 'baby blocks' tessellation. Prop. XIX-XX, pp. 51-56 & four plates (three missing in my facsimile). Finds the 8 further Archimedean tessellations and 7 of the 10 further ways to fill 360 degrees with corners of regular polygons. He misses 3,7,42; 3,8,24; 3,9,18 despite computing, e.g., that a triangle and a heptagon would leave a gap of 40/21 of a right angle. Field, p. 109, notes that Kepler doesn't clearly have all vertices the same in some pictures -- e.g. he has both 3366 and 3636 patterns in his figure R.

Koloman Moser. Ver Sacrum. 1902. This Viennese art nouveau drawing is considered to be the first tessellation using life-like figures. It has trout and the pattern has symmetry (or wallpaper) group pg and isohedral type IH2.

Branko Grünbaum & G. C. Shephard. Tilings and Patterns. Freeman, 1986. I haven't examined this thoroughly yet, but it clearly is the definitive work and describes everything known to date.

6.AT.6.a. TESSELLATING WITH CONGRUENT FIGURES

This is a popular topic which I have just added. Gardner's article and addendum in Time Travel gives most recent results, so I will just give just some highlights. The facts that any triangle and any quadrilateral will tile the plane must be very old, perhaps Greek, but I have no early references. Generally, I will consider convex polygons and most items only deal with the plane.

David Hilbert. Mathematische Probleme. Göttinger Nachrichten (= Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Klasse) 3 (1900) 253-297. This has been reprinted and translated many times, e.g. in the following.

R. Bellman, ed. A Collection of Modern Mathematical Classics -- Analysis. Dover, 1961. Pp. 248-292 [in German].

Translated by M. W. Newson. Bull. Amer. Math. Soc. 8 (1902) 437-479. Reprinted in: F. E. Browder, ed. Mathematical Developments Arising from Hilbert Problems. Proc. Symp. Pure Math. 28 (1976) 1-34.

Problem 18: Aufbau des Raumes aus kongruenten Polyedern [Building up of space from congruent polyhedra]. "The question arises: Whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of space is possible." Hilbert also asks two other questions in this problem.

The problem is discussed by John Milnor in his contribution to the Symposium, but he only shows non-convex 8- & 10-gons which fill the plane.

K. Reinhardt. Über die Zerlegung der Ebene in Polygone. Dissertation der Naturwiss. Fakultät, Univ. Frankfurt/Main, Borna, 1918. ??NYS -- cited by Kershner. Finds the three types of hexagons and the first five types of pentagons which fill the plane.

Max Black. Reported in: J. F. O'Donovan; Clear thinking; Eureka 1 (Jan 1939) 15 & 20. Problem 2: which quadrilaterals can tile the plane? Answer: all!

R. B. Kershner. On paving the plane. AMM 75:8 (Oct 1968) 839-844. Says the problem was posed by Hilbert. Gives exhaustive lists of hexagons and a list of pentagons which he claimed to be exhaustive. Cites previous works which had claimed to be exhaustive, but he has found three new types of pentagon.

J. A. Dunn. Tessellations with pentagons. MG 55 (No. 394) (Dec 1971) 366-369. Finds several types and asks if there are more.

M. M. Risueño, P. Nsanda Eba & Editorial comment by Douglas A. Quadling. Letters: Tessellations with pentagons. MG 56 (No. 398) (Dec 1972) 332-335. Risueño's letter replies to Dunn by citing Kershner. Eba constructs a re-entrant pentagon. [This is not cited by Gardner.]

Gardner. On tessellating the plane with convex polygon tiles. SA (Jul 1975). Much extended in Time Travel, chap. 13.

Ivan Niven. Convex polygons that cannot tile the plane. AMM 85 (1978) 785-792. n-gons, with n > 6, cannot tile the plane.

Doris Schattschneider. In praise of amateurs. In: The Mathematical Gardner; ed. by David A. Klarner; Wadsworth, Belmont, California, 1981, pp. 140-166 & colour plates I-V between 166 & 167. Surveys history after Kershner, describing contributions of James & Rice.

Gardner. On tessellating the plane with convex polygon tiles. [Originally: SA (Jul 1975).] Much extended in Time Travel, 1988, chap. 13. The original article generated a number of responses giving new pentagonal tilings, making 14 types in all. Good survey of the recent literature.

6.AT.7. PLAITING OF POLYHEDRA

New section.

John Gorham. A System for the Construction of Crystal Models on the Type of an Ordinary Plait: Exemplified by the Forms Belonging to the Six Axial Systems in Crystallography. E. & F. N. Spon, London, 1888. Gorham's Preface says he developed the idea and demonstrated it to the Royal Society some 40 years earlier.

A. R. Pargeter. Plaited polyhedra. MG 43 (No. 344) (May 1959) 88-101. Cites and quotes Gorham. Extends to plaiting dodecahedron, icosahedron and some archimedean, dual and stellated examples.

J. Brunton. The plaited dodecahedron. MG 44 (No. 347) (Feb 1960) 12-14. With comment by Pargeter. Obtains a 3-plait which almost completes the dodecahedron.

6.AT.8. DÜRER'S OCTAHEDRON

New section -- I know of other articles claiming to 'solve' the problem.

Albrecht Dürer. Melencolia I. 1514. Two impressions are in the British Museum. In the back left is an octahedron whose exact shape is the subject of this section. It looks like a cube truncated at two opposite corners, but the angles do not quite look like 90o.

Albrecht Dürer. Dresden Sketchbook. Facsimile as The Human Figure, the complete Dresden Sketchbook; Dover, NY, 1972. ??NYS -- cited by Sharp. This has a sketch of the solid with hidden lines indicated, so the combinatorial shape is definitely known and is a hexahedron of six equal faces, truncated at two opposite corners.

E. Schröder. Dürer Kunst und Geometrie. Birkhäuser, Basel, 1980. ??NYS -- cited by Sharp and MacGillavry.

Caroline H. MacGillavry. The polyhedron in A. Dürer's Melencolia I An over 450 years old puzzle solved? Koninklijke Nederlandse Akademie van Wetenschappen Proc. B 84:3 (28 Sep 1981) 287-294. The rhombohedral angle, i.e. the angle between edges at the truncated top and bottom vertices of the rhombohedron, was estimated as 72o by Grodzinski. She determines it is 79o ± 1o. She then built and photographed such a polyhedron and then computed its projection, both of which seem identical to Dürer's picture. Crystallographers believe Dürer was drawing an actual crystal, with a form of calcite having rhombohedral angle of 76o being the closest known shape, though it is not known to have been studied in Dürer's time, so others have suggested fluorite, though fluorite has two standard forms, neither of this form, but Dürer's 'hybrid' artistic version could have been derived from them.

Terence Lynch. The geometric body in Dürer's engraving Melencolia I. J. Warburg and Courtauld Institutes 45 (1982) 226-232 & plate a on p. 37?. Lots of references to earlier work. Notes that perspective was not sufficiently advanced for Dürer to construct a general drawing of such an object. After many trials, he observes that a parallel projection of the solid fits onto a 4 x 4 grid -- like the magic square in the picture -- and that symmetry then permits the construction with straight edge and compass (which are both shown in the picture). This shows that the original faces are rhombuses whose diameters are in the ratio 2 : (3. And the dihedral angle between the triangular faces and the cut off rhombuses is 30o Further, the actual drawing can then be made by one of the simplified perspective techniques known to Dürer. However, Dürer has taken a little bit off the top and bottom of the figure and this distortion has misled many previous workers.

John Sharp. Dürer's melancholy octahedron. MiS (Sep 1994) 18-20. Asserts that the shape was first determined by Schröder in 1980 and verified by Lynch.

6.AT.9. OTHER POLYHEDRA

New section.

Stuart Robertson. The twenty-two cuboids. Mathematics Review 1:5 (May 1991) 18-21. This considers polyhedra with six quadrilateral faces and determines what symmetries are possible -- there are 22 different symmetry groups.

6.AU. THREE RABBITS, DEAD DOGS AND TRICK MULES

See S&B, p. 34.

Loyd's Trick Mules has two mules and two riders which can only be placed correctly by combining each front with the other rear.

Earlier forms showed two dead dogs which were brought to life by adding four lines. The resulting picture is a pattern, generally called 'Two heads, four children' and can be traced back to medieval Persian, Oriental and European forms.

The three rabbits problem is: "Draw three rabbits, so that each shall appear to have two ears, while, in fact, they have only three ears between them." Until about 1996, I only knew this from the 1857 Magician's Own Book and the many books which copied from it. Someone at a conference at Oxford in 1996 mentioned that the pattern occurs in a stained glass window at Long Melford, Suffolk. Correspondence revealed that the glass is possibly 15C and the pattern was apparently brought from Devon about that time. More specifically, it comes from the east side of Dartmoor and inquiries there have turned up numerous examples as roof bosses from 13-16C. Totally serendipitously, I was reading a guide book to Germany in 1997 and discovered the pattern occurs in stonework, possibly 16C, at Paderborn, Germany. A letter led to receiving a copy of Schneider's article (see below) which described the pattern occurring at Dunhuang, c600. I am indebted to Miss Y. Yasumara, the Art Librarian at the School of Oriental and African Studies, for directing me to several works on Dunhuang. However, I have not examined all these works in detail (the largest is five large volumes), so I may not have found all the examples of this pattern. Miss Yasumara also directed me to Roderick Whitfield, of the School of Oriental and African Studies, who tells me there is no other example of this pattern in Chinese art, and to Susan Whitfield (no relation), head of the International Dunhuang Project at the British Library. However, Greeves (see his articles, below) has found other examples of the pattern in Europe, Iran and Tibet and found that modern carpets with the pattern are being made in China. A student of his recently went to Dunhuang and the locals told her that the pattern came from 'the West', meaning India, which opens up a whole new culture to examine.

In 1997, I visited the Dartmoor area, seeing several examples and finding a reference (Hambling, below) to Tom Greeves' article. In 2000, I again visited the area, seeing more churches and meeting Tom Greeves and his associate Sue Andrew. Later in 2000, I visited Paderborn. Later in 2000, I showed this material to Wei Zhang and Peter Rasmussen, leading collectors of Chinese puzzles and they have begun to investigate the Chinese material much more thoroughly than I have done -- see below. In 2001, I went again to the area.

I have read that rabbits were introduced to England in 1176 by the Normans and became common in the 13C, though I believe they weren't really wild for some time after that. E.g. [J. A. R. Pimlott; Recreations; Studio Vista, 1968, p. 18] says: "The rabbit was not known in Britain until the thirteenth century and did not become plentiful until the fifteenth", citing Elspeth M. Veale; The English Fur Trade in the Later Middle Ages; 1966. I have read that hares were introduced between -500 and +500, but I have just seen a mention that bones of a hare found in Ireland have been carbon dated to -26,000. So it is probable that the animals in many of the images are hares, though some of the images are distinctly more rabbit-like than hare-like.

The material in this section has grown so much that it is now divided into seven subsections: China; Other Asia; Paderborn; Medieval Europe; Modern Versions of the Three Rabbits Puzzle; Dead Dogs; Trick Mules.

I have about a dozen letters and emails which have not yet been processed.

Rabbits going clockwise: Dunhuang (14 caves -- all except 407 & 420); Goepper; St. Petersburg; Iranian tray; Paderborn; Münster; Bestiary; Lyon; Throwleigh; Valentine; Clyst Honiton; Hasloch am Main; Michelstadt; Collins; Greeves letterhead; Urumqi;

Rabbits going anti-clockwise: Dunhuang (2 caves -- 407 & 420); Corbigny; Lombard's Gloss; North Bovey; Long Crendon; Chester; Widecombe; Long Melford; Chagford; Best Cellars, Chagford; Tavistock; Broadclyst; Sampford Courtenay; Spreyton; Paignton; South Tawton; Valentine; Schwäbisch Hall; Baltrušaitis Fig. 97; Child; Magician's Own Book et al; Warren Inn; Best Cellars, Chagford; Newman; Lydford; Trinity Construction Services;

The choice of clockwise versus anticlockwise seems to be random! Except the Chinese clearly preferred clockwise.

FOUR RABBITS versions. Baltrušaitis; Wilson; Goepper. Bestiary; Andrew/Lombard; Lyon; Hamann-MacLean;

MORE FIGURES. Chichester Cathedral. Boxgrove Priory.

CHINA

Jurgis Baltrušaitis. Le Moyen Age Fantastique Antiquités et exotismes dans l'art gothique. (A. Colin, Paris, 1955, 299 pp.) Revised, Flammarion, Paris, 1981, 281 pp. Thanks to Peter Rasmussen for telling me about this. Supposedly an English edition was published in 1998, but I have found an entry in the Warburg catalogue for The Fantastic in the Middle Ages, published by Boydell & Brewer, Woodbridge, Suffolk, 2000, marked 'order cancelled' -- so the publication seems to have never happened. This is a major source used for When Silk Was Gold, below. Pp. 132-139 of the 1981 edition have many examples of three and four rabbits, four boys, etc. He gives a small illustration (Fig. 96B) from Dunhuang (Touen-houang) (6C-10C), the oldest example he knows, and many others. See other sections for more details.

[Huang Zu'an ?? -- Schneider, below, gives this author, but there is no mention of an author in the entire issue.] Dunhuang -- Pearl of the Silk Road. China Pictorial (1980:3) 10-23 with colour photo on p. 22. 9th article in a series on the Silk Road. Colour photo of the three rabbits pattern with caption: "A ceiling design. The three rabbits with three ears and the apsarases seem to be whirling. Cave 407. Sui Dynasty." The Sui dynasty was from 581 to 618, so we can date this as c600. The image is rather small, but the three hares can be made out. There is no discussion of the pattern in the article.

Chang Shuhong & Li Chegxian. The Flying Devis of Dunhuang. China Travel and Tourism Press, Beijing, 1980. Unpaginated. In the Preface, we find the following. "What is particularly novel is the full-grown lotus flower painted in the centre of the canopy design on the ceiling of Cave 407. In the middle of the flower there are three rabbits running one after the other in a circle. For the three rabbits only three ears are painted, each of them borrowing one ear from another. This is an ingenious conception of the master painter." From this, it seems that this pattern is uncommon. The best picture of the pattern that I have located is in this book, in the section on the Sui period. I have now acquired a copy with its dust jacket and find a painting of the pattern is on the back of the dust jacket -- this is the painting also given in Li Kai et al, below, p, 27. [Incidentally, a devi or apsaras is a kind of Buddhist angel. The art of Dunhuang is quite lovely.]

The Dunhuang Institute for Cultural Relics. The Mogao Grottos of Dunhuang. 5 vols. + Supplement. Heibonsha Ltd., Tokyo, 1980-1982. (In Japanese, with all the captions given in English at the end of the Supplementary volume. Fortunately the plate and cave numbers are in western numerals. A Chinese edition was planned.) Vol. 2, plate 94, is a double-page spread of the ceiling of Cave 407, with the page division running right through the middle of the rabbits pattern! Vol. 2, plate 95, is a half-page plate of the ceiling of Cave 406, and shows the rabbits pattern, but it seems rather faded. The English captions simply say "Ornamental ceiling decoration".

R. Whitfield & A. Farrer. Caves of the Thousand Buddhas. British Museum, 1990, esp. pp. 12 & 16. Though cited by Greeves, these pages only have general material on Dunhuang and the book does not mention any of the relevant caves.

Duan Wenjie. Dunhuang Art Through the Eyes of Duan Wenjie. Indira Gandhi National Centre for the Arts, Abhinav Publications, New Delhi, 1994. This gives much more detail about the caves. Pp. 400-401 describes Caves 406-407. Peter Rasmussen examined the book in detail and found it mentions 12 other caves with the pattern. Peter has found that the book is accessible on-line at ignca.nic.in/ks_19.htm. This provides the facility to download a font which will display diacritical marks and this is worth doing before you start to browse.

In late 2002, Peter Rasmussen and Wei Zhang were able to inspect 10 caves that they hadn't seen before. Peter sent notes of their impressions of 12 caves on 24 Nov 2002 and I will add some of Peter's comments and additions in [ ] and marked PR. This will make the following list the basic list of all sixteen of the caves.

Cave No. 127. Late Tang (renovated in Five Dynasties and Qing). "The ceiling exhibits lotus and three rabbits (joining as one) in the centre." [PR: tan on turquoise, going clockwise.]

Cave No. 139. Late Tang. "The ceiling shows the three rabbits (joining as one) and lotus designs in the centre." [PR: this is a small cave off the entry to Cave 138. Tan on light green, going clockwise, in excellent condition. "Rabbits beautifully drawn in pen-like detail, showing toes, eyes (with eyeballs!), all four legs, tail, nose, mouth, outline of thigh muscle, and hair on stomach, breast, legs and top of head." Peter says this is by far the finest of the images; he is applying to get it photographed.]

Cave No. 144. Middle and Late Tang (renovated during the Five Dynasties and Qing). "The centre of the ceiling shows the three rabbits (joining as one) and floral designs." [PR: white on aqua green, going clockwise.]

Cave No. 145. Late Tang (renovated during the Five Dynasties and Song). "The ceiling of the niche on the west wall shows lotus and three rabbits (joining as one), chess-board and floral patterns." [Zhang & Rasmussen's letter of 1 Jun 2001 says the rabbits are going clockwise.]

Cave No. 147. Late Tang. "Main Hall: The ceiling shows three rabbits (joining as one) and lotus designs in the centre". [PR: tan or turquoise green, going clockwise. Paint on ears has peeled off.]

Cave No. 200. Middle Tang. "The ceiling shows three rabbits (joining as one) and round petalled lotus designs in the centre". [PR: white on turquoise green, going clockwise.]

Cave No. 205. Early and High Tang (renovated during Middle Tang and the Five Dynasties). Main Hall: "The ceiling has the three rabbits (joining as one) designs drawn in Early Tang". [Zhang & Rasmussen's letter of 1 Jun 2001 says the rabbits are going clockwise.]

Cave No. 237. Middle Tang (renovated in Western Xia and Qing). Main Hall: "The ceiling shows the three rabbits (joining as one) and round petalled lotus designs in the centre." [Zhang & Rasmussen's letter of 1 Jun 2001 says the rabbits are going clockwise.]

[PR: Cave No. 305. Sui (renovated during the Five Dynasties and Qing). "Rabbit paint is gone (only white silhouette remains on faint rusty red background)." See: Decorative Patterns in the Dunhuang Art; Li Kai et al; Zhang & Rasmussen's letter of 1 Jun 2001, which says the rabbits are going clockwise]

Cave No. 358. Middle Tang (renovated during the Five Dynasties, Western Xia and Qing). "Main Hall: The caisson ceiling shows the three rabbits (joining as one) and round petalled lotus in the centre." [PR: white on turquoise, going clockwise, faded.]

Cave No. 383. Sui (renovated during Song, Western Xia and Qing Dynasties). "Main Hall: The centre of the caisson ceiling shows the three rabbits (joining as one) and lotus flower designs". [PR: brown with white outlines, going clockwise, fair condition.]

Cave No. 397. Sui and Early Tang (renovated during the Five Dynasties and Qing). "Main Hall: The caisson ceiling shows the three rabbits (joining as one) and lotus in the centre". [PR: white on peeled-off aqua green, going clockwise. Poor to fair condition.]

Cave No. 406. Sui (renovated in Song and Qing). "The centre of the caisson ceiling shows four designs of a set of three rabbits (joining as one) and lotus". I don't quite understand his phrasing -- there is a picture of a pattern of three rabbits in the centre of a lotus, as in Cave 407, but perhaps there are other patterns which are not reproduced?? [PR: white on tan, going clockwise, faded, fair condition. Peter says nothing to clarify Duan's text.]

Cave No. 407. Sui (renovated in Song and Qing). "Main Hall: The caisson ceiling is covered with the three rabbits, lotus designs and flying figures drawn in Sui." [PR: Black with white outline on turquoise green background, going anticlockwise, good to excellent condition.]

[PR: Cave No. 420. Sui. Zhang & Rasmussen's letter of 1 Jun 2001 says the rabbits are going anticlockwise.]

Cave No. 468. Middle Tang (renovated during the Five Dynasties). "Main Hall: The centre of caisson ceiling has three rabbits (joining as one) and a lotus design". [PR: white on turquoise, going clockwise.]

CHRONOLOGY Tang is 618-907, but there is a Later Tang (923-936).

Sui (581-618, so c600): 305, 383, 406, 407, 420.

Sui and Early Tang (c620): 397.

Early/Mid Tang (7-8C): 205.

Mid Tang (8C): 200, 237, 358, 468.

Mid/Late Tang (8-9C): 144.

Late Tang (9-10C): 127, 139, 145, 147.

Roderick Whitfield. Dunhuang Caves of the Singing Sands. (Revision of a Japanese book by NHK, 1992.) Textile & Art Publications, London, 1995. On pp. 59 & 238, plates 66 & 361-362, are pictures of the roof of cave 420 which may be showing a three rabbit pattern, but it is too faded and too small to really be sure.

Decorative Patterns in the Dunhuang Art. 1996. This has no English text except for the book title. The dust jacket has a colour picture of a painting of the ceiling of cave 407. Otherwise it gives only black and white drawings of patterns. P. 31 is an introduction to a section and has an unidentified three rabbits pattern, probably from cave 407. P. 32 is cave 305. P. 33 is cave 397.

Li Kai, chief designer; Zhao Le Nin & Luo Ke Hua, eds. The Selections of Copied Art Works of Dun Huang Sunk Panel. 1997. This presents various artists' paintings of the ceiling panels.

Plate 21 is: Sui Dynasty "three rabbits and lotus flower" sunk panel (cave number 305).

Plate 26 is: Sui Dynasty "three rabbit and flying Apsarase" sunk panel (cave number 407) copied by Duan Wen Jie.

Plate 27 is: Sui Dynasty "three rabbits and flying Apsarase" sunk panel (cave number 407) copied by Guo Shi Qing and Chang Sha Na. This is reproduced on the cover of the book, but it has been reversed in printing in both places. The difference in coloration and even in the outlines show the difficulty of seeing what is present in a rather faded and damaged cave where the early artists had only daylight. Neither of these is the same as the cover of the previous item! See also Shuhong & Chegxian above for another version.

Plate 32 is: Sui Dynasty sunk panel (the cave number is not recorded).

Plate 26 is: Tang Dynasty sunk panel (cave number 205).

Wei Zhang & Peter Rasmussen. Letter of 1 Jun 2001 reporting on their research and visit to Dunhuang. They have made good contacts there and were given a lengthy special tour. They have now found the three rabbits pattern occurs in 16 caves: 127, 139, 144, 145, 147, 200, 205, 237, 305, 358, 383, 397, 406, 407, 420 and 468, all in the Mogao caves and all as central ceiling panels. They were able to see 145 (late Tang, 848-906), 205 (Early Tang, 618-704), 237 (mid Tang, 781-847), 305 (Sui, 581-618), 407 (Sui, 581-618), 420 (Sui, 581-618). In the first four, the rabbits are going clockwise, in the last two, they are going anticlockwise. They also visited the Western Thousand Buddha Caves, but there was no sign of the three rabbits there. However, none of the researchers there has investigated the three rabbits pattern. Unfortunately, photography is not allowed, but they sent the previous two books and two reproductions from an otherwise unidentified book: Dunhuang of China (2000).

P. 20 is: Hall With Inverted Funnel Shaped Ceiling Shape of Cave 305 (Sui Dynasty) and gives a good impression of the shape of these caves -- this one is roughly cubical with several statues, apparently life-size, on a central plinth, with a decorated ceiling with the Three Rabbits in the centre. The central part of the ceiling usually is a panel which is sunk into the ceiling, i.e. higher than the rest of the ceiling, but some have two steps and some are more rounded. [Greeves (2001)] suggests this is to represent a cloth canopy.

Pp. 40-41 is: Pattern (ceiling) Cave 407 (Sui Dynasty), showing the whole ceiling and the tops of the walls.

OTHER ASIA

Anna Filigenzi, of the Istituto Universitario Orientale di Napoli and the Italian Archaeological Mission in Pakistan reports and has sent an image of a three rabbits plaque found at Bir-kot-ghwandai, Swat, Pakistan and dating from 9C-11C. The publication is: P. Callieri et al.; Bir-kot-Ghwandai 1990-92 A Preliminary Report; Supplemento n. 73 of Annali dell'Istituto Universitario Orientale de Napoli 52:4 (1992) 45 -- ??NYS. [emails of 30 Oct & 10 Nov 2001.]

In the Museum für Islamische Kunst, Berlin, is a glass medallion with the Three Hares pattern, attributed to Afghanistan. It is 52mm in diameter. The museum purchased it from a dealer and there is no record of its origin. Peter Rasmussen & Wei Zhang were shown it in mid 2002 by Dr. Jens Kröger, the Curator of Islamic Art at the museum. [Email from Rasmussen, 17 Aug 2002.] There is a description of it in Kröger's book and in Stefano Carboni; Glass from Islamic Lands; Thames & Hudson, 2001, pp. 272-280. This also describes a fragmentary glass piece with four rabbits in the al-Sabah Collection in Kuwait, supposed to come from Ghazni, Afghanistan. [Email from Rasmussen, 20 Aug 2002.]

Eva Wilson. Islamic Patterns (British Museum Pattern Books). British Museum Publications, 1988. Plate 42, bottom picture, is a four rabbits pattern. The notes on p. 19 say: Engraved design in the centre of a brass plate. Diameter 7cm. Iran, 12th century. British Museum, London (1956 7-26.12).

[Greeves (2000)] reports a light blue glass seal from Afghanistan, c1200.

In the Hermitage Museum, St. Petersburg, is an oriental silver flask of 12/13C with the three rabbits on the base. Discussed and illustrated in Hamann-MacLean, below. A small illustration is in Baltrušaitis (Fig. 96D), cf under China, above, who says it comes from near Perm and has a Kufic inscription on it. Cf next entry.

Vladislav P. Darkevich. Khudozhestvennyi metall Vostoka VII-XIII. Nauka, Moskva, 1976, 195 pp. ??NYS – described by Peter Rasmussen [email of 8 Jan 2002] and photocopy of pl. 34 sent by him. See pp. 16, 17, 115 and Таблуца 34: Серебряные чаша у флакон 5 - 8: Селянино Озеро (No. 19) [Tablutsa 34: Serebryanye chasha u flakon: 5 - 8: Selyanino Ozero (Plate 34: Silver basin and flagon: 5 - 8: Salt(?) Lake)]. This is the Hermitage Museum silver flask. Fig. 5 is an overall view; fig. 6 is the bottom, which is rounded, rather like part of a sphere and has the three rabbits pattern; figs. 7 & 8 are the Kufic inscriptions.

[Greeves (2000)] reports a fine metal tray from Iran, c1200, in the Keir Collection and gives a fine photo of it. He notes that Sassanian culture is believed to have spread outward and could have been the source for the Himalayan and Chinese versions as well as the western versions. Sue Andrew tells me that similar trays are in the Victoria and Albert Museum, London.

[Greeves (2000)] reports that the pattern has been found in Nepalese temples and in c1200 wall paintings at the temple complex of Alchi in Ladakh, Jammu & Kashmir. I can't find Alchi on maps, but a guide book says it is on the Indus about 1½ hours drive from Leh.

Roger Goepper. Alchi: Buddhas, Göttinnen, Mandalas: Wandmalerei in einem Himalaya-Kloster. DuMont Buchverlag, Köln, 1982, 110 pp. ??NYS -- photocopies sent by Peter Rasmussen. Another version appeared in 1996 -- ??NYS, described by Rasmussen [email of 21 Dec 2001], see below -- but the earlier book has different pictures. Rasmussen says this is about Sumtsek, the Buddhist temple at Alchi, Ladakh, presumably the temple mentioned by Greeves, above. Goepper dates the temple as c1200, but another author suggests 11C or 12C [Lionel Fournier; The Buddhist Paradise: The Murals at Alchi; 1982]. On p. 127 of the 1996 book is a photo of a 4.63 m high statue of Maitreya (the Buddha) wearing a dhoti on which Rasmussen finds no less than 50 (fifty!) examples of the three hares pattern. A close-up on p. 128 shows three complete examples. Goepper refers to these examples as 'three or four deer-like animals, the three long ears being shared ...', but Rasmussen could not see any examples with four animals and the text only refers to three ears. However, plate 7 of the 1982 book shows the same two roundels as on page 128 of the 1996 Alchi, but this illustration shows FOUR rabbits (sorry, hares) or deer or bulls (take your pick) in the upper left corner above Maya that were cropped out of the 1996 illustration, so Goepper's reference on page 126 of his 1996 book to "three or four deer-like animals" was correct. In a note on p. 278 of the 1996 book, Goepper says the occurrence of the same pattern at Paderborn is 'hardly anything more than a coincidence'. Rasmussen finally notes that Goepper said the photographer, Jaroslav Poncar, and his group took about 3000 transparencies, a 'virtually complete documentation of the Alchi murals', but only 300 occur in the 1996 book. I have seen a B&W copy of Plate 7 of the 1982 edition. This has two images, each of a central character in a roundel surrounded by four frames in the form of a Greek cross with extra squares at the corners. (The colour image from the 1993 book, see below, shows these are adjacent, indeed overlapping, images.) These frames contain patterns of three and, in one case, four animals, but the identity of the animals is not clear. The four-fold pattern seems to be rabbits sharing ears, but others seem to be horses (or bulls) sharing ears or deer sharing antlers or possibly bulls sharing horns. All the animals are going clockwise.

Sue Andrew has been in contact with Goepper & Poncar. Goepper said "just about a week ago I hinted at the 'intercultural' character of this strange motif during my lectures at Cologne University".

Pratapaditya Pal. A Buddhist Paradise: The Murals of Alchi Western Himalayas. Ravi Kumar for Visual Dharma Publications Ltd., Hong Kong, 1982, 67 pp. ??NYS -- described by Peter Rasmussen [email of 4 Jan 2002]. Plate S9 shows Maitreya's full dhoti, S10 and S11 are closeups of the individual legs, and S12 and S13 are closeups of details. Lionel Fournier's photography is very poor in comparison to Poncar's. On pp. 51-52, Pal states "Interspersed with the rondels [sic] are little cruciform blocks adorned with leaping bulls, whose exact function is not clear, but which remind one of similar though more naturalistic bulls on the ceilings of Ajanta." So Goepper's deer are Pal's BULLS!!! But the interesting part of the sentence is the reference to the ceilings of Ajanta. It's not clear whether Pal means the Ajanta bulls are leaping in threes, but this statement reminded us that Terese Bartholomew of the San Francisco Asian Art Museum told us she thought she remembered the three-animal motif being somewhere in the Ajanta caves. In an email of 8 Jan, Rasmussen reports that he has gone through all the Ajanta books at UC Berkeley and could only find an image of four deer sharing a single head -- details are in the Dead Dogs section.

Roger Goepper. The 'Great Stupa' at Alchi. Artibus Asiae 53:1/2 (1993) 111-143. ??NYS -- photocopy sent by Peter Rasmussen. Figure 9: Deities in outer triangles of the ceiling. Photo by Poncar. This shows a number of three and four beast roundels forming a decorative band going out of the picture. as with the other Alchi images, it is hard to tell whether the beasts are rabbits, hares, bulls, etc., but they are clearly sharing ears here. all the beasts are going clockwise.

James C. Y. Watt & Anne E. Wardwell. When Silk Was Gold Central Asian and Chinese Textiles. The Metropolitan Museum of Art in Cooperation with The Cleveland Museum of Art, dist. by Abrams, nd [mid 1990s?, after 1993]. Section 45: Cloth of gold with rabbit wheels, p. 158, with a colour plate opposite (no number on my copy). This shows a square array of circles with four rabbits in each circle. The direction of the rabbits alternates from one row of circles to the next. It is from the 'Eastern Iranian world, second quarter to mid-13th century', i.e. c1240. The cloth has a green-gold colour, while the patterns are outlined with red silk, giving red-gold lines. A footnote says to consult Roes 1936-37, pp. 85-105 for the history of the motif of the animal wheel. Other notes cite Dunhuang, metalwork in Khurasan (1150-1225), the dhoti of the Maitreya at Sumtsek Temple in Alchi and the ceiling paintings of the Great Stupa in Alchi. The section goes on to discuss the 'two heads, four boys' motif - see under Dead Dogs, below.

Sue Andrew, via Peter Rasmussen [email of 26 Dec 2001], reported finding this, but only gives Wardwell as author - perhaps she wrote this part, but this is not indicated on the photocopies sent by Rasmussen. Peter Rasmussen [email of 4 Jan 2002] says Wardwell's main source was Baltrušaitis, under China, above. However, Baltrušaitis doesn't mention several of the cited areas.

[Greeves (2001)] reports a pre-Islamic Mongol coin from north Iran, dated 1281, with the three hares on one side.

Jurgis Baltrušaitis. Le Moyen Age Fantastique,.... Op. cit. under China, above. Pp. 132-139 of the 1981 edition have many examples of three and four rabbits, four boys, etc. After discussing Dunhuang, he says the motif was taken on by Islam and cites the Petersburg cup. He says there is a Mogul (school of Akbar) miniature with the pattern.

Roger Goepper. Alchi Ladakh's Hidden Buddhist Sanctuary. Serindia Publications, London, nd [1993?]. Colour image sent by Peter Rasmussen showing the two overlapping images of the dhoti of Maitreya shown in his 1982 book, above. However, this image is more centralised and hence shows only one of the three animals patterns completely, with parts of several others.

In Cairo, the Museum of Islamic art has a fragment from the bottom of a bowl with a three rabbits pattern using three colours! It is their item 6939/1, coming from Egypt or Syria in the 12-13C. In 2001-2002, it was a featured item at the exhibition: L'Orient de Saladin L'Art des Ayyoubides at the Institut du Monde Arabe, Paris. It is reproduced in the catalogue: Éric Delpont et al. L'Orient de Saladin L'Art des Ayyoubides. [Catalogue for] Exposition présentée à l'institut du monde arabe, Paris du 23 octobre 2001 au 10 mars 2002. Institut du monde arabe / Éditions Gallimard, Paris, 2001. Item 111, p. 123, is 'Tesson aux trois lièvres'. A black on yellow version was adopted as the logo of the exhibition and hence appeared on many other items associated with the exhibition and on the advertisements for it.

PADERBORN

In the cloister (Kreuzgang) of the Cathedral (Dom) of Paderborn, Nordrhein-Westfalen, Germany, is the "Three Hares Window" (Dreihasenfenster), with hares instead of rabbits. This faces the outside, i.e. into the central garden of the cloister. I learned of this from the Michelin Green Guide - Germany (Michelin et Cie, Clermont-Ferrand, 1993, p. 229) and wrote a letter of enquiry. A response from Dr. Heribert Schmitz in the Archbishopric states that the present form of the cloister dates from the early 16C and this is the date given on a postcard he included (and in a local guidebook). ([Greeves (2000); Greeves (2001)] says probably 15C.) He also included a guide to the Cathedral, a poster and a copy of the parish magazine with the article by Schneider, see below. On 12 Jul 2000, I was able to visit Paderborn, see the window, meet Dr. Schmitz and obtain much more material. The image is actually carved stone tracery in the arch over one of the triple windows of the cloister, and is about 3 ft (1 m) across. The central stone image is supported only by the three rear feet of the hares which are on a circular rim -- the intermediate spaces are filled with leaded glass. The local guidebook refers to the mason as 'crafty', perhaps implying that he saved having to carve three more ears. Several of the photos show the bodies of the hares supported by metal rods, but there are presently no rods. Later inquiry revealed that the original version is now in the Cathedral Museum and the version in the cloister is a recent copy. The Cathedral guidebook refers to the 'well known' window and says the symbol is an old land-mark of the city and the poster describes it as a famous landmark to be studied and developed in a workshop for children. One guide book shows three people dressed as hares who are a regular feature of parties and celebrations. The Cathedral guidebook says the 'motif is also to be found in other buildings, but elsewhere is mostly smaller and less conspicuous', but no references are given and Dr. Schmitz's letter says that he knows of no other examples than Long Melford and the article by Schneider. Greeves, below, notes that St. Boniface, the Apostle of the Germans, came from Crediton, Devon, some 10 mi east of Dartmoor! Further, he consecrated a bishop at Paderborn.

Hans Schneider. Symbolik des Hasenfensters in Nordwestchina entdeckt. Die Warte (Heimatzeitschrift für die Kreise Paderborn und Höxter) 32 (Dec 1981) 9. This was kindly provided by Dr. Schmitz of the Archbishopric of Paderborn. First Schneider gives various interpretations of the symbolism of the three hares pattern: old German fertility symbols from the myths of the gods; the Easter rabbit as a symbol of the eternal power of nature; a symbol of the Trinity. In recent years, it has been connected to the patron saint of Paderborn, St. Liborius, by viewing his name as Leporius, which means 'hare man'. But Schneider has discovered the article of Huang and gives a B&W reproduction of the picture. Schneider notes that Paderborn had connections with the Islamic world -- e.g. Achmed el Taruschi and a delegation came to Paderborn in 970, [and we know Charlemagne's court had contacts with Constantinople, Córdoba and Baghdad]. Hence it is possible that the Chinese symbol could have been transmitted to Paderborn [and elsewhere].

A small illustration is in Baltrušaitis (Fig. 96D), under China, above.

On 12 Jul 2000 I was able to visit Paderborn and meet Dr. Schmitz. The way to the Dreihasenfenster is clearly signposted in the Cathedral and we found images of it elsewhere in the town, and the local guidebook mentions further locations, e.g. the Drei Hasen restaurant at 55 Königstrasse. I got an English version of the Cathedral guide and a children's guide to the Cathedral. I obtained two more postcards featuring the window and several multi-image postcards of Paderborn with the window as one image. I also bought a stained glass roundel of the pattern, 225mm (8 3/4") across. The city information office has the pattern on many of their guide books and I also got stickers, transfers, etc.

I had met Michael Freude from Münster and he had recalled there was an example in Münster and that there was a children's rhyme about it, though he could not remember it, nor could his family. We stayed with Michael Freude and Hanno Hentrich in Münster and they had located the example in Münster, which is a roof boss in the southwest corner of the south transept of the Dom (Cathedral) (St. Paul's), over the organ. It is very high and I was unable to get a good picture of it. [Greeves (2000)] notes that it is stunning, but he told me that he also had been unable to get a good picture. Since the Dom was much restored after the War, I thought it might be a post-war addition, but Hentrich checked in a Münster history and photos showed this part of the Dom had survived. [Greeves (2000)] says it is early 16C. I told Dr. Schmitz of this example when I visited him and he did not know of it. Freude & Hentrich thought it likely that the Paderborn example was a replacement after the War.

[Greeves (2000)] says there is an example in the cloisters of a former monastery at Hardehausen, S of Paderborn (but I can't locate this on my maps).

Theodore Fockele & Ewald Regniet. Domführer für Jungen und Madchen. Metropolitan-Kapitel, Paderborn, (1982), 6th ptg, 1999. On p. 21 is a brief description and a drawing with caption being the rhyme: "Der Hasen und der Löffel drei, / und doch hat jeder Hase zwei." [The hares and ears are three, / and yet each hare has two[, you see].] This rhyme also occurs in the local guidebook and on one of the available postcards.

Verkehrsverein Paderborn [Paderborn Tourist Information]. Paderborn A short guide to the old city. Paderborn, 1998, p. 13 and back cover. This says the window is early 16C and gives an English version of the rhyme: "Count the ears. There are but three. But still each hare has two, you see?" and I have now inserted 'you see' into my translation above. The pattern is printed on the outside covers of this booklet

One of the postcards available in Paderborn has a longer poem.

Viribus Auribusque Unitis (Mit vereinten Kräften und Ohren)

Jedweder Hase hat zwei Ohren.

Und hier ging jedem eins verloren.

Das Soll ist sechs, das Ist nur drei.

Und Schein und Sein sind zweierlei.

Was führt der Steinmetz wohl im Schilde?

Welch ein Gedanke liegt im Bilde?

Die Ohren sitzen an der Stirne,

Gehörtes fliess in drei Gehirne.

Drittselbst wird hier somit bedacht,

was Sorgen oder Freude macht.

Vereint geht manches leichtes eben

im Hasen- wie in Menschenleben.

Und überdies ist, was ihr seht,

'ne Spielart von der Trinität.

[With united powers and ears Every hare has two ears. And here each has lost one. There should be six, there are only three. And appearance and being are different. How can the stone mason make an emblem? What thought is in the picture? The ears sit on the forehead, which flow into three heads. A third itself is here thus considered, which makes fear or joy. United, many things go easily even in the life of hares as in the life of men. And moreover this is, as you see, a playful image of the Trinity.]

Annemarie Schimmel. The Mystery of Numbers. (As: Das Mysterium der Zahl; Eugen Diederichs Verlag, Munich, 1984. Based on: Franz Carl Endres (1878-1954); Das Mysterium der Zahl, last edition of 1951. It's not clear when Schimmel's work was done -- the Preface is dated Sep 1991 and her © is dated 1993, so perhaps 1984 refers to the last printing of the original Endres book??) OUP, 1993, p. 63 has a drawing of the Paderborn three hares, but with no indication of the puzzle aspect. "Hares, symbols of the tri-unity that is always awake, seeing and hearing everything. Their ears form a triangle.".

MEDIEVAL EUROPE

There are 17 churches with 28 roof bosses of the Three Rabbits in Devon. The dating of these is not very exact and is not always given in the church guides. I have now discovered Cave's 1948 book which mentions many of these. See also Jenkins' book of 1999.

Carl Schuster & Edmund Carpenter. Patterns That Connect Social Symbolism in Ancient & Tribal Art. Abrams, NY, 1996. Pp. 158-159, fig. 453, is a 12C European wind chart with a central demon face with four mouths, but it has four pairs of eyes.

A stone boss supposed to come from a church demolished c1200 is built into a cellar of a house in Corbigny, Nièvre, Bourgogne, about 50km NE of Nevers. [Greeves (2000) with colour photo.] [Greeves (2001)] changes the name to Corbenay, but this is not in my French atlas.]

[Greeves (2000); Greeves (2001)] says the pattern occurs on a bell from the early 13C in the great abbey church at Kloster Haina. I can't locate this on my maps, but there is a Haina, Hessen, about 40km SW of Kassel.

Bestiary. MS Bodley 764, 1220-1250. ??NYS. Translated by Richard Barber as: Bestiary Being an English version of the Bodleian Library, Oxford M.S. Bodley 764 with all the original miniatures reproduced in facsimile; Folio Society, 1992; The Boydell Press, Woodbridge, Suffolk, 1993; PB, 1999. Pp. 66-67, the entry for hare is preceded by a 'four rabbits' pattern. There were many medieval versions of the bestiary and the BL MS Harley 4751 is very similar to this. However, Barber [p. 13] notes that the entry for the hare is not in earlier texts and rarely reappears in later texts. Thanks to Sue Andrew for this reference and a copy of the photo she had done from the original MS.

Sue Andrew also showed me an illuminated initial Q, also with four rabbits, from a c1285 MS, a French copy of Peter Lombard's Gloss on the Psalter, Bodleian MS Auct.D.2.8. f. 115r, commentary on Psalm 51.

A roof boss in the south choir aisle of Chichester Cathedral, dated to the first half of the 13C, shows six 'Green Men' sharing eyes. The Green Men have foliage coming out of their mouths. My thanks to Marianna Clark for noticing this and sending me an example of the colour postcard of it which is labelled: Six heads with six eyes between them. Colin Clark, the Chief Guide to the Cathedral, told me of the example in Boxgrove Priory. See Cave, 1930 & 1948.

A roof boss in Boxgrove Priory is roughly contemporary with that in Chichester Cathedral, but shows eight faces sharing eyes. Photo 300 in Cave, who includes it on a page of Foliate Heads. On p. 184, he says alternate heads have a stem from the mouth, but this is very small, leading one to wonder if the foliage has been broken off. [Jenkins, pp. 686-687] says: "The second boss from the altar is so crafted that each of eight faces comes complete with two eyes, yet there are only eight eyes in all." See Cave, 1948.

A roof boss in the Chapter House of a former Benedictine Abbey, now the sacristy of the church of Saints Peter and Paul, Wissembourg, Bas-Rhin, Alsace, is dated to c1300. Nearby bosses include a Green Man, one of the common figures in Devon bosses. [Greeves (2000); Greeves (2001).]

On the right side of the southern west doorway of St. John's Cathedral (Primatiale St-Jean) in Lyon, there is a quatrefoil panel with four rabbits, from about 1315. The rabbits are going clockwise. Discussed and illustrated in Hamann-MacLean, below, and in Baltrušaitis (Fig. 96A), cf under China, above. I photographed this in Lyon and my picture is much better than that in Hamann-MacLean, possibly because his photograph was taken before the facade was cleaned in 1982. However, there is no postcard of the pattern and I could not find a picture in any of the material available -- but see the next item.

John Winterbottom & Diana Hall have sent a photo of a similar four rabbits pattern, but going anti-clockwise, at Bord du Forêt, near Lyon. Here the rabbits are leaping upward, so the central square of ears is smaller and tilted by about 30o.

N. Reveyron. Primatial Church of Saint-John-the-Baptist, Cathedral of Lyon. Translated by Valérie Thollon & Diana Sarran. Association Lyon Cathédrale, Lyon, nd [c2000]. Pp. 22-29 describe the western doorways, saying the sculptures were made during 1308-1332. The South doorway is described on pp. 28-29, and the third section, on p. 28, headed Bestiary, includes: "The four hares, arranged like a swastika." This is the only time I have seen the four rabbits pattern compared to a swastika and I don't think there is any similarity!

In the Church of St. John the Baptist, North Bovey, Devon, there is a carved wood roof boss of the Tinners' Hares which possibly dates to the 13C. The leaflet guide to the Church says the pattern was an emblem of the tin-miners of the 14C and is thought by some to refer to the Trinity. The symbol is used by a number of local firms as a logo -- though I didn't see any on my visit in 1997. Thanks to Harry James, Churchwarden, for his letter of 25 Apr 1997, his drawing of the pattern and a copy of the guide.

In the Church of St. Mary the Virgin, Long Crendon, Buckinghamshire, about halfway between Aylesbury and Oxford, is a mid-14C tile showing the Three Rabbits pattern. About a third of it is missing. It is just at the altar step, which has preserved it somewhat from wear. The church guide says the tile was made in nearby Penn and is unique. This is an encaustic tile with the rabbits in yellow on a pale orange background. (This church is generally locked; try telephoning the Vicar on 01844-208363 or the Churchwarden on 01844-208665 if you want to get in. My thanks to Avril Neal, the Churchwarden, for letting me in.) The pattern is reproduced in B&W in Carol Belanger Grafton; Old English Tile Designs; Dover, 1985 [selected from Haberly, cf below], p. 91, but there is no text or indication of the source of it. My thanks to Sue Andrew for finding the tile and the Grafton reproduction. She has persuaded a potter to make facsimiles of the tile. The potter is Diana Hall, Anne's Cottage, Wimborne St. Giles, Dorset, BH21 5NG; tel: 01725-517475. I have now purchased some of these and they are very fine.

Loyd Haberly. Mediaeval English Pavingtiles. Blackwell for Shakespeare Head Press, Oxford, 1937. Shows the complete pattern in red and white on the TP. Shows it in B&W as fig. CXIII on p. 168, saying it occurs in Long Crendon Church and Notley. "The design is also found elsewhere in glass and carved on stone. Some say it symbolizes the Trinity, others the Trivium, or three Liberal Arts of Grammar, Rhetoric and Logic. The conies, these say, represent the scholars, a feeble folk, who have an ear for each of the three Arts. One writer thinks the donor of the this design was therefore a man of academic distinction." Thanks to John Winterbottom and Diana Hall for copies of this material.

[Greeves (2000)] discusses this and says it is the earliest known British example, presumably because the North Bovey example is not precisely dated. [Greeves (2001)] thinks this is roughly contemporary with the Chester tile, see below, and another tile in Anglesey.

In the Grosvenor Museum, Chester, is a floor tile from the Cathedral with an inlaid design of the Three Rabbits pattern. Sue Andrew has found that this is shown in Jane A. Wight; Mediaeval Floor Tiles; John Baker, London, 1975 and she has kindly sent a photocopy. On p. 48 is fig. 15: "Trinity Rabbits: Narrow inlaid design of linked rabbits, symbolising the Holy Trinity, in Chester Cathedral. (About 5⅜ inches square.)" These are just outline rabbits, looking much as though Matisse had drawn them. The outer edges of the ears are curved to produce a circle, so the usual delta shape is here very curved. On pp. 12-13, Wight discusses the pattern.

Some signs are ambiguous, hovering between religion and magic, like the three rabbits or hares linked by their shared ears, that may act as a symbol of the Holy Trinity. In The Leaping Hare by George Ewart Evans and David Thomson (1972) it is pointed out that these are correctly hares, 'joined in a kind of animated Catherine-wheel' and 'another instance of a pre-Christian symbol being adopted by the church'. (On roof bosses in the Dartmoor churches, financed by money from the stanneries or mines, the creatures appear as the craft-badge of the tin-miners.) This 'Holy Trinity' is found on tiles in Chester Cathedral and in Buckinghamshire.

It is clear that Wight and Evans & Thomson have no knowledge of the puzzle aspect of the pattern. The connection with the tin miners is now known to be a modern myth -- see Greeves, below. Hares certainly have pre-Christian associations, but I don't know of any pre-Christian example of the Three Rabbits pattern except at Dunhuang and possibly St. Petersburg, see above. [Greeves (2000)] mentions this example and gives more non-Christian examples. [Greeves (2001)] suggests this is early 14C, roughly contemporary with the Long Crendon example and gives a photo of the actual tile on p. 62 -- this is slightly damaged in the middle, but does not have the circularity I noted above which is thus an artist's liberty. Greeves, Andrew & Chapman recently visited Chester, but the only tiles shown to them were from excavations in the Cathedral nave in the 1990s, which makes us wonder if the tile described by Wight is still somewhere in the Cathedral??

Sue Andrew (Nov 2002) reports that Diana Hall has learned of a 14C tile from the Stadion's Prebendary Court in Constance, now in the Zurich Schweizerisches Landesmuseum.

In St. Pancras Church, Widecombe, Devon, is a roof boss of the three rabbits pattern, probably from the late 14C. The guide book to the church says it is "a symbol of the Trinity connected with tin-mining." This is shown on a postcard by Judges of Hastings, card number c11933X, where it is called the Hunt of Venus. A separate guide to the roof bosses also calls it the Hunt of Venus and suggests the tinners took the imagery from either the mines being like rabbit burrows, or from Venus, the goddess of Cyprus, the island which produced the copper that the tin was combined with. Thanks to the Rector of the Church, Derek Newport, for the material. [Jenkins, 1999, p. 142] says: "This rare symbol of the Trinity is formed of three animal heads sharing just three ears."

At The Great Church of the Holy Trinity, Long Melford, Suffolk, there is an example of the three rabbits pattern in the 15C(?) stained glass. Christopher Sansbury, the Rector, wrote on 3 Jun 1996 that the motif is common on the east side of Dartmoor and that it may have been brought to Suffolk by the Martyn family c1500. He says it is old glass, older than the church, which was built in 1484, but doesn't specify a date for it, nor does the commercial postcard (Jarrold & Sons, Norwich, no. CKLMC 6). The pattern is considered to be an emblem of the Trinity. In a later latter, he cited Chagford, North Bovey and Widecombe as churches with the pattern near Dartmoor and cites Greeves' article. Greeves, below, says the pattern seems to be about 5.5cm in diameter, but I wonder if he is measuring a picture as when I visited the Church, it seemed to be perhaps 5" or 6". Sansbury wrote that it was less than a foot across. Greeves also says that the Cornish merchant family of Martin came to Long Melford c1490, so the glass is more likely to be 16C. [Jenkins, 1999, pp. 658-659.]

The Church of St. Michael the Archangel in Chagford, Devon, has a roof boss in the chancel depicting the Tinners' Rabbits, from the 15C. My photo is not very clear. The Rector, P. Louis Baycock, states that there are one or two other bosses in the wooden ceilings, but they are dark and were obscured by the lighting so I was unable to locate them. On a later visit, I managed to see one at the top of the aisle vault a bit to the left of the entrance door -- one needs to shield one's eyes from the light bulb near by. There is a kneeler in the church with the pattern and it figures at the lower right of a large modern embroidery 'Chagford through the ages' hanging at the back of the church (postcard and explanatory leaflet available). The leaflet says that a 'rabbit' was a tool used in tin-mining. On a later visit, I found also a pew seat cover with four examples of the pattern on it. James Dalgety tells me that there was a gift shop called Tinners' Rabbits in Chagford some years ago. [Mary Gray; Devon's Churches; James Pike Ltd, St. Ives, 1974, pp. 7 & 15] only mentions the pattern at Chagford, describing it as 'the old tinners' design'.

The local wine shop, called Best Cellars, has the pattern on its window. On my 2001 visit, the owner said the shop had previously been the Tinners' Rabbits. The Newsagent's in the main square sells a Chagford plate featuring the Tinners Rabbits and with the usual mythology on the back. There is a new building, Stannary Place, with a modern crest of three rabbits over the doorway, on New Street going away from the Church.

The 15C house of Cardinal Jouffroy at Luxeuil-les-Bains, Haute-Saône, Franche-Comté, about 40km NW of Belfort, has the pattern carved under a balcony. [Greeves (2000); Greeves (2001).]

The church of St. Andrew, Sampford Courtenay, Devon, has two roof bosses in the Three Rabbits pattern. A letter from D. P. Miles says these date from c1450. Cf [Jenkins, 1999, pp. 137-138.]

St. Michael's Church, Spreyton, Devon, has a roof boss of the Three Rabbits pattern. The sign at the entrance to the churchyard is a large and beautiful version of the pattern painted by Helen Powlesland, the nicest I have seen. Thanks to Rev. John Withers for information. Mrs. Powlesland informs me that the roof boss is 15C. [Greeves (2000); Greeves (2001)] says the roof is dated 1451. [Greeves (2001)] has a photo. Spreyton has produced a Millennium cup showing the school and an inscription surrounded by small squares containing the three rabbits pattern in white on blue and blue on white -- sent by Helen Powlesland. In 2003, they produced a cup for the Church with one side having the three rabbits pattern -- kindly sent by Helen Powlesland. Cave, p. 211, says there are two bosses, one in the chancel and one in the nave, and that there is an inscription on the roof of the chancel dated c1450.

The Chapel of Saints Cyr and Julitta (now called St. Anne and St. Catherine), Cotehele House, Cornwall, was built in 1485-1489 and has an example of the pattern as a roof boss -- see [Greeves (2000); Greeves (2001)]. The boss is on the midline, one in from the E end of the Chapel. A chandelier hangs from it. It is unpainted and so dark that one really needs binoculars to make out the pattern. One can get a better view from a squint off the South Room on the first floor.

In Scarborough, North Yorkshire, there is a c1350 building in Sandgate (or Sandside) on the seafront of South Bay called the King Richard III Inn because he is reputed to have stayed there. On the ceiling of one of the upper rooms is a 'Three Rabbits' pattern, but this is in the landlord and landlady's rooms and the landlady was unwilling to let me see it. Inquiry to the Scarborough Museums and Gallery Officer elicited a photo held by the Planning Department in which the pattern can just be discerned (though I can't see which way it is going) and the information that it is in 16C plasterwork apparently done by Italian workers. [Greeves (2000)] describes this and notes that Richard III's wife's family (the Nevills) owned the manor of North Bovey in the early 16C. Greeves, Andrew & Chapman have been to see and photograph it -- the pattern has been painted red.

[Greeves (2000)] reports two occurrences of plaster ceilings with the pattern in Devon. A 16C example at Treasbeare Farm, near Clyst Honiton, and a mid-17C(?) example at Upcott Barton, Cheriton Fitzpaine. [Greeves (2001)] says the Clyst Honiton example is 17C and gives a fine photo of it.

Baltrušaitis. Op. cit. below. Fig. 97 is a 1576 Dutch engraving of three rabbits.

In Throwleigh, Devon, a few miles from Chagford, there is a roof boss of the Tinners' Rabbits in the 16C north aisle of the Church of St. Mary the Virgin. (Thanks to the Rector of Chagford and Throwleigh, P. Louis Baycock, for directing me to this site.) Photo in Sale, below, p. 63.

[Greeves (2000)] has a picture of the example in the parish church, Tavistock, Devon, but only dated as medieval.

[Greeves (2000)] reports there is a fine stone lintel from Charmois-L'Orgueilleux (I can't locate this) in the Musée d'Art Ancien et Contemporain in Epinal, Vosges, Lorraine, dated as 16C, but Greeves thinks the carving may be more primitive than 16C. He also reports two medieval stone roof bosses at Ingwiller, Bas-Rhin, Alsace, about 35km NW of Strasbourg, and Xertigny, Vosges, Lorraine, about 15km S of Epinal. Greeves says there are other French examples: in the chapel of the Hotel de Cluny (= Cluny Museum), Paris ([Greeves (2001)] says this is 15C; I have a good photo); at Vienne, Isère, Rhône-Alpes; at St. Bonnet le Chateau. Loire, Rhône-Alpes; and that potters at Soufflenheim, Bas-Rhin, Alsace, use the pattern in current production.

In 1997, an old trunk and crates in the Statens Museum for Kunst (National Gallery), Copenhagen were opened and found to contain over 20,000 prints which had been stored during re-organization in the 1830s. One of these is a three rabbits engraving, very similar to the 1576 Dutch engraving shown in Baltrušaitis, below. Here the rabbits are going clockwise. This material is the basis of an exhibition continuing until 16 Feb 2003. This picture is being used to advertise the exhibition and appears in Copenhagen This Week for Jan 2003 with a short English text, on ctw.dk/Sider/Articles.html . The museum's site is smk.dk , but I cannot see an English catalogue available. Information and photo from Diana Hall and John Winterbottom -- John's son saw it in Copenhagen on a stopover at the airport.

The Church of St. John the Baptist, Broadclyst, Devon, has nine roof bosses of the Three Rabbits pattern. (Greeves (1991) erroneously has eight.) They were made in 1833 but are said to be careful plaster copies of the medieval examples, but my sources give no date for the originals. There is one three rabbits boss in the central aisle -- Chris Chapman thinks some of the bosses in the central aisle may be originals, but this one has been recently painted and doesn't look old to me.

[Greeves (2000); Greeves (2001)] mentions two early 17C plaster ceilings with the pattern at Burg Breuberg in the Odenwald (I can't locate this on my maps, but the Odenwald is where Hessen, Baden-Württemberg and Bayern meet) and at Seligenstadt, Hessen, about 20km SE of Frankfurt.

Basil Valentine. (He may be catalogued as Basilius Valentinus (or Valentis) and entered under B rather than V.) De Macrocosmo, oder von der grossen Heimlichkeit der Welt und ihrer Artzney, dem Menschen zugehorig. c1600. ??NYS -- reproduced and briefly discussed in Greeves. This shows the Hunt of Venus, with three hares going clockwise, but with each hare pursued by an unconnected dog and Greeves notes that the dogs are an essential part of a hunt symbol. Inside the triangle of ears is the astrological/alchemical symbol for Mercury, which is claimed to be similar to a symbol used for tin (or Jupiter) -- but I have now looked at a book on alchemy and there is no similarity; further both symbols are included in the surrounding text and they are clearly distinct. The pattern is drawn inside a circle of text which contains the astrological symbols of all seven planets and hence is rather cryptic -- see Roob, below, for the text. The top of the picture has a flaming heart pierced by an arrow and the legend VENUS. So there is no real connection with tin, though this is the source cited by the first book on Dartmoor to mention the symbol, in 1856. Greeves notes that the three ears produce a delta-shape and this has connotations of fertility, both as the Nile delta and as the female pubic triangle.

Valentine is a (semi-?) mythical character. He was allegedly a Benedictine monk of the early 15C, but no trace of his writings occurs before the 17C. (However, de Rola (below) asserts that Antonius Guainerius (d. 1440) praised Valentine and that Valentine himself says he was a Benedictine monk at the monastery of St. Peter in Erfurt. But his real name is unknown and so he cannot be traced in the records of the monastery or at Erfurt. De Rola quotes a 1675 report that Valentine was at St. Peter's in 1413.) Legend says his works were discovered when a pillar in the Cathedral of Erfurt split open (a variant of a story often used to give works a spurious age) -- cf the next item. Despite their uncertain origin, the works were well received and remained popular for about 200 years, with the pictures of his 'Die zwölff Schlüssel' (The Twelve Keys) being used to the present day. He was probably an alias of Johann Thölde (fl. 1595-1625). The work cited may occur in his Last Will and Testament (1st English ed of 1657) or in his Chymische Schriften (Gottfried Liebezeite, Hamburg, 1700).

Basilius Valentinus. The Last Will and Testament of Basil Valentine, Monke of the Order of St. Bennet ... To which is added Two Treatises: .... Never before Published in English. Edward Brewster, London, 1671. ??NYS -- seen in a bookdealer's catalogue, 2003.

Basilius Valentinus. Letztes Testament / Darinnen die Geheime Bücher vom groffen Stein der uralten Weifen / und anderen verborgenen Geheimnüssen der Natur Auss dem Original, so zu Erfurt in hohen Altar / unter einem Marmorsteineren Täflein gefunden worden / nachgeschrieben: Und nunmehr auf vielfältiges Begehren / denen Filiis Doctrinæ zu gutem / neben angehengten XII. Schlüsseln / und in Kupffer gebrachten Figuren ie. deffen Innhalt nach der Vorrede zu sehen / zum vierdtenmahl ans Liecht gegehen / deme angehänget ein Tractätlein von der ALCHIMIE, Worinnen von derselben Usprung / Fortgang und besten Scriptoribus gehandelt / auff alle Einwürffe der Adversariorum geantwortet / und klar bewiesen wird / dass warhafftig durch die Alchimie der rechte Lapis Philosophorum als eine Universal Medicin Könne bereitet werden / von Georg Philips Nenter. Johann Reinhold Dulssecker, Strasburg, 1712. This has several parts with separate pagination. About halfway through is a new book with TP starting: Von dem Grossen Stein der Uhralten / Daran so viel tausend Meister anfangs der Welt hero gemacht haben. Neben angehängten Tractätlein / derer Inhalt nach der Vorrede zu finden. Den Filis doctrinæ zu gutem publicirt / und jetzo von neuen mit seinen zugehörigen Figuren in Kupffer and Leight gebracht. Strassburg / Im Jahr M. DCC. XI. Part V. of this is: Von der grossen Heimligkeit der Welt / und ihrer Artzney / dem Mensche zugehörig -- from Greeves, it appears this is also called Macrocosmo. On p. 140 is a smaller and simpler version of the three hares picture, with 'folio 222' on it. This has the hares going anticlockwise, the only example of Basilius' picture with this feature that I have seen. Further, the word 'recht' is missing from the text around the picture and has been written in. Below the picture is a lengthy poem, starting "Ein Venus-Jagt ist angestallt". I translate the first part as: "A Hunt of Venus is prepared. The hound catches, so the hare will not now grow old. I say this truly that Mercury will protect well when Venus begins to rage, so there occur fearfully many hares. Then Mars guards with your [sic, but must be his] sword, so that Venus does not become a whore."

Part II. of Von dem Grossen Stein der Uhralten is Die zwölff Schlüssel ... and Der neundte Schlüssel is on p. 51, with 'folio 70' on it. In the centre of the circular part of this is a pattern of three hearts with serpents growing out of them and biting the next heart. Above this are a man and a woman, each nude and in a sitting position, with bottoms almost touching. Baltrušaitis mentions this in his discussion of 'two heads, four bodies' pictures, but I don't feel this is really a picture of that type. A different and less clear version of this picture in reproduced in the following.

Stanislas Klossowski de Rola. The Golden Game Alchemical Engravings of the Seventeenth Century. Thames & Hudson, 1998. This reproduces the entire Twelve Keys from: Michael Maier; Tripus aureus; Lucas Jennis, Frankfurt, 1618, and gives some discussion. P. 119 is the title page of the Valentine; p. 103 includes the Ninth Key; pp. 125-126 give explications of the Keys.

Roob (below), p. 678, from: D. Stolcius v. Stolcenberg; Viridarium chymicum; Frankfurt, 1624.

Adam McLean. The Silent Language The Symbols of Hermetic Philosophy Exhibition in the Bibliotheca Philosophica Hermetica. In de Pelikaan, Amsterdam, 1994, p. 47, reproduced from: Johann Grasshoff; Dyas chymica tripartita; Lucas Jennis, Frankfurt, 1625.

Baltrušaitis, above, Fig. 103, but he cites: B. Valentino; Les Sept Clefs de Sagesse; vers 1413; via an 1891 book.

This three heart/serpent pattern is known as the Ouroboros or Ouroborus. McLean explains that man has three hearts: physical, soul and spiritual.

Basilius Valentius. Chymische Schriften. Hamburg, 1717. ??NYS -- reproduced in: Alexander Roob; Alchemie & Mystik Das Hermetische Museum; Taschen, Köln, et al., 1996, p. 676. Same picture as in De Macrocosmo, with some descriptive text. The circle of text is translated as: "Sol and Luna and Mars mit Jupiter jagen / Saturnus muss die Garne trage / Steltt Mercurius recht nach dem Wind / so wird Frau Venus Kind" (Sun and Moon and Mars hunt with Jupiter / Saturn must carry the net (or decoy) / Place Mercury correctly according to the Wind / then Frau Venus captures a child) and Roob says this is an alchemical description of the preparation of the 'Universal Medicine' from copper vitriol, which Basilius called the highest of all salts. Externally it is green, but inside it is fiery red from its father Mars, an oily balsam. He then gives lines 5-8 of the poem and says the hare is a known symbol for the fleetness of Mercury.

At Schwäbisch Hall, Baden-Württemberg, about 60km NE of Stuttgart, the pattern appears painted on the ceiling of a synagogue, from 1738/9 [Greeves (2000); Greeves (2001), with picture.] Sol Golomb notes that the commandment against making graven images is interpreted as referring to solid images and excepts decorative patterns even if they use animals.

The town of Hasloch (am Main), Bayern, about 70km SE of Frankfurt, uses the pattern as its town crest (colour photo of an 1842 example in [Greeves (2000)]).

C. J. P. Cave. The roof bosses in Chichester Cathedral. Sussex Archaeology 71 (1930) 1-9 & plate opp. p. 1. Photo 11 is a B&W picture, but not as sharp or clear as the current postcard. Discussion on p. 6 says the Boxgrove example is 'more boldly and better carved'. See Cave, 1948.

Jurgis Baltrušaitis. Le Moyen Age Fantastique,.... Op. cit. under China, above. Pp. 132-139 of the 1981 edition have many examples of three and four rabbits, four boys, etc. Fig. 96 gives small rabbit illustrations from Dunhuang (10th c.), Islamic vase from the Hermitage Museum (12th-13th c.), Lyon Cathedral (1310-20), Paderborn (15th c.), and Fig 97 is a large Dutch engraving, Lièvres a Oreilles Communes, of 1576 in the BN, Paris, with rabbits going anti-clockwise. He cites examples at Saint-Maurice de Vienne (15C); at the Hôtel de Cluny, at Saint-Benoit-le-Château (Loire) and says it is frequent in the east of France (Thiélouse, Xertigny), Switzerland (Abbey of Muototal), and in Germany (Munster, Paderborn). He cites a 1928 book for the pattern being a symbol of the Trinity. By the beginning of the 16C, he says it was used as a vignette by the printer Jacques Arnollet and as the sign of the Three Rabbits Inn (L'Hostellerie aux trois lapins). He gives some lines from a poem of the time:

Tournez et retournez et nous tournerons aussi,

Afin qu'a chacun de vous nous donnions du plaisir.

Et lorsque nous aurez tournés faites compte de nos oreilles,

C'est là que, sans rien déguiser, vous trouverez une merveille.

and says these or similar occur frequently in prints of the 16C and 17C, referring to the large Dutch engraving which has similar lines in Dutch and French around the border - but his reproduction (or the original) is truncated on the left. He cites an 1879 history of inn signs and seems to say this inn is in Lyon and that the author had also seen a version with three deer.

See Dead Dogs for further material.

C. J. P. Cave. Roof Bosses in Medieval Churches An Aspect of Gothic Sculpture. CUP, 1948. This has several hundred B&W plates. I have a reference to Cave's collection of photographs of roof-bosses and an index thereof at the Society of Antiquaries, so I looked up Cave at the Warburg and found this book.

Chapter VI: Beasts, birds and fish, pp. 69-75. On p. 71, he says: "There are a few rabbits or hares, it is difficult to tell which; the most curious are the three rabbits with only three ears between them, each rabbit sharing an ear with its neighbours; this device is found at Broadclyst, North Bovey (49), Chagford, Sampford Courtenay (182), Spreyton, South Tawton, Tavistock, and Widecombe-in-the-Moor, all villages on or not far from Dartmoor; at Selby there is a similar arrangement, but there is a fourth rabbit unconnected with the three.1

1 The three rabbits occur in stained glass at Long Melford. The same motif occurs in Paderborn Cathedral in Germany, on a window known as the Hare Window; see Walter Hotz, Mittelalterliche Groteskplastic (Leipsig), p. 47 (text) and p. 68 (plate); this window is even mentioned in Baedeker's Guide to North Germany."

Plate 1. Tickencote. This has three heads in a circle, but there is no interconnection.

Plate 30. Bristol Cathedral, north transept: Fish. This has three fish arranged in a triangular pattern, each overlapping the next.

Plate 49. North Bovey; rabbits.

Plate 182. Sampford Courtenay. Rabbits.

Plate 200. Selby. Triple face. [This has three faces looking forward, a bit to the right and a bit to the left, sharing four eyes. I have a photo of a similar keystone in Citta di Castello where the side faces are facing directly right and left. This seems to be a development from the Roman double-sided heads of Janus.]

Plate 300. Foliate Heads. Boxgrove.

Appendix I List of churches containing roof bosses. Pp. 181-222. He begins with a note: This list does not aim at completeness, .... Names of places which I have not visited personally are in square brackets.] He lists 208 sites, including: Bovey (North); Boxgrove Priory; Broadclyst ('evidently modern copies of old designs', but he doesn't say how many there are); Chagford ('two bosses ..., one much restored'); Cheriton Bishop; Chichester Cathedral (cites his: The roof bosses in Chichester Cathedral, Sussex Archaeological Collections, vol. LXXI, 1940, above); Havant ('Two early thirteenth-century bosses in the chancel with affinities to those as Chichester and Boxgrove.' ??); Oxford, Christ Church ('four lion's bodies joined to one head'); Portsmouth Cathedral ('Two bosses in the north aisle of the quire in the Chichester-Boxgrove style.' ??); Portsmouth, SS John and Nicholas (Garrison Chapel) ('The first and third bosses are early conventional foliage in the Chichester-Boxgrove style.' ??); Sampford Courtenay (cites two examples, in the chancel and the nave); Selby Abbey (says many bosses are medieval, but says nothing about the Triple face (plate 200)); Spreyton (cites two examples, in the chancel and the nave); Tavistock; Tawton (South); Widecombe-in-the-Moor. The nine three rabbits examples were known, but Chichester and Boxgrove are new and there are a number of possibles that seem worth investigating, but Plates 1 and 30, now added above, are the only previously unnoted pictures of marginal interest.

Richard Hamann-MacLean. Künstlerlaunen im Mittelalter. IN: Friedrich Möbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann Böhlaus Nachfolger, Weimar, 1987, pp. 385-452. The material of interest is on pp. 400-403. He discusses and illustrates: the Paderborn example, saying it is 15C; the silver flask in the Hermitage; the four rabbit pattern at Lyon. He notes that the last cannot be considered a symbol of the Trinity! He mentions a cave painting in Chinese Turkestan, 10C?, which is probably the Dunhuang example. He gives a number of references to earlier articles -- ??NYS.

Tom Greeves. The Tinners' Rabbits -- chasing hares?. Dartmoor Magazine 25 (Winter 1991) 4-6. This was certainly the most informative discussion of the topic until his later articles. Greeves is an archaeologist and an authority on the Dartmoor tin industry. He has now been joined in this research by Sue Andrew and the photographer Chris Chapman, comprising the Three Hares Project. Greeves says that it is claimed to be the emblem of the medieval tinners, and various connections between tinners and rabbits have been adduced, e.g. it is claimed that the pattern was the medieval alchemical symbol for tin. It is also called the Hunt of Venus and/or an emblem of the Trinity. However, the earliest reference to the pattern on Dartmoor is an 1856 description of Widecombe Church which only says that the roof was connected with the tinners and that the pattern had an alchemical connection. Later guides to Dartmoor are still pretty vague, e.g. a 1956 writer connects the symbol with copper, not tin. It is not until 1965 that the symbol is specifically called The Tinners' Rabbits. See his 2000 article for more early references.

There is no particular Dartmoor mythology connected with rabbits, but there is much mythology of hares. See the note at the beginning of this section about the introduction of rabbits to Britain. Three unconnected rabbits do occur in some English crests. Greeves reproduces and discusses the c1600 Valentine picture -- see above.

Since no list of occurrences of the pattern had ever been compiled, Greeves examined almost all the churches in the area and discovered roof bosses with the pattern in 12 churches -- see [Greeves (2000)] for a longer list). These are all on the east side of Dartmoor or to the north, except Tavistock is on the west side and Broadclyst is some 20 mi further east. Bridford, Iddesleigh, Sampford Courtenay and Spreyton have no significant tin-mining connections. No examples are known from the much more important tin-mining area of Cornwall, but Greeves has since found an example at Cotehele, just over the border into Cornwall.

Greeves then discusses the examples at Long Melford and Paderborn, giving comments which are mentioned above. He then briefly describes the Dunhuang example, citing Whitfield & Farrar, and then the St. Petersburg example. He then notes some modern versions: a wooden teapot stand from Scandinavia and a Victorian(?) carving in Holy Trinity, Fareham, Hampshire.

Early Christianity took over many pagan symbols and the three hares or rabbits (like the Green Men) could have been adapted to represent the Trinity.

In a letter of 3 Jun 1997, Greeves says he has located further examples of the three rabbits in Cheriton Bishop and Paignton in Devon, Cotehele in Cornwall and in Wales, Scarborough (North Yorkshire), France, Germany, Switzerland, Bohemia and modern China, where the pattern is still woven into carpets. See [Greeves (2000); Greeves (2001)] for more details of these.

See below for the continuation articles [Greeves (2000); Greeves (2001)].

Paul Hambling. The Dartmoor Stannaries. Orchard Publications, Newton Abbott, 1995, pp. 38-39. This gives a short summary of Greeves' work. He adds that a story is that the tinners adopted the rabbit as their emblem in allusion to their common underground mode of life. Tinners are also said to have been responsible for some rabbit warrens, but there were lots of other warrens and they would have been too common to be specifically associated with the tinners. He notes that the symbol of three intertwined fishes was a common Christian symbol.

Simon Jenkins. England's Thousand Best Churches. Allen Lane, 1999; slightly revised, Penguin, 2000. This mentions the Three Rabbits pattern in several churches.

Boxgrove, pp. 686-687. "The second boss from the altar is so crafted that each of eight faces comes complete with two eyes, yet there are only eight eyes in all."

Long Melford, pp. 658-659. "... three rabbits sharing three ears, representing the Trinity."

Sampford Courtenay, pp. 136-137.

Widecombe in the Moor, pp. 142. "This rare symbol of the Trinity is formed of three animal heads sharing just three ears."

Richard Sale. Dartmoor The Official National Park Guide. Photos by Chris Chapman. Pevensey Guides (David & Charles), 2000. Pp. 63-64 says the symbol of 'three rabbits each with only one ear' was adopted by the tinners and may be an allusion to the Holy Trinity. He also says the rabbits became a pest and myxomatosis was introduced in 1954 and rabbit warrening was abolished in 1956. P. 63 has a photo of the boss at Throwleigh.

Tom Greeves. Chasing three hares. Dartmoor Magazine No. 61 (Winter 2000) 8-10. P. 11 contains several advertisements for hotels and tours featuring the three rabbits motif. This reports on information discovered since his previous article -- see above. Many bits of information are incorporated above, citing this as Greeves (2000).

The earliest known connection of the pattern with the tinners is given in the Torquay Directory (25 Nov 1925), but the first popular usage seems to be Sylvia Sayers' The Outline of Dartmoor's Story (1925), p. 24.

He now has found 17 Devon churches with 28 examples of the pattern: Ashreigney, Bridford, Broadclyst (9 examples from 1833 said to be careful copies of medieval bosses -- Greeves (1991) erroneously has 8), Chagford (2 examples), Cheriton Bishop, Iddesleigh, Ilsington, Kelly (2 examples, one a modern copy), Newton St. Cyres, North Bovey, Paignton, Sampford Courtenay (2 examples), South Tawton, Spreyton (2 examples), Tavistock, Throwleigh, Widecombe(-in-the-Moor). Excepting the recent copies, there are 19 medieval (i.e. pre-1500) examples, all wooden roof bosses. These stretch well beyond the Dartmoor area into mid and east Devon. Many of these sites have no significant tin-mining connections. No examples are known from the much more important tin-mining area of Cornwall, except an example at Cotehele, Cornwall, just over the Cornish border. There are also examples, probably 15C, on the roof of the Lady Chapel in St. David's Cathedral, St. David's, Pembrokeshire, and on an arch of St. Aidan's church, near Llawhaden, Dyfed.

Greeves writes that his group are organising an exhibition on the pattern at the High Moorland Visitor Centre, Princetown, Devon, for several weeks from 22 Nov 2001.

Tom Greeves. Three hares -- a Medieval Mongol Mystery. Devon Today (Apr 2001) 58-63. Many bits of information are incorporated above, citing this as Greeves (2001). Notes that Easter is believed to derive from the festival of the pagan goddess Eastre, whose familiar spirit was the hare. Reports a possibly 15C example at Corfe Mullen, Dorset. Says the pattern was used in 18C Slovakian pottery. Gives photos of examples at Paignton and South Tawton and of a modern carpet from Urumqi.

MODERN VERSIONS OF THE THREE RABBITS PUZZLE

Child. Girl's Own Book. Puzzle 10. 1833: 163; 1839: 143; 1842: 264; 1876: 221. "Can you draw three rabbits, so that they will have but three ears between them; yet each will appear to have the two that belongs to it?" (1839, 1842 and 1876 have belong instead of belongs.)

Magician's Own Book. 1857. Prob. 7: The three rabbits, pp. 269 & 293. "Draw three rabbits, so that each shall appear to have two ears, while, in fact, they have only three ears between them." The drawing is similar to, but reasonably different than that in Girl's Own Book.

Book of 500 Puzzles. 1859. Prob. 7: The three rabbits, pp. 83 & 107. Identical to Magician's Own Book.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 3, pp. 395 & 436. Identical to Magician's Own Book, prob. 7.

Boy's Own Conjuring Book. 1860. Prob. 6: The three rabbits, pp. 230 & 255. Identical to Magician's Own Book.

The Drei Hasen hotel, Michelstadt, Hessen, about 40km NE of Heidelberg, uses 19C and later versions of the pattern. [Greeves (2000); Greeves (2001) has a photo of a 19C stained glass window.]

Hanky Panky. 1872. P. 87: The one-eared hares. Very similar to Magician's Own Book.

Wehman. New Book of 200 Puzzles. 1908. The three rabbits, p. 21. c= Magician's Own Book.

Collins. Book of Puzzles. 1927. The Manx rabbit puzzle, p. 153. Says it was invented by a Manxman. Shows three rabbits, each with two ears, and one has to assemble them to have just three ears.

The Warren House Inn is at one of the highest passes over Dartmoor, Devon, on the B3212 about halfway between Princetown and Moretonhampstead. The pub sign shows the Three Rabbits and they sell a polo shirt with the pattern. [Thanks to Tom Greeves for directing me to this.]

Marjorie Newman. The Christmas Puzzle Book. Hippo (Scholastic Publications, London, 1990. Kangaroos' ears, pp. 69 & 126. Like Magician's Own Book, prob. 7, but with kangaroos.

The Castle Inn, Lydford, Devon, has a fine stained glass window of the Three Rabbits by James Paterson (1915-1986). ??NYS -- described and illustrated in colour in Greeves (2000).

Jan Misspent (??sp). Design Sources for Symbolism. Batsford, 1993, p. 18. Shows the three rabbits, going anticlockwise, among other examples of three-fold rotational symmetry. Sent by Diana Hall.

Tom Greeves (see above) uses a three rabbits logo as his letterhead.

Holy Trinity Church, Long Melford, Suffolk, uses a version of their stained glass as a letterhead.

Trinity Construction Services, London and Essex, uses a three rabbits logo as their letterhead. A former director saw the pattern in Devon and liked it.

Laurie Brokenshire reports that the chaplain at HMS Raleigh, the naval training station near Plymouth, has a vestment with the three rabbits emblem. He saw it many years ago in the area and thought it was an excellent symbol of the Trinity and had the vestment made, probably for use on Trinity Sunday.

Martin and Philip Webb run a company called Fine Stone Miniatures () which makes miniatures of medieval beasties from cathedrals, etc. They have recently introduced two versions of the three rabbits.

DEAD DOGS

G. Yazdani. Ajanta Monochrome Reproductions of the Ajanta Frescoes Based on Photography Part I. 1930; reprinted by Swati Publications, Delhi, 1983. Photocopies sent by Peter Rasmussen. P. 3 has the following. "A good example of the artistic fancy of the sculptors of Ajanta is the delineation of four deer on the capital of a column in this cave (Plate XLb). They have been so carved that the one head serves for the body of any of the four. The poses of the bodies are most graceful and absolutely realistic, showing close study of nature combined with high artistic skill." He dates the cave to the end of the 5C. Footnote 1 says the motif of the four deer also occurs in a cave at Ghatotkach, which have an inscription dating them to the end of the 5C. Plate IV is a general view of Cave I, but I cannot recognise the image in it. Plate XLb is "Four deer with a common head" but very unclear.

Narayan Sanyal. Immortal Ajanta. Hrishikes Barik, Calcutta, 1984. Photocopies sent by Peter Rasmussen. P. 18, fig. 4.1, is a Plan of Cave I with Exhibits. P. 19 has a List of Exhibits in Cave I -- entry 17 is "Four deer with one head" and notes that it is shown in Yazdani. On the plan, 17 points to the two middle columns on the right side. P. 38 describes this: "On another [pillar-capital] there are four deer with a common head. The local guide would often invite the attention of the tourist to this and similar freaks. But the magnanimity of Ajanta lies not in such frivolities ...."

Benoy K. Behl, text and photographs. The Ajanta Caves Artistic Wonder of Ancient Buddhist India. Abrams, NY, 1998, 255pp. [= Benoy K. Behl, text and photographs. The Ajanta Caves: Ancient Paintings of Buddhist India. Thames and Hudson, London, 1998, 256pp.] P. 18 has a colour photo of a relief of four deer sharing a single head. Peter Rasmussen sent a B&W copy, which is not good, and then an enlarged colour picture which does show the effect, but it is still not very good. Behl says " Inside the main hall on the right-hand side one of the capitals is subtly carved to create the illusion of four recumbent and standing deer sharing a single head." This is apparently the only example at Ajanta as other authors refer to it as 'artistic fancy', 'freaks', 'frivolities'. The Ajanta Caves date from -2C to 6C.

Carl Schuster & Edmund Carpenter. Patterns That Connect Social Symbolism in Ancient & Tribal Art. Abrams, NY, 1996. P. 34, fig. 68, shows a painted pottery pattern from Panama with two heads and four legs, but rather more like Siamese twins than our 'two heads, four boys' pattern.

The Peterborough Psalter, Brussels. c1310. At the bottom of f. 48v, Psalm 68, is a somewhat crude example of 'two heads, four horses', where the two vertical horses have their front and back hooves touching so they are very curved while the horizontal horses have bellies on the ground. B&W in: Lucy Freeman Sandler; The Peterborough Psalter in Brussels and Other Fenland Manuscripts; Harvey Miller, London, 1974, plate 45, p. 28 (see p. 9 for the date of the MS). Also in Baltrušaitis. below.

Anna Roes. "Tierwirbel" in JPEK: Jahrbuch für prähistorische und ethnographische Kunst: Jahrgang 1936/37. De Gruyter, Berlin, 1937, 182 pp. ??NYS -- described by Peter Rasmussen [email of 8 Jan 2002]. Pp. 85-105 gives a history of the animal wheel motif, with many illustrations, but none with three hares.

Jurgis Baltrušaitis. Le Moyen Age Fantastique,.... Op. cit. under China, above. Pp. 132-139 of the 1981 edition have many examples of three and four rabbits, four boys, etc.

He discusses the animal wheel motif where several animals share the same head. This sort of image is rather more common than either the Three Rabbits or the Dead Dogs type of image and I have not tried to chronicle it. But I will include a few early examples. Baltrušaitis shows (Fig. 98) several examples of three fish with one head: from Egypt (XVIII-XX dynasty, i.e. c-1500); by Villard de Honnecourt, c1235; a pavement at Hérivaux, 13C; an Arabised plate from Paterna, 13C-14C. He cites examples in Italian and French ceramics.

He then goes on to 'two heads, four animals' patterns, showing (Fig. 99) a version of 'two heads, four horses' by the Safavid artist Riza Abbasi, signed and dated 20 Oct 1616, though the authenticity of the signature and date have been disputed. He notes that another version is in the Boston Museum of Fine Arts (cf below) and there are definite differences in the two versions.

He then shows (Fig. 99) the example of 'two heads, four horses' in the Peterborough Psalter. His note 160 cites J. van den Gheyn, op. cit., pl. xxi. Considerable searching finds the citation as part of note 68: J. van den Gheyn; Le Psautier de Peterborough; Haarlem, nd, ??NYS. There are a number of Peterborough Psalters, including one in the Society of Antiquaries in London, one now MS. 12 in the Fitzwilliam Museum, Cambridge, and the famous example in Brussels (which seems to be the one studied by van den Gheyn).

Baltrušaitis then shows (Fig. 100) an example of 'two heads, four men' from the Library Portal of the Cathedral of Rouen, 1290-1300. This is a relief, nicely enclosed in a quatrefoil frame. In his text, the author says the pattern also occurs on a misericord in the Cathedral and on the Palace of Justice in Rouen. These are now the earliest known examples of this idea. Baltrušaitis cites an expert on Iranian art who says the tradition is very old in Iran and could well have inspired these examples. Baltrušaitis says the pattern also occurs: in the Church at Rosny, Aube (15C); on a cathedral stall at Vendôme; at New College, Oxford.

He then shows the Two Apes on Horseback (Fig. 101), apparently the form 1985n of Schreiber. He says several engravings of the 15C show these musical apes with the movable centre and his note cites similar examples with 'amours' and men by Cornelius Reen (1560) and Adrien Herbert (1576) and very frequently in the 17C. He then has a photo (Fig. 102) of a oriental scrollweight, and says it undoubtedly comes from the same Islamic source as the other compositions with interchangeable elements. He then shows the Ninth Key of Basil Valentine, cf under Medieval Europe, above, but I don't feel it is a picture of the present type. He then reproduces (Fig. 104) Reen's picture, Symbol d'un Amour Inconstant, from the BN, dated as c1561, with two cherubs. And then he gives (Fig. 105) a 'two heads, four men' picture with a drinking man and a harlequin horn-player. This is another 1576 Dutch engraving in the BN, in a circular frame with French and Dutch text.

Zwei Affen als Kunstreiter (Verwandlungsbild). This has two apes on a horse with a small bit of paper on a pivot which shifts the mid-body connections so either head is connected to either legs. Two versions described in W. L. Schreiber; Handbuch der Holz- und Metallschnitte des XV.Jahrhunderts. Band IV: Holzschnitte, Nr. 1783-2047. Verlag Karl W. Hiersemann, Leipzig, 1927. P. 120. Schreiber pasted up examples of all the woodcuts and metalcuts described in his Handbuch in four mammoth volumes. This unique set was presented to the Warburg Institute, where they are RR 240 1-4. The following plates are in plate vol. 3. Schreiber describes coloured examples of each version, but he only has uncoloured examples in his plate volume.

1985m. This was supposedly found in Ulm when a small church was demolished. Schreiber says the painting points to a Swiss origin and guesses a date of 1460-1480. His example shows the rotating piece in one of its positions. He describes a coloured example in the Germanisches Museum, Nürnberg.

[This picture is used in: Jasia Reichardt, ed.; Play Orbit [catalogue of an exhibition at the ICA, London, and elsewhere in 1969-1970]; Studio International, 1969, p. 45. It is described as "Paper toy from an Ulm woodcut, 1470." No further details are given in Reichardt and she tells me that it was found by a research assistant, probably at the V&A. However, it seems likely it was found at the Warburg, though it is possible that the V&A also has an example of the same print.]

[Ray Bathke [email of 20 Aug 1998] says the Ulm woodcut, 1470, appears in: Karl Gröber; Children's Toys of Bygone Ages; Batsford, 1928, 1932. ??NYS.]

1985n. Basically the same picture, but elaborated and with much more decorative detail, clearly added to an earlier version (an extra column lacks a base), again probably from Switzerland, but Schreiber makes no estimate of a date. His copy lacks the rotating piece and shows no indication of its existence. He describes a coloured version at Zürich Zentralbibliothek. Baltrušaitis, above, gives a version as Fig. 100.

Wolfgang Brückner. Populäre Druckgraphik Europas Deutschland Vom 15. bis zum 20.Jahrhundert. (As: Stampe Popolari Tedesche; Verlag Electra, Milan, 1969); Verlag Georg D. W. Callwey, München, 1969. Pp. 24-25 (Abb. 17 is on p. 25), 203. Coloured example of 1985n from Zürich Zentralbibliothek, described as Swiss, 1460/80. Cites Schreiber. He says there is a replica at Nürnberg, but this must be confusing it with 1985m. This example is slightly different than Schreiber's picture in that the circle where the rotating piece would rotate does show, but the print is pasted to another sheet and one back line of an ape has been drawn in on the backing sheet. I have a slide.

Thomas Eser. Schiefe Bilder Die Zimmernsche Anamorphose und andere Augenspiele aus den Sammlungen des Germanischen Nationalmuseums [catalogue of an exhibition in 1998]. Germanisches Museum, Nürnberg, 1998, pp. 86-87. The picture is a B&W version of a coloured example of 1985m, from the Museum Graphische Sammlung, H5690, Kapsel 8. This is probably the same example described by Schreiber, but is described as a coloured woodcut, Swiss or Schwabian, 1460/70 and probably the oldest surviving example of a picture which the viewer can change. He cites Schreiber and Brückner.

Barbara Maria Stafford & Frances Terpak. Catalog for the exhibition Devices of Wonder, at the Getty Museum, early 2002. ??NYS - information sent by William Poundstone. This shows (in colour) and discusses an oil painting of Hermes and Aphrodite, like the above apes image, with a rotating piece in the middle which covers one of the pairs of waists. It dates from early 17C Bohemia, probably the school of Prague some time after Rudolf II's death in 1612.

James C. Y. Watt & Anne E. Wardwell. When Silk Was Gold Central Asian and Chinese Textiles. The Metropolitan Museum of Art in Cooperation with The Cleveland Museum of Art, dist. by Abrams. Section 45: Cloth of gold with rabbit wheels, p. 158. After discussing the four rabbits textile (see above under Other Asia), this goes on to discuss the 'two heads, four boys' motif, with Figure 73 showing a Ming example in the Shanghai Museum dated to 16-17 C. These were often used as toggles. This says that the pattern is called 'two boys make four images' and is a rebus for part of a famous line from the Yi-ching: "The primal 'one' [taiji] begets the two opposites [yi], the two opposites beget the four elements [xiang], and the four elements beget the eight trigrams [gua]." The word yi is a homonym for the word for 'boy' in some pronunciations and xiang means both 'element' and 'image'.

Rza (or Riza) Abbasi (1587-1628). Drawing: Four Horses. Drawing given in: Kh. S. Mamedov; Crystallographic Patterns. Comp. & Maths. with Appls. 12B:1/2 (1986) [= I. Hargittai, ed., Symmetry -- Unifying Human Understanding, as noted in 6.G.] 511-529, esp. 525-526. Two heads and four bodies. This seems to be an outline made from the original, probably by Mamedov? See below for a possible original version. Cf Baltrušaitis, above.

Early 17C Persian drawing: Four Horses: Concentric Design. Museum of Fine Arts, Boston. Reproduced in: Gyorgy Kepes; The New Landscape in Art and Science; Paul Theobold, Chicago, 1956, fig. 44 on p. 53 with caption on p. 52; and in: S&B, p. 34. This picture and the drawing above differ in the position of the feet and other small details, so it is not clear if the above has been copied from this picture. Seckel, 1997 (op. cit. in 6.AJ), reproduces it as 6( and says it is 18C. Baltrušaitis (Fig. 99), above, has a similar, but different version.

Itsuo Sakane. A Museum of Fun (in Japanese). Asahi Shimbun, Tokyo, 1977. Chap. 54-55, pp. 201-207. Seven examples, but I haven't had the text translated.

Itsuo Sakane et al. The Expanding Visual World -- A Museum of Fun. Catalogue of a travelling exhibition with some texts in English. Asahi Shimbun, Tokyo, 1979. Section IV: Visual Games, no. 12-15, pp. 96-99, with short texts on p. 170. Seven examples, mostly the same as in the previous book, including the following. I give the page numbers of the previous book in ( ) when the picture is in both books.

IV-12, p. 96 (p. 205), both B&W. Sadakage Gokotei. Five Children, Ten Children. Edo era (= 1603-1867). Seckel, 1997 (op. cit. in 6.AJ), reproduces it in colour as 5( with the same data and Seckel, 2000 (op. cit. in 6.AJ), reproduces it in colour as fig. 44, p. 55 (= 2002a, fig, 44, p. 55), with no data. However, the same picture is reproduced in colour in Julian Rothenstein & Mel Gooding; The Paradox Box; Redstone Press, London, 1993; with a caption by James Dalgety, saying that it is a painting by Yamamoto Hisabei, c1835, based on an earlier Chinese image, and giving the title as Ten Children with Five Heads.

Unnumbered, p. 97 (pp. 202-203, figs. 2, 3, 4), both B&W, but the titles have no English versions. Though no credit is given, the top item is is the early 17C Persian drawing in the Museum of Fine Arts, Boston.

The second item seems to be from Renaissance Europe and can be called 'three heads, seven children'. There are three heads and upper bodies alternating with three legs and lower bodies in a flattened hexagonal pattern so the the top head can also connect to the bottom legs giving a third arrangement of the pieces, though the other two children occur in the previous arrangements, so one gets a total of seven children rather than nine. I have only recently discovered this is a European item and Bill Kalush has an example of a lead medallion with the same pattern which is dated to c1610 Prague -- I have a photo. Edward Hordern's collection has a wooden box with this pattern on the cover, dated as 16C, but it looks later to me, though I have only seen photos.

The third item is a version of the 'five heads, ten chldren' picture described above. However, the references to this chapter mention Lietzmann and I have found it there, where it is stated to be a Japanese matchbox -- see below.

IV-14, p. 99 (colour) (p. 207, B&W). Kuniyoshi Ichiyusai [= Utagawa Kuniyoshi]. Stop Yawning. Late Edo era [mid 19C].

IV-15, p. 98 (colour). Anon. Four Heads, Twelve Horses. Probably Persian. This item belongs to Martin Gardner, having been left to him by M. C. Escher. It seems to be a leather cushion cover, probably Persian. Seckel, 1997 (op. cit. in 6.AJ), reproduces it as the 6( and says it is 18C. Seckel, 2000 (op. cit. in 6.AJ), reproduces it as fig. 88, pp. 99 & 122 (= 2002b, fig. 86, pp. 97 & 120) and says sometime in the 17C.

Metal scrollweights of the 'two heads, four children' pattern have been made in China since at least the 17C. I have three modern examples which Peter Hajek obtained for me in Hong Kong. Edward Hordern's collection has a version from c1680 and a porcelain version, about six or eight inches across, among other examples -- I have the date somewhere. James Dalgety also has an example of the porcelain version.

"Three Boys -- Nine Torsos". Anonymous painting on silk from 1700-1710 in City Palace Museum, Jaipur, India. Edward Hordern's collection has a modern replica. This is similar to the Three Heads, Seven Children version mentioned above. I have photos from Hordern's replica and a copy of his information sheet on it and another painting. Reproduced from Hordern's example in: Rothenstein & Gooding, below, p. 16.

Mohammad Bagheri has sent me some souvenir material from the Museum, now named the Maharaja Sawai Man Singh II Museum, City Palace, Jaipur. This shows a different version of the pattern, with the same basic geometry but very different figures and colours.

Family Friend 1 (1849) 148 & 178. Practical puzzles -- No. 3 -- Dead or alive? "These dogs are dead you well may say:-- Add four lines more, they'll run away!" Answer has "See now the four lines. "Tally-ho!" We've touch'd the dogs, and away they go!"

Julian Rothenstein & Mel Gooding. The Playful Eye. Redstone Press, London, 1999. They include a number of Japanese prints. There are brief notes on p. 100.

P. 16. "Three Boys -- Nine Torsos". Photo from Hordern's facsimile, see above.

P. 20. Three heads, six bodies. Woodblock print, 1860s.

P. 20. Five heads, ten bodies, similar to the version given by Sakane, above, but with different costumes. Woodblock print, 1860s.

P. 21. Two or three bodies, one head. Woodblock print, c1855. This has six groups of bodies where one head or hat can be associated with two or three of the bodies. There is a set of four upper bodies which can be associated with one pair of legs.

P. 22. Five heads, ten bodies, almost identical to the version given by Sakane, above, but in colour with MADE IN JAPAN on the bottom edge. The notes say it is a Japanese matchbox label, c1900.

P. 22. Six heads, twelve bodies. Two rings of three heads, six bodies, with elephants. Japanese matchbox label, c1900.

P. 24. Five heads, ten bodies, similar to the version given by Sakane, above, but with women. Woodblock print, c1850.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 5, p. 177 (1868: 188): Alive or Dead. "These dogs are dead, perhaps you'll say; Add four lines and then they'll run away."

Magician's Own Book. 1857. Prob. 27: The dog puzzle, pp. 275 & 298. "The dogs are, by placing two lines upon them, to be suddenly aroused to life and made to run. Query, How and where should these lines be placed, and what should be the forms of them?" S&B, p. 34.

Book of 500 Puzzles. 1859. Prob. 27: The dog puzzle, pp. 89 & 112. Identical to Magician's Own Book.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 33: Dead or Alive, pp. 401 & 441 "These dogs are dead you well may say:-- Add four lines more, they'll run away!"

Boy's Own Conjuring Book. 1860. Prob. 26: The dog puzzle, pp. 237 & 261. Identical to Magician's Own Book.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 583-7, p. 285. Dead dogs.

Magician's Own Book (UK version). 1871. The solution drawing is given at the bottom of p. 231, apparently to fill out the page as there is no relevant text anywhere. The drawing is better than in the 1857 US book of the same name.

Elliott. Within-Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 6: The dog puzzle, pp. 28 & 31. "By connecting the dogs with four lines only they will suddenly start into life, and commence running. Where should the lines be placed?" However, he omits to give a picture!

Hoffmann. 1893. Chap. 10, no. 32: The two dogs, pp. 348 & 387 = Hoffmann-Hordern, pp. 244-245. No poetry, but the solution notes that you have to view the dogs sideways.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 9:5 (Mar 1903) 490-491. Trick donkeys. "Here are two apparently very dead donkeys. To bring them to life it is only necessary to fill in the dotted lines and then turn the page half way round."

Benson. 1904. The dead dogs puzzle, pp. 256-257. Prose version.

Pearson. 1907. Part III, no. 83: Rousing dead dogs -- A good old puzzle, p. 83 . "These dogs are dead, we all should say; Give them four strokes, they run away."

Wehman. New Book of 200 Puzzles. 1908. The dog puzzle, p. 22. c= Magicians Own Book.

W. Lietzmann. Lustiges und Merkwürdiges von Zahlen und Formen. 1922. I can't find it in the 2nd ed. of 1923. 4th ed, F. Hirt, Breslau, 1930, p. 208, unnumbered figure, shows the 'five heads, ten children' pattern mentioned as the third item on p. 97 of the Sakane book above, labelled: Japanese Matchbox How many people, how many heads, how many legs, how many arms are in this picture?

The material is also in the 6th ed., 1943, p. 200, fig. 46; 7th & 10th eds., 1950 & 1969, p. 196, fig. 37.

Collins. Book of Puzzles. 1927. The dead dogs puzzle, p. 152.

A two bodied woman. In Seckel, 2002a, op. cit. in 6.AJ, fig. 2, pp. 11 & 44 (= 2002b, fig. 144, pp. 161 & 194). A real photo of Lady Bird Johnson greeting a woman friend which shows just one head on two embracing bodies.

TRICK MULES

Loyd. P. T. Barnum's Trick Mules. 1871. Loyd registered this in 1871 and sold it to Barnum shortly thereafter. Barnum used it in his Advance Courier. See S&B, p. 34, for an illustration of Barnum's version and two recent versions. See SLAHP: Out for a gallop, pp. 65 & 110. See Gardner, SA (Aug 1957) c= 1st Book, chap. 9. In a 1907 interview, it was stated that thousands of millions of copies of the puzzle had been printed, with Loyd taking orders for a million at a time!

Gaston Tissandier. Jeux et Jouets du jeune age Choix de récréations amusantes & instructives. Ill. by Albert Tissandier. G. Masson, Paris, nd [c1890]. Le mulet rigolo, pp. 36-37, with elegant coloured plate. No reference to its history.

Mel Stover. 1980s?? Trick zebras puzzle. This has two identical cards with two zebras and two riders. The instructions say to cut one card into three parts along the dotted lines and put the riders on the zebras. However, one zebra is facing the opposite way to the usual case and it takes some time to realise how to solve the problem.

Seckel, 2000 (op. cit. in 6.AJ), gives a nice colour version as fig. 105, pp. 116 & 122 (= 2002b, fig. 103, pp. 114 & 120), but only says it is due to Loyd.

6.AV. CUTTING UP IN FEWEST CUTS

Mittenzwey. 1880. Prob. 191, pp. 36 & 88-89; 1895?: 216, pp. 40 & 91; 1917: 216, pp. 37 & 87. Cut a 2 x 4 into eight unit squares with three cuts. First cuts into two squares, then overlays them and then cuts both ways.

Perelman. FFF. 1934.

Sectioning a cube. Not in the 1957 ed. 1979: prob. 122, pp. 170-171 & 182. MCBF: prob. 122, pp. 171-172 & 186. How many cuts to cut a cube into 27 cubes?

More sectioning. Not in the 1957 ed. 1979: prob. 123, pp. 171 & 182-183. MCBF: prob. 123, pp. 172-173 & 186. How many cuts to cut a chessboard into 64 squares?

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. Prob. k, pp. 39, 189 & 191. Cut a cube into 27 cubelets.

6.AW. DIVISION INTO CONGRUENT PIECES

Polyomino versions occur in 6.F.4.

Quadrisecting a square is 6.AR.

See also: 6.AS.1, 6.AT.6.a, 6.AY, 6.BG.

For solid problems, see: 6.G.3, 6.G.4, 6.AP, 6.AZ?, 6.BC.

See: Charades, Enigmas, and Riddles, 1862, in 6.AW.2 for a quadrisection with pieces not congruent to the original.

6.AW.1. MITRE PUZZLE

Take a square and cut from two corners to the centre to leave ¾ of the square. The problem is to quadrisect this into four congruent parts.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has an entry "8½ a Prob of figure" followed by the L-tromino. 8½ b is the same with a mitre and there are other dissection problems adjacent -- see 6.F.3, 6.F.4, 6.AQ, 6.AY -- so it seems clear that he knew this problem.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 15, pp. 26 & 86 & plate II, fig. 11. 2 squares, one double the size of the other, to be cut into four pieces to make a mitre. Just cut each along the diagonal.

No. 18, pp. 27 & 87 & plate II, fig. 14. Six equal squares to form a mitre. Cut each diagonally. [Actually you only need to cut three of the squares.]

Endless Amusement II. 1826? Prob. 5, pp. 192-193. Mitre puzzle -- says the pieces are not precisely equal. = New Sphinx, c1840, pp. 131-132.

Magician's Own Book. 1857.

Prob. 12: The quarto puzzle, pp. 269 & 294. Solution is a bit crudely drawn, but the parts are numbered to make it clear how they are combined. = Illustrated Boy's Own Treasury, 1860, No. 41, pp. 403 & 442.

Prob. 28: Puzzle of the two fathers, pp. 275-276 & 298. One father has L-tromino (see 6.F.4), the other has the mitre. Solution carefully drawn and shaded. c= Landells, Boy's Own Toy-Maker, 1858, pp. 148-149.

Book of 500 Puzzles. 1859.

Prob. 12: The quarto puzzle, pp. 83 & 108. Identical to Magician's Own Book.

Prob. 28: Puzzle of the two fathers, pp. 89-90 & 112. Identical to Magician's Own Book..

Boy's Own Conjuring Book. 1860.

Prob. 11: The quarto puzzle, pp. 231 & 257. Identical with Magician's Own Book.

Prob. 27: Puzzle of the two fathers, pp. 237-238 & 262. Identical to Magician's Own Book.

Hanky Panky. 1872. The one-quarterless square, p. 132

Hoffmann. 1893. Chap. X, no. 29: The mitre puzzle, pp. 347 & 386 = Hoffmann-Hordern, pp. 243 & 247. Photo on p. 247 shows Enoch Morgan's Sons Sapolio Color-Puzzle. This says to arrange the blocks 'in four equal parts so that each part will be the same size color and shape.' It appears that the blocks are isosceles right triangles with legs equal to a quarter of the original square. There are 6 blue triangles, 6 yellow triangles and 12 red triangles. I think there were originally 6 red and 6 orange, but the colors have faded, and I think one wants each of the four parts having one color.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 5: The mitre puzzle. Similar to Hoffmann. No solution.

Loyd. Origin of a famous puzzle -- No. 19: The mitre puzzle. Tit-Bits 31 (13 Feb & 6 Mar 1897) 363 & 419. Nearly 50 years ago someone told him to quadrisect ¾ of a square into congruent figures. The L-tromino was intended, but young Loyd drew the mitre shape instead. He says it took him nearly a year to solve it. But see Dudeney's comments below.

Clark. Mental Nuts. 1904, no. 31. Dividing the land. Quadrisect an L-tromino and a mitre.

Pearson. 1907. Part II, no. 87: Loyd's mitre puzzle, pp. 87 & 178.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He says he has traced it back to 1835 (Loyd was born in 1841) and that "strictly speaking, it is impossible of solution, but I will give the answer that is always presented, and that seems to satisfy most people." See also the solution to AM, prob. 150, discussed in 6.AY. Can anyone say what the 1835 source might be -- a version of Endless Amusement??

Wehman. New Book of 200 Puzzles. 1908.

P. 43: Puzzle of the two fathers. c= Magician's Own Book, with cruder solution.

P. 47: The quarto puzzle. c= Magician's Own Book, without the numbering of parts.

Loyd. Cyclopedia. 1914. A tailor's problem, pp. 311 & 381. Quadrisect half of a mitre. This has a solution with each piece similar to to the half mitre.

Loyd Jr. SLAHP. 1928. Wrangling heirs, pp. 35 & 96. Divide mitre into 8 congruent parts -- uses the pattern of Loyd Sr.

Putnam. Puzzle Fun. 1978. No. 106: Divide the shape, pp. 16 & 39. "Divide the given shape into four pieces, such that each and every piece is the same area." This is much easier than the usual version. Put two half-size mitres on the bottom edge and two trapeziums are left.

6.AW.2. REP-TILES

Here one is cutting a shape into congruent pieces similar to the original shape. Section 5.J is a version with similar but non-congruent pieces.

Charades, Enigmas, and Riddles. 1862: prob. 34, pp. 137 & 143; 1865: prob. 578, pp. 109 & 156. Divide isosceles right triangle and a hatchet-shaped 10-omino into four congruent pieces. In the first case, they are congruent to the original, but in the second case, the pieces are right trapeziums of height 1 and bases 2 and 2½.

Dudeney. The puzzle realm. Cassell's Magazine ?? (May 1908) 713-716. No. 3: An easy dissection puzzle. Quadrisect a right trapezium in form of square plus half a square. = AM, 1917, prob. 146, pp. 35 & 170.

C. Dudley Langford. Note 1464: Uses of a geometric puzzle. MG 24 (No. 260) (Jul 1940) 209-211. Quadrisects: L-tromino, P-pentomino, right trapezium (trapezoid), L-tetromino, isosceles trapezium, another right trapezium, two squares joined at a corner. Gives some 9-sections and asks several questions, including asking about 3-D versions.

R. Sibson. Note 1485: Comments on Note 1464. MG 24 (No. 262) (Dec 1940) 343. Says some of Langford's 4-sections also give 9-sections. Mentions some 3-D versions and 144-sections.

Howard D. Grossman. Fun with lattice points. 13. A geometric puzzle. SM 14 (1948) 157-159. Cites Langford & Sibson. Gives two 9-sections obtained from Langford's 4-sections as asserted by Sibson. Gives alternative 4- & 9-sections. Gives a method of generating infinitely many examples on both square and triangular lattices.

Gardner. SA (May 1963) = Unexpected, chap. 19. Says Golomb started considering rep-tiles in 1962 and wrote three private reports on them (??NYS). Gardner describes the ideas in them.

Solomon W. Golomb. Replicating figures on the plane. MG 48 (No. 366) (Dec 1964) 403-412. Cites Langford and adds numerous examples. Defines rep-k and shows all k can occur.

Roy O. Davis. Note 3151: Replicating boots. MG 50 (No. 372) (May 1966) 175.

Rochell Wilson Meyer. Mutession: a new tiling relationship among planar polygons. MTg 56 (1971) 24-27. A and B mutually tessellate if each tiles an enlargement of the other.

6.AW.3. DIVIDING A SQUARE INTO CONGRUENT PARTS

In the 1960s, a common trick was to give someone a number of quadrisection problems where the parts happen to be congruent to the original figure -- e.g. the quadrisection of the square or the L-tromino. Then ask her to divide a square into five congruent parts. She usually tries to use square pieces in some way and takes a long time to find the obvious answer. c1980, Des MacHale told me that it was a serious question as to whether there was any non-trivial dissection of a square into five or even three congruent pieces. Sometime later, I found a solution -- slice the square into 6 equal strips and say part A consists of the 1st and 4th strips, part B is the 2nd and 5th, part C is the 3rd and 6th. However this is not what was intended by the problem though it leads to other interesting questions. Since then I have heard that the problem has been 'solved' negatively several times on the backs of envelopes at conferences, but no proof seems to have ever appeared. Very little seems to be published on this, so I give what little I know. Much of this applies to rectangles as well as squares. QARCH is an occasional problem sheet issued by the Archimedeans, the Cambridge (UK) student mathematics society.

Gardner, in an article: My ten favorite brainteasers in Games (collected in Games Big Book of Games, 1984, pp. 130-131) says the dissection of the square into five congruent parts is one of his favorite problems. ??locate

David Singmaster. Problem 12. QARCH III (Aug 1980) 3. Asks if the trisection of the square is unique.

David Singmaster. Response to Problem 12 and Problem 21. QARCH V (Jan 1981) 2 & 4. Gives the trisection by using six strips and unconnected parts. In general, we can have an n-section by cutting the square into kn strips and grouping them regularly. For n = 2, k = 4, there is an irregular dissection by using the parts as strips 1, 4, 6, 7 and 2, 3, 5, 8. If p is an odd prime, are there any irregular p-sections?

John Smith, communicated in a letter from Richard Taylor, editor of QARCH, nd, early 1981? Smith found that if you slice a square into 9 strips, then the following parts are congruent, giving an irregular trisection. 1, 2, 6; 3, 7, 8; 4, 5, 9.

David Singmaster. Divisive difficulties. Nature 310 (No. 5977) (9 Aug 1984) 521 & (No. 5979) (23 Aug 1984) 710. Poses a series of problems, leading to the trisections of the cube. No solutions were received.

Angus Lavery asserts that he can trisect the cube, considered as a 3 x 3 x 3 array of indivisible unit cubes. I had sought for this and was unable to find such a trisection and had thought it impossible and I still haven't been able to do it, but Angus swears it can be done, with one piece being the mirror image of the other two.

George E. Martin. Polyominoes -- A Guide to Puzzles and Problems in Tiling. MAA Spectrum Series, MAA, 1991. Pp. 29-30. Fig. 3.9 shows a 5 x 9 rectangle divided into 15 L-trominoes. Shrinking the length 9 to 5 gives a dissection of the square into 15 congruent pieces which are shrunken L-trominoes. Prob. 3.10 (very hard) asks for a rectangle to be dissected into an odd number of congruent pieces which are neither rectangles nor shrunken L-trominoes. He doesn't give an explicit answer, but on p. 76 there are several rectangles filled with an odd number of L, P and Y pentominoes. One might argue that the L and P have the shape of some sort of shrunken L-tromino, but the Y-pentomino is certainly not. Prob. 3.11 (unsolved) asks if a rectangle can be dissected into three congruent pieces which are not rectangles. Prob. 3.12 is a technical generalization of this and hence is also unsolved.

On 19 Jun 1996, I proposed the trisection of the square and Lavery's problem on NOBNET. Michael Reid demonstrated that Lavery's problem has no solution and someone said Lavery had only conjectured it. Reid also cited the following two proofs that the square trisection is impossible.

I. N. Stewart & A. Wormstein. Polyominoes of order 3 do not exist. J. Combinatorial Theory A 61 (1992) 130-136. ??NYS -- Reid says they show that if a rectangle is dissected into three congruent polyominoes, then each is a rectangle.

S. J. Maltby. Trisecting a rectangle. J. Combinatorial Theory A 66 (1994) 40-52. ??NYS -- Reid says he proves the result of Stewart & Wormstein without assuming the pieces are polyominoes.

Martin Gardner. Six challenging dissection tasks. Quantum (May/Jun 1994). Reprinted, with postscript, in Workout, chap. 16. Trisecting the square into congruent parts is his first problem. Cites Stewart & Wormstein. Then asks if one can have three similar parts, with just two, or none, congruent. Then asks the same questions about the equilateral triangle. All except the first question are solved, but the solutions for the fifth and sixth are believed to be unique. In the postscript, Gardner says Rodolfo Kurchan and Andy Liu independently suggested the problems with four parts and cites Maltby.

6.AW.4. DIVIDING AN L-TROMINO INTO CONGRUENT PARTS

See also 6.F.4.

F. Göbel. Problem 1771: The L-shape dissection problem. JRM 22:1 (1990) 64-65. The L-tromino can be dissected into 2, 3, or 4 congruent parts. Can it be divided into 5 congruent parts?

Editorial comment -- The L-shaped dissection problem. JRM 23:1 (1991) 69-70. Refers to Gardner.

Comments and partial solution by Michael Beeler. JRM 24:1 (1992) 64-69.

Martin Gardner. Tiling the bent tromino with n congruent shapes. JRM 22:3 (1990) 185-191.

6.AX. THE PACKER'S SECRET

This requires placing 12 unit discs snugly into a circular dish of radius 1 + 2 (3 = 4.464.

Tissandier. Récréations Scientifiques. 5th ed., 1888, Le secret d'un emballeur, pp. 227-229. Not in the 2nd ed. of 1881 nor the 3rd ed. of 1883. Illustration by Poyet. He shows the solution and how to get the pieces into that pattern. No dimensions given. = Popular Scientific Recreations; [c1890]; Supplement: The packer's secret, pp. 855-856.

Hoffmann. 1893. Chap. X, no. 48: The packer's secret, pp. 356 & 394 = Hoffmann-Hordern, p. 255. Says the problem is of French origin. Gives dimensions 3½ and ¾, giving a ratio of 14/3 = 4.667. "The whole are now securely wedged together ...." [I think this would be a bit loose.]

"Toymaker". The Japanese Tray and Blocks Puzzle. Work, No. 1447 (9 Dec 1916) 168. Says to make the dish of radius 7 and the discs of radius 1½, again giving a ratio of 14/3 = 4.667. Makes "a firm immovable job ...."

6.AY. DISSECT 3A x 2B TO MAKE 2A x 3B, ETC.

This is done by a 'staircase' cut. See 6.AS.

Pacioli. De Viribus. c1500. Ff. 189v - 191r. Part 2. LXXIX. Do(cumento). un tetragono saper lo longare con restregnerlo elargarlo con scortarlo (a tetragon knows lengthening and contraction, enlarging with shortening ??) = Peirani 250-252. Convert a 4 x 24 rectangle to a 3 x 32 using one cut into two pieces. Pacioli's

description is cryptic but seems to have two cuts, making d c

three pieces. There is a diagram at the bottom of f. 190v, badly k f e

redrawn on Peirani 458. Below this is a inserted note which Peirani

252 simply mentions as difficult to read, but can make sense. The g  

points are as laid out at the right. abcd is the original 4 x 24 h a o b

rectangle. g is one unit up from a and e is one unit down from c.

Cut from c to g and from e parallel to the base, meeting cg at f. Then move cdg to fkh and move fec to hag. Careful rereading of Pacioli now seems to show he is using a trick! He cuts from e to f to g. then turns over the upper piece and slides it along so that he can continue his cut from g to h, which is where f to c is now. This gives three pieces from a single cut! Pacioli clearly notes that the area is conserved.

Although not really in this topic, I have put it here as it seems to be a predecessor of this topic and of 6.P.2.

Cardan. De Rerum Varietate. 1557, ??NYS. = Opera Omnia, vol. III, p. 248 (misprinted 348 and with running head Lib. XII in the 1663 ed.). Liber XIII. Shows 2A x 3B to 3A x 2B and half of 3A x 4B to 4A x 3B and discusses the general process.

Kanchusen. Wakoku Chiekurabe. 1727. Pp. 11-12 & 26-27. 4A x 3B to 3A x 4B, with the latter being square. Solution asserts that any size of paper can be made into a square: 'fold lengthwise into an even number and fold the width into an odd number' -- cf Loyd (1914) & Dudeney (1926) below.

Minguet. 1733. Pp. 117-119 (1755: 81-82; 1822: 136-137; 1864: 114-115). 3 x 4 to 4 x 3. Shows a straight tetromino along one side moved to a perpendicular side so both shapes are 4 x 4.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has an entry "8½ c Prob of figure" followed by a staircase piece. F. 145-146 show two pieces formed into both rectangles. There are other dissection problems adjacent on F. 4v -- see 6.F.3, 6.F.4, 6.AQ, 6.AW.1.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 7, pp. 24 & 83-84 & plate I, fig. 4. 9 x 16 to 12 x 12.

No. 12, pp. 25 & 85 & plate I, fig. 9. 4 x 9 to 6 x 6.

No. 14, pp. 26 & 86 & plate I, fig. 10. 10 x 20 to 13 1/3 x 15.

Endless Amusement II. 1826?

Prob. 1, p. 188. 5A x 6B to 6A x 5B and to 4A x 7B with two A x B projections. The 6A x 5B looks to be square. = New Sphinx, c1840, p. 136.

Prob. 22, pp. 200-201. 16 x 9 to fill a 12 x 12 hole. Does it by cutting in four pieces -- one 12 x 9 and three 4 x 3. = New Sphinx, c1840, pp. 136-137.

Prob. 33, pp. 210-211. Take a rectangle of proportion 2 : 3 and cut it into two pieces to make a square. Uses cut from 4A x 5B to 5A x 4B, but if we make the rectangle 4 x 6, this makes A = 1, B = 6/5 and the 'square' is 5 x 24/5. = New Sphinx, c1840, p. 140.

Nuts to Crack II (1833), no. 124. 9 x 16 to fill a 12 x 12 hole using four pieces. = Endless Amusement II, prob. 22.

Young Man's Book. 1839. Pp. 241-242. Identical to Endless Amusement II, prob. 22.

Boy's Own Book. 1843 (Paris): 436 & 441, no. 6. 5A x 6B to 6A x 5B and to 4A x 7B with two A x B projections. The 6A x 5B looks to be square. = Boy's Treasury, 1844, pp. 425 & 429. = de Savigny, 1846, pp. 353 & 357, no. 5. Cf de Savigny, below.

de Savigny. Livre des Écoliers. 1846. P. 283: Faire d'une carte un carré. View a playing card as a 5A x 4B rectangle and make a staircase cut and shift to 4A x 5B, which will be nearly square. [When applied to a bridge card, 3.5 x 2.25 in, the result is 2.8 x 2.8125 in.]

Magician's Own Book. 1857. Prob. 2: The parallelogram, pp. 267 & 291. Identical to Boy's Own Book, 1843 (Paris).

The Sociable. 1858. Prob. 19: The perplexed carpenter, pp. 292 & 308. 2 x 12 to 3 x 8. = Book of 500 Puzzles, 1859, prob. 19, pp. 10 & 26. = The Secret Out, 1859, p. 392.

Book of 500 Puzzles. 1859.

Prob. 19: The perplexed carpenter, pp. 10 & 26. As in The Sociable.

Prob. 2: The parallelogram, pp. 81 & 105. Identical to Boy's Own Book, 1843 (Paris).

Charades, Enigmas, and Riddles. 1860: prob. 29, pp. 60 & 64; 1862: prob. 30, pp. 136 & 142; 1865: prob. 574, pp. 107 & 155. 16 x 9 to 12 x 12. All the solutions have an extraneous line in one figure.

Boy's Own Conjuring Book. 1860. Prob. 2: The parallelogram, pp. 229 & 254. Identical to Boy's Own Book, 1843 (Paris).

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 584-4, pp. 286 & 404. Looks like 3 x 4 to 2 x 6. The rectangles are formed by trimming a quarter off a playing card. The diagrams are not very precise, but it seems that the card is supposed to be twice as long as wide. If we take the card as 4 x 8, then the problem is 3 x 8 to 4 x 6.

Hanky Panky. 1872. The parallelogram, p. 107. "A parallelogram, ..., may be cut into two pieces, by which two other figures can be formed." Shows 5A x 4B cut, but no other figures.

Mittenzwey. 1880. Prob. 253 & 255, pp. 45-46 & 96-97; 1895?: 282 & 284, pp. 49 & 98-99; 1917: 282 & 284, pp. 45 & 93-94. 4 x 9 to 6 x 6. 16 x 9 to 12 x 12.

Cassell's. 1881. The carpenter's puzzle, p. 89. = Manson, 1911, p. 133. 3 x 8 board to cover 2 x 12 area.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. Cut 4 x 3 to 3 x 4. Discusses general method for nA x (n+1)B to (n+1)A x nB and notes that the shape can be oblique as well as rectangular.

Lemon. 1890. Card board puzzle, no. 58, pp. 11-12 & 99. c= The parallelogram puzzle, no. 620, pp. 77 & 120 (= Sphinx, no. 706, pp. 92 & 121). Same as Boy's Own Book, 1843 (Paris). In the pictures, A seems to be equal to B.

Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136, no. 8. 9 x 15 to 12 x 12. No solution.

Tom Tit, vol. 2. 1892. Les figures superposables, pp. 149-150. 3 x 2 to 2 x 3.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Chinese Geometrical Puzzles No. 1, pp. 108 & 111. Same as Boy's Own Book, 1843 (Paris).

Hoffmann. 1893.

Chap. III, no. 8: The extended square, pp. 91 & 124-125 = Hoffmann-Hordern, p. 80. As in Boy's Own Book, 1843 (Paris), but A is clearly not equal to B.

Chap. III, no. 31: The carpenter's puzzle -- no. 2, pp. 103 & 137 = Hoffmann-Hordern, p. 101. 12 x 36 to 18 x 24.

Clark. Mental Nuts. 1897, no. 27. The leaking ship. 12 x 12 to 9 x 16.

Benson. 1904.

The extended square, p. 190. 5A x 6B square, but the other two figures are 8A x 7B and 8A x 5B with two A x B projections.

The carpenter's puzzle (No. 1), pp. 190-191. = Hoffmann, p. 103.

Anon [possibly Dudeney??] Breakfast Table Problems No. 331: A carpenter's dilemma. Daily Mail (31 Jan & 1 Feb 1905) both p. 7. 16 x 9 to 12 x 12.

Pearson. 1907. Part II, no. 1: The carpenter's puzzle, pp. 1-2 & 185-186. 2 x 12 to 3 x 8.

Wehman. New Book of 200 Puzzles. 1908.

P. 6: The perplexed carpenter. 2 x 12 to 3 x 8. c= The Sociable.

P. 9: The carpenter's puzzle. "A plank was to be cut in two: the carpenter cut it half through on each side, and found he had two feet still to cut. How was it?" This is very vague and can only be recognised as a version of our present problem because the solution looks like cutting a 2 x 6 to make a 3 x 4.

P. 16: The parallelogram. Identical to Boy's Own Book, 1843 (Paris).

P. 17: Another parallelogram. Takes a parallelogram formed of a square and a half a square and intends to form a square. He cuts 5A x 4B and makes 4A x 5B. But for this to be a square, it must be 20 x 20 and then the original was 25 x 16, which is not quite in the given shape.

M. Adams. Indoor Games. 1912. A zigzag puzzle, p. 349, with figs. on 348. 5A x 6B square, but the other two figures are 5A x 4B and 7A x 4B with two A x B projections.

Loyd. Cyclopedia. 1914.

The smart Alec puzzle, pp. 27 & 342. (= MPSL1, prob. 93, pp. 90-91 & 153-154.) Cut a mitre into pieces which can form a square. He trims the corners and inserts them into the notch to produce a rectangle and then uses a staircase cut which he claims gives a square using only four pieces. Gardner points out the error, as carefully explained by Dudeney, below. Since Dudeney gives this correction 1n 1911, he must have seen it in an earlier Loyd publication, possibly OPM?

The carpenter's puzzle, pp. 51 & 345. Claims any rectangle can be staircase cut to make a square. Shows 9 x 4 to 6 x 6 and 25 x 16 to 20 x 20. Cf Kanchusen (1727) and Dudeney (1926).

Dudeney. Perplexities. Strand Magazine 41 (No. 246) (Jun 1911) 746 & 42 (No. 247) (Jul 1911) 108. No. 45: Dissecting a mitre. "I have seen an attempt, published in America, ..." Sketches Loyd's method and says it is wrong. "At present no solution has been found in four pieces, and one in five has not apparently been published."

Dudeney. AM. 1917. Prob. 150: Dissecting a mitre, pp. 35-36 & 170-171. He fully describes "an attempt, published in America", i.e. Loyd's method. If the original square has side 84, then Loyd's first step gives a 63 x 84 rectangle, but the staircase cut yields a 72 x 73½ rectangle, not a square. Dudeney gives a 5 piece solution and says "At present no solution has been found in four pieces, and I do not believe one possible."

Dudeney. MP. 1926. Prob. 115: The carpenter's puzzle, pp. 43-44 & 132-133. = 536, prob. 338, pp. 116-117 & 320-321. Shows 9 x 16 to 12 x 12. "But nobody has ever attempted to explain the general law of the thing. As a consequence, the notion seems to have got abroad that the method will apply to any rectangle where the proportion of length to breadth is within reasonable limits. This is not so, and I have had to expose some bad blunders in the case of published puzzles ...." He discusses the general principle and shows that an n-step cut dissects n2 x (n+1)2 to a square of side n(n+1). Gardner adds a note referring to AM, prob. 150. Cf Kanchusen (1727) & Loyd (1914)

Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 299. Mentions 12 x 12 to 9 x 16.

Harry Lindgren. Geometric Dissections. Van Nostrand, 1964. P. 28 discusses Loyd's mitre dissection problem and variations. He also thinks a four piece solution is impossible.

6.AY.1. O'BEIRNE'S STEPS

This is a cube dissected into 6 pieces which form 6 cuboids, each of which can be 'staircased' in two ways. There is a 6-cycle through the cuboids, with relative sizes: 12 x 12 x 12, 8 x 12 x 18, 8 x 9 x 24, 12 x 6 x 24, 16 x 6 x 18, 16 x 9 x 12. I have a fine example using six different woods, that had been made for Tom O'Beirne, from Mrs. O'Beirne.

Richard K. Guy. Op. cit. in 5.H.2. 1960. Pp. 151-152 describes O'Beirne's invention.

T. H. O'Beirne. Puzzles and paradoxes -- 9: A six-block cycle for six step-cut pieces. New Scientist 9 (No. 224) (2 Mar 1961) 560-561.

6.AY.2. SWISS FLAG PUZZLE

This appears to be a 7 x 5 flag with a Greek cross X X X X X O O X X X X X

of 5 cells removed from the middle as in the first figure X X X O O O X X X O O

at the right. One has to cut it into two pieces to make a X X O O X X O O O

perfect square. This is done by cutting along a 'staircase' X X X O O O X X X O O

as shown. However, this seems to produce a 5 x 6 flag, X X O O O O O X X O O O

not a square. But there is usually a swindle -- the diagram O O O O O

is not drawn with the unit cells square, but instead the unit

cells are 6/5 as wide as they are tall. Normally the reader would not recognize this and the diagrams are often rather imprecise.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 5: The flag puzzle. Starts with a square and asks to make a Swiss flag. The square is actually 31.5 mm = 1¼ in on an edge and the flag is 26 x 37 mm = 1 x 1½ in with the Greek cross formed from squares of edge 5 mm = ¼ in, so the areas do not add up! No solution.

Loyd. Problem 4: The Swiss flag puzzle. Tit-Bits 31 (31 Oct 1896) 75 & (21 Nov 1896) 131. c= Cyclopedia, pp. 250 & 373. Swiss flag puzzle with the flag at an angle and a slight wiggle in the edges, so the solution requires an extra cut to make it square.

Loyd. Cyclopedia, 1914, pp. 14 & 341: A Swiss puzzle -- part 2. = MPSL2, no. 144, pp. 101 & 166. = SLAHP, pp. 48 & 102: How was this flag made? Starts with the Swiss flag which is 47 x 30 mm and the Greek cross has cells 6.5 x 5.5 mm, so this is approximately the correct shape to make a square, but the resulting square is not drawn.

6.AZ. BALL PYRAMID PUZZLES

This section is largely based on Gordon's Notes, cf below. See also 6.AP.2 for dissections of a tetrahedron in general. This section is now expanding to consider all polysphere puzzles.

See also S&B, p. 42, which mentions Hein and some other versions.

Piet Hein. Pyramystery. Made by Skjøde of Skjern, Denmark, 1970. With leaflet saying it was "recently invented by Piet Hein.... Responding to numerous requests, the inventor has therefore obliged the many admirers of the puzzle by also inventing its history". He then gives a story about Cheops. Peter Hajek and Jerry Slocum have different examples!!

Hajek's example has four planar rectangular pieces: 1 x 4, 2 x 3, 3 x 2, 4 x 1 rectangles. It is the same as Tut's Tomb -- see below. It has a 4pp English leaflet marked © Copyright Piet Hein 1970.

Slocum's example has 6 planar pieces: 4 3-spheres and 2 4-spheres. The leaflet is 34pp (?? -- Slocum only sent me part of it) with 3pp of instructions in each of 9 languages and then 6pp of diagrams of planar and 3-D problems. It is marked © 1970 Aspila, so perhaps this is a later development from the above?? The story part of the text is very similar to the above, but slightly longer. The pieces make an order 4 tetrahedron or two order 3 tetrahedra or two order 4 triangles and one can also divide them into two groups of three pieces such that one group makes an order 3 tetrahedron, but the other does not.

Advertising leaflet for Pyramystery from Piet Hein International Information Center, ©1976, describes the puzzle as having six pieces.

Mag-Nif Inc. Tut's Tomb. c1972. Same as the first Pyramystery.

Akira Kuwagaki & Sadao Takenaka. US Patent 3,837,652 -- Solid Puzzle. Filed: 1 May 1973; patented: 24 Sep 1974. 2pp + 4pp diagrams. Four planar 3-spheres and a 2-sphere to make a square pyramid of edge 3. 11 planar 4-spheres to make an octahedron shape of edge 4. Cites a 1936 Danish patent -- Hein ??NYS

Len Gordon. Perplexing Pyramid. 1974. Makes a edge 4 tetrahedron with 6 planar right-angled pieces: domino; straight and L trominoes; I, L, Y tetrominoes.

Patrick A. Roberts. US Patent 3,945,645 -- Tangential Spheres Geometric Puzzle. Filed: 28 Jun 1976; patented: 29 Nov 1977. 3pp + 3pp diagrams. 8 4-spheres and a 3-sphere to make a tetrahedron of side 5. 5 of the 4-spheres are non-planar.

Robert E. Kobres. US Patent 4,060,247 -- Geometric Puzzle. Filed: 28 Jun 1976; patented: 29 Nov 1977. 1p + 2pp diagrams. 5 pieces which make a 4 x 5 rhomboid or a tetrahedron. Two pieces have the form of a 2 x 3 rhombus; two pieces are 2-spheres and the last piece is the linear 4-sphere.

Len Gordon. Some Notes of Ball-Pyramid and Related Puzzles. Revised version, 10 Jul 1986, 14pp. Available from the author, 2737 N. Nordic Lane, Tucson, Arizona, 85716, USA.

Ming S. Cheng. US Patent 4,988,103 -- Geometric Puzzle of Spheres. Filed: 2 Oct 1989; patented: 29 Jan 1991. Front page, 5pp diagrams, 4pp text. A short version is given in Wiezorke, 1996, p. 64. 7 planar 5-spheres to make a tetrahedron; a hexagon with sides 3, 4, 4, 3, 4, 4; an equilateral triangle lacking one vertex.

Bernhard Wiezorke. Puzzling with Polyspheres. Published by the author (Lantzallee 18, D-4000 Düsseldorf 30, Germany), Mar 1990, 10pp.

Bernhard Wiezorke. Compendium of Polysphere Puzzles. (1995); Second Preliminary Edition, as above, Aug, 1996. 64pp, reproducing the short versions of the above patents. Despite Wiezorke's searches, nothing earlier than Hein's 1970 puzzles has come to light.

Torsten Sillke & Bernhard Wiezorke. Stacking identical polyspheres. Part 1: Tetrahedra. CFF 35 (Dec 1994) 11-17. Studies packing of tetrahedra with identical polysphere pieces, with complete results for tetrahedra of edges 4 - 8 and polyspheres of 3, 4, 5 spheres. Some of the impossibility results have only been done by computer, but others have been verified by a proof.

6.BA. CUTTING A CARD SO ONE CAN PASS THROUGH IT

Ozanam. 1725. 1725: vol. IV, prob. 34, pp. 436-437 & fig. 40, plate 12 (14).

Minguet. 1733. Pp. 115-117. (1755: 83-84; 1864: 112-113; not noted in 1822, but it's likely to be at p. 138.) Similar to Ozanam, 1725.

Alberti. 1747. Art. 34, p. 208-209 (110) & fig. 42, plate XI, opp. p. 210 (109). Copied from Ozanam, 1725.

Family Friend 3 (1850) 210 & 241. Practical puzzle -- No. XVII. Shows a 3 inch by 5 inch card. Repeated as Puzzle 15 -- The wonder puzzle in (1855) 339 with solution in (1856) 28.

Magician's Own Book. 1857. Prob. 10: The cardboard puzzle, pp. 269 & 294. Problem shows 3 inch by 5 inch card. Answer calls it "the cut card puzzle". c= Landells, Boy's Own Toy-Maker, 1858, p. 142. = Book of 500 Puzzles, 1859, pp. 83 & 108. = Boy's Own Conjuring Book, 1860, prob. 9, pp. 230 & 256.

Indoor & Outdoor. c1859. Part II, prob. 7: The cardboard puzzle, p. 129. No diagram, so the solution is a bit cryptic.

The Secret Out. 1859. How to Cut a Visiting Card for a Cat to Jump through it, p. 382.

Illustrated Boy's Own Treasury. 1860. No. 27, pp. 400 & 440. Identical to Magician's Own Book, but solution omits the sentence: "A laurel leaf may be treated in the same manner."

Magician's Own Book (UK version). 1871. To cut a card for one to jump through, p. 124. He adds: "The adventurer of old, who, inducing the aborigines to give him as much land as a bull's hide would cover, and made it into one strip by which acres were enclosed, had probably played at this game in his youth." See 6.AD.

Elliott. Within-Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 2: The cardboard puzzle, pp. 27 & 30-31. No diagram, so the solution is a bit cryptic.

Lemon. 1890. Cardboard puzzle, no. 140, pp. 23 & 102. = Sphinx, no. 467, pp. 65 & 113.

J. B. Bartlett. How to walk through a laurel leaf. The Boy's Own Paper 12 (No. 587) (12 Apr 1890) 440.

Hoffmann. 1893. Chap. X, no. 28: The cut playing-card, pp. 346 & 385-386 = Hoffmann-Hordern, p. 243.

Benson. 1904. The elastic cardboard puzzle, pp. 200-201.

Dudeney. Cutting-out paper puzzles. Cassell's Magazine ?? (Dec 1909) 187-191 & 233-235. With photo of Dudeney going through the card.

Collins. Book of Puzzles. 1927. Through a playing card, pp. 16-17.

6.BB. DOUBLING AN AREA WITHOUT CHANGING ITS HEIGHT OR

WIDTH

The area is usually a square, but other shapes are possible. If one views it as a reduction, one can reduce the area to any fraction of the original!

The Sociable. 1858. Prob. 41: The carpenter puzzled, pp. 298 & 316. 3 x 3 square of wood with holes in it forming a 4 x 4 array with the corner holes at the corners of the board. Claims one can cut 1/4 of the board out of the centre without including any holes. But this only gets 2/9 of the area -- double the central square. = Book of 500 Puzzles, 1859, prob. 41, pp. 16 & 34. = Secret Out, 1859, pp. 386-387.

Indoor & Outdoor. c1859. Part II: prob. 14: The carpenter puzzled, pp. 133-134. Almost identical with The Sociable.

Hanky Panky. 1872. P. 226 shows the same diagram as the solution in The Sociable, but there is no problem or text.

Lewis Carroll. Letter of 15 Mar 1873 to Helen Feilden. = Carroll-Collingwood, pp. 212-215 (Collins 154-155), without solution. Cf Carroll-Wakeling, prob. 28: The square window, pp. 36-37 & 72. Halve the area of a square window. Wakeling and Carroll-Gardner, p. 52, give the surname as Fielden, but it is Feilden in Carroll-Collingwood and in Cohen, who sketches her life. Wakeling writes that Feilden is correct.

Mittenzwey. 1880. Prob. 216-217, pp. 38-39 & 90; 1895?: 241-242, pp. 43 & 92; 1917: 241-242, pp. 39 & 88. Divide a rectangle or square into two pieces with the same height and width as the square. Solution is to draw a diagonal.

Lemon. 1890. A unique window, no. 444, pp. 58 & 114. The philosopher's puzzle, no. 660, pp. 82 & 121.

Don Lemon. Everybody's Scrap Book of Curious Facts. Saxon, London, 1890. P. 82 quotes an article from The New York World describing this as 'an excellent, if an old, puzzle'.

Hoffmann. 1893. Chap. IX, no. 28: A curious window, pp. 319 & 327 = Hoffmann-Hordern, pp. 211-212. Notes that either a diamond or a triangle in appropriate position can be so doubled.

Clark. Mental Nuts. 1897, no. 40. The building lot. "Have a lot 50 x 100. Want to build a house 50 x 100 and have the yard same size. How?" Solution shows 50 x 100 with a diagonal drawn.

Pearson. 1907. Part II, no. 79: At a duck pond, pp. 79 & 176. A square pond is to be doubled without disturbing the duckhouses at its corners.

Wehman. New Book of 200 Puzzles. 1908. The carpenter puzzled, p. 39. = The Sociable.

Will Blyth. Handkerchief Magic. C. Arthur Pearson, London, 1922. Doubling the allotment, pp. 23-24.

Hummerston. Fun, Mirth & Mystery. 1924. Some queer puzzles, Puzzle no. 76, part 6, pp. 164 & 183. Solution notes that a window in the shape of a diamond or a right triangle or an isosceles triangle can be doubled in area without changing its width or height.

King. Best 100. 1927.

No. 2, pp. 7-8 & 38. Same as Indoor & Outdoor, with the same error.

No. 4, pp. 8 & 39. Halve a square window. See Foulsham's.

Foulsham's Games and Puzzles Book. W. Foulsham, London, nd [c1930]. No. 2, pp. 5 & 10. Double a window without changing its height or width. (This is one of the few cases where the problem is not quite identical to King.)

M. Adams. Puzzle Book. 1939. Prob. B.117: Enlarging the allotment, pp. 86 & 110. Double a square allotment without disturbing the trees at the corners.

6.BC. HOFFMAN'S CUBE

This consists of 27 blocks, a x b x c, to make into a cube a+b+c on a side. It was first proposed by Dean Hoffman at a conference at Miami Univ. in 1978. See S&B, p. 43. The planar version, to use 4 rectangles a x b to make a square of side a + b is easy. These constructions are proofs of the inequality of the arithmetic and geometric means. Sometime in the early 1980s, I visited David Klarner in Binghamton and Dean Hoffman was present. David kindly made me a set of the blocks and a three-sided corner to hold them.

D. G. Hoffman. Packing problems and inequalities. In: The Mathematical Gardner, op. cit. in 6.AO, 1981. Pp. 212-225. Includes photos of Carl Klarner assembling the first set of the blocks. Asks if there are analogous packings in n dimensions.

Berlekamp, Conway & Guy. Winning Ways. 1982. Vol. 2, pp. 739-740 & 804-806. Shows all 21 inequivalent solutions.

6.BD. BRIDGE A MOAT WITH PLANKS

In the simplest case, one has a 3 x 3 moat with a 1 x 1 island in the centre. One wants to get to the island using two planks of length 1 or a bit less than 1. One plank is laid diagonally across the corner of the moat and the second plank is laid from the centre of the first plank to the corner of the island. If the width of the moat is D and the planks have length L, then the method works if 3L/2 > D(2, i.e. L > 2(2 D/3 = .94281.. D. One really should account for the width of the planks, but it is not clear just how much overlap is required for stability. Depew is the only example I have seen to use boards of different lengths. With more planks, one can reach across an arbitrarily large moat, but the number of planks needed gets very large. In this situation, the case of a circular moat and island is a bit easier to solve.

The Magician's Own Book (UK version) version is quite different and quite erroneous.

Magician's Own Book (UK version). 1871. The puzzle bridge, p. 123. Stream 15 or 16 feet across, but none of the available planks is more than 6 feet long. He claims that one can use a four plank version of the three knives make a support problem (section 11.N) to make a bridge. However the diagram of the solution clearly has the planks nearly as long as the width of the stream. In theory, one could build such a bridge with planks slightly longer than half the width of the stream, but to get good angles (e.g. everything crossing at right angles or nearly so), one needs planks somewhat longer than (2/2 of the width. E.g. for a width of 16 ft, 12 ft planks would be adequate.

Mittenzwey. 1880. Prob. 298, pp. 54 & 105; 1895?: 330, pp. 58 & 106; 1917: 330, pp. 52-53 & 101. 4 m gap bridged with two boards of length 3¾ m. He only gives a diagram. In fact this doesn't work because the ratio of lengths is 15/16 = .9375

Lucas. RM2. 1883. Le fossé du champ carré. Bridge the gap with two planks whose length is exactly 1. Notes this works because 3/2 > (2.

Hoffmann. 1893. Chap. VII, no. 9, pp. 289 & 295. Matchstick version = Hoffmann-Hordern, p. 193.

Benson. 1904. The moat puzzle, p. 246. Same as Hoffmann, but the second plank is shown under the first!!

Dudeney. CP. 1907. No. 54: Bridging the ditch, pp. 83-85 & 204. Eight 9' planks to cross a 10' ditch where it makes a right angle.

Pearson. 1907. Part I, no. 34: Across the moat, pp. 122 & 186.

Blyth. Match-Stick Magic. 1921. Boy Scouts' bridge, p. 21. Ordinary version done with matchsticks.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Boy Scouts' bridge, pp. 68-69. As in Blyth.

Depew. Cokesbury Game Book. 1939. Crossing the moat, pp. 225-226. Square moat 20 feet wide to be crossed with boards of width 18 and 15 ft. In fact this doesn't work -- one needs L1 + L2/2 > D(2.

"Zodiastar". Fun with Matches and Matchboxes. Op. cit. in 4.B.3. Late 1940s? The bridge, pp. 66-67 & 83. Matchstick version of the square moat & square island problem.

F. D. Burgoyne. Note 3106: An n plank problem. MG 48 (No. 366) (Dec 1964) 434-435. The island is a point in the centre of a 2 x 2 lake. Given n planks of length s, can you get to the island? He denotes the minimal length as s(n) and computes s(1) = 1, s(2) = 2 (2/3, s(3) = .882858... and says s(() = (2/2, [but I believe it is 0, i.e. one can get across an arbitrarily large moat with a fixed length of plank].

Jonathan Always. Puzzling You Again. Tandem, London, 1969.

Prob. 10: A damsel in distress, pp. 15 & 70-71. Use two planks of length L to reach a point in the centre of a circular moat of radius R. He finds one needs L2 ( 4R2/5.

Prob. 11: Perseus to the rescue again, pp. 15-16 & 71-72. Same with five planks. The solution uses only four and needs L2 ( 2R2/3.

C. V. G.[?] Howe? Mathematical Pie 75 (Summer 1975) 590 & 76 (Autumn 1975) 603. How big a square hole can be covered with planks of unit length? Answer says there is no limit, but the height of the pile increases with the side of the square.

Highlights for Children (Columbus, Ohio). Hidden Picture Favorites and Other Fun. 1981. Brain Buster 4, pp. 12 & 32: Plink, plank, kerplunk? Two children arrive at a straight(!) stream 4m wide with two planks 3m long. Solution: extend one plank about 1¼ m over the stream and one child stands on the land end. The second child carries the other plank over the stream and extends it to the other side and crosses. He then pulls the plank so it extends about 1¼ m over the stream. The first child now extends her plank out to rest on the second plank and crosses, pulling up her plank and taking it with her. I theory this technique will work if L > ⅔D, but one needs some overlap space, and the children may not have the same weight.

Richard I. Hess. Puzzles from Around the World. The author, 1997. (This is a collection of 117 puzzles which he published in Logigram, the newsletter of Logicon, in 1984-1994, drawn from many sources. With solutions.) Prob. 64. Usual problem with D = 10, but he says the board have width W = 1 and so one use the diagonal of the board in place of L. In my introduction, we saw that the standard version leads to L2 ( 8D2/9, so Hess's version leads to L2 ( 8D2/9 - W2 and we can get across with slightly shorter planks, but we have to tread very carefully!

6.BE. REVERSE A TRIANGULAR ARRAY OF TEN CIRCLES

One has a triangle of ten coins with four on an edge. Reverse its direction by moving only three coins. New section -- I'm surprised not to have seen older examples.

Sid G. Hedges. More Indoor and Community Games. Methuen, London, 1937. The triangle trick, p. 54. Uses peas, buttons or nuts.

M. Adams. Puzzle Book. 1939. Prob. B.34: Pitching camp, pp. 66 & 103. Array of tents.

Evelyn August. The Black-Out Book. Op. cit. in 5.X.1. 1939. The General inspects the balloons, pp. 106 & 214. Array of 10 barrage balloons.

Leopold. At Ease! 1943. 1, 2, 3 -- shift!, pp. 20 & 198. Thanks to Heinrich Hemme for the lead to this.

Joseph Leeming. Games with Playing Cards Plus Tricks and Stunts. Franklin Watts, 1949. ??NYS -- but two abridged versions have appeared.

Games and Fun with Playing Cards. Dover, NY, 1980. This contains everything except the section on bridge.

Tricks and Stunts with Playing Cards Plus Games of Solitaire. Gramercy Publishing, NY, nd [1960s?]. This includes all the tricks, stunts, puzzles and solitaire games.

25 Puzzles with Cards, 8th puzzle. Tricky triangle. Dover: pp. 154-155 & 172. Gramercy: pp. 45-46 & 65. Both have fig. 25 & 42.

Young World. c1960. P. 7: fifteen coin problem. Reverse a triangle with five on a side by moving five coins.

Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. Birds in flight, p. 34. Says this problem is described by Gardner, but gives no specific source.

Maxey Brooke. (Fun for the Money, Scribner's, 1963); reprinted as: Coin Games and Puzzles, Dover, 1973. Prob. 4: Bottoms up, pp. 15 & 75. On p. 6, he acknowledges Leopold as his source. Thanks to Heinrich Hemme for this reference.

D. B. Eperson. Triangular (old) pennies. MG 54 (No. 387) (Feb 1970) 48-49. The number of pennies which must be moved to reverse a triangle with n on a side is [T(n)/3], where T(n) is the n-th triangular number, which is the number in the array.

James Bidwell. The ten-coin triangle. MTg 54 (1971) 21-22. How many coins must be moved to reverse the triangle with n on an edge? His students find the same value as Eperson, but they weren't sure they had proved it.

Putnam. Puzzle Fun. 1978. No. 25: Triangular reverse, pp. 6 & 31. Usual 10 coin triangle.

6.BF. PYTHAGOREAN RECREATIONS

6.L might be considered as part of this section. There are some examples of problems with ladders which look like crossed ladders, but are simple Pythagorean problems.

See also 6.AS.2 for dissection proofs of the theorem of Pythagoras. I will include here only some interesting ancient examples. See Elisha Scott Loomis; The Pythagorean Proposition; 2nd ed., NCTM, 1940, for many proofs.

Aryabhata I, v. 17, states the Theorem of Pythagoras and the related theorem that if ABC is a diameter of a circle and LBM is a chord perpendicular to it, then LB2 = AB x BC; Bhaskara I's commentary applies the latter in several forms where modern algebra would make it more natural to use the former. Brahmagupta, v. 41, states LM2 = AB x BC.

I had overlooked the examples in Mahavira -- thanks to Yvonne Dold for pointing them out.

Fibonacci. 1202. Pp. 397-398 (S: 543-544) looks like a crossed ladders problem but is a simple right triangle problem.

Vyse. Tutor's Guide. 1771?

Prob. 9, 1793: p. 178, 1799: p. 189 & Key p. 224. A ladder 40 long in a roadway can reach 33 up one side and, from the same point, can reach 21 up the other side. This is actually a simple right triangle problem. There is a misprint of 9 for 6 in the answer.

Prob. 17 (in verse), 1793: 179, 1799: p. 190 & Key p. 228. A variation of the Broken Bamboo problem, cf below, with D = 30, H - X = 63, which is a simple right triangle problem.

Hutton. A Course of Mathematics. 1798?

Prob. VIII, 1833: 430; 1857: 508. = Vyse, prob. 19.

Prob. IX, 1833: 430; 1857: 508. = Vyse, prob. 17 with D = 15, H - X = 39.

6.BF.1 THE BROKEN BAMBOO

A bamboo (or tree) of height H breaks at height X from the ground so that the broken part reaches from the break to the ground at distance D from the foot of the bamboo. In fact the quadratic terms drop out of the solution, leaving a linear problem. This may be of Babylonian origin?? The hawk and rat problems of 6.BF.3 are geometrically the same problem viewed sideways.

In all cases below, H and D are given and X is sought, so I will denote the problem by (H, D).

See Tropfke, p. 620.

Chiu Chang Suan Ching (Jiu Zhang Suan Shu). c-150? Chap. IX, prob. 13, p. 96. [English in Mikami, p. 23 and in Swetz & Kao, pp. 44-45, and in HM 5 (1978) 260.] (10, 3).

Bhaskara I. 629. Commentary to Aryabhata, chap. II, v. 17, part 2. Sanskrit is on pp. 97-103; English version of the examples is on pp. 296-300. The material of interest is examples 4 and 5. In the set-up described under 6.BF.3, the bamboo is BOC which breaks at O and the point C reaches the ground at L.

Ex. 4: (18, 6). Shukla notes this is used by Chaturveda.

Ex. 5: (16, 8).

Chaturveda. 860. Commentary to the Brahma-sphuta-siddhanta, chap. XII, section IV, v. 41, example 2. In Colebrooke, p. 309. Bamboo: (18, 6).

Mahavira. 850. Chap. VII, v. 190-197, pp. 246-248.

v. 191. (25, 5), but the answer has H - X rather than X.

v. 192. (49, 21), but the answer has H - X rather than X.

v. 193. (50, 20), but with the problem reflected so the known leg is vertical rather than horizontal.

v. 196. This modifies the problem by imagining two trees of heights H and h, separated by D. The first, taller, tree breaks at height X from the ground and leans over so its top reaches the top of the other tree. If we subtract h from X and H, then │X - h│ is the solution of the problem (H - h, D). Because the terms are squared, it doesn't matter whether X is bigger or smaller than h. He does the case H, h, D = 23, 5, 12.

Bhaskara II. Lilavati. 1150. Chap. VI, v. 147-148. In Colebrooke, pp. 64-65. (32, 16).

Bhaskara II. Bijaganita. 1150. Chap. IV, v. 124. In Colebrooke, pp. 203-204. Same as Lilavati.

Needham, p. 28, is a nice Chinese illustration from 1261.

Gherardi. Libro di ragioni. 1328. Pp. 75-76: Regolla di mesura. (40, 14).

Pseudo-dell'Abbaco. c1440. No. 166, p. 138 with B&W reproduction on p. 139. Tree by stream. (60, 30). I have a colour slide of this.

Muscarello. 1478. F. 96v, pp. 224-225. Tree by a stream. (40, 30).

Calandri. Aritmetica. c1485. Ff. 87v-88r, pp. 175-176. Tree by a stream. (60, 30). = Pseudo-dell'Abbaco.

Calandri. Arimethrica. 1491. F. 98r. Tree by a river. (50, 30). Nice woodcut picture. Reproduced in Rara, 48.

Pacioli. Summa. 1494. Part II, f. 55r, prob. 31. (30, 10). Seems to say this very beautiful and subtle invention is due to Maestro Gratia.

Clark. Mental Nuts. 1897, no. 78. The tree and the storm. (100, 30). [I have included this as this problem is not so common in the 19C and 20C as in earlier times.]

N. L. Maiti. Notes on the broken bamboo problem. Gaņita-Bhāratī [NOTE: ņ denotes an n with an underdot] (Bull. Ind. Soc. Hist. Math.) 16 (1994) 25-36 -- ??NYS -- abstracted in BSHM Newsletter 29 (Summer 1995) 41, o/o. Says the problem is not in Brahmagupta, though this has been regularly asserted since Biot made an error in 1839 (probably a confusion with Chaturveda -- see above). He finds eight appearances in Indian works, from Bhaskara I (629) to Raghunath-raja (1597).

6.BF.2. SLIDING SPEAR = LEANING REED

A spear (or ladder) of length H stands against a wall. Its base moves out B from the wall, causing the top to slide down D. Hence B2 + (H - D)2 = H2.

The leaning reed has height H. It reaches D out of the water when it is straight up. When it leans over, it is just submerged when it is B away from its upright position. This is identical to the sliding spear turned upside down.

In these problems, two of H, B, D are given and one wants the remaining value. I will denote them, by e.g. H, D = 30, 6.

See Tropfke, p. 619 & 621.

BM 85196. Late Old Babylonian tablet in the British Museum, c-1800. Transcribed, translated and commented on by O. Neugebauer; Mathematische Keilschrift-texte II; Springer, Berlin, 1935, pp. 43++. Prob. 9 -- translation on pp. 47-48, commentary on p. 53. Quoted in B. L. van der Waerden; Science Awakening; OUP, 1961, p. 76. See also: J. Friberg; HM 8 (1981) 307-308. Sliding beam(?) with H, D = 30, 6 and with H, B = 30, 18.

BM 34568. Seleucid period tablet in the British Museum, c-300. Transcribed, translated and commented on by O. Neugebauer; Mathematische Keilschrift-texte III, Springer, Berlin, 1937, pp. 14-22 & plate 1. Prob. 12 -- translation on p. 18, commentary on p. 22. Quoted in van der Waerden, pp. 76-77. See also: J. Friberg, HM 8 (1981) 307-308. Sliding reed or cane, B, D = 9, 3.

Papyri Cairo J. E. 89127-30 & 89137-43. c-260. Shown and translated in: Richard A. Parker; Demotic Mathematical Papyri; Brown Univ. Press, Providence, 1972; pp. 1, 3-4, 35-40 & Plates 9-10.

Prob. 24-26: H, B = 10, 6; 14½, 10; 10, 8.

Prob. 27-29: H, D = 10, 2; 14½, 4; 10, 4.

Prob. 30-31: B, D = 6, 2; 10, 4.

Chiu Chang Suan Ching (Jiu Zhang Suan Shu). c-150? Chap. IX.

Prob. 6, p. 92. [English in Mikami, p. 22.] Leaning reed, B, D = 5, 1.

Prob. 7, p. 93. Version with a rope hanging and then stretched giving B, D = 8, 3.

Prob. 8, p. 93. [English in Mikami, p. 22 and in Swetz & Kao, pp. 30-32 and in HM 4 (1977) 274.] Ladder, but with vertical and horizontal reversed. B, D = 10, 1. Mikami misprints the answer as 55 rather than 50.5.

Bhaskara I. 629. Commentary to Aryabhata, chap. II, v. 17, part 2. Sanskrit is on pp. 97-103, with reproductions of original diagrams on pp. 101-102; English version of the examples is on pp. 296-300. The material of interest is examples 6 and 7. In the setup of 6.BF.3, the lotus is OBA and LBM is the water level.

Ex. 6: B, D = 24, 8. Shukla notes this is used by Chaturveda.

Ex. 7: B, D = 48, 6.

Chaturveda. 860. Commentary to the Brahma-sphuta-siddhanta, chap. XII, section IV, v. 41, example 3. In Colebrooke, pp. 309-310. Leaning lotus: B, D = 24, 8.

Tabari. Miftāh al-mu‘āmalāt. c1075. P. 124, no. 42. ??NYS - cited by Tropfke 621.

Bhaskara II. Lilavati. 1150. Chap. VI, v. 152-153. In Colebrooke, pp. 66. Leaning lotus: B, D = 2, ½.

Bhaskara II. Bijaganita. 1150. Chap. IV, v. 125. In Colebrooke, p. 204. Same as Lilavati.

Fibonacci. 1202. P. 397 (S: 543). Sliding ladder: H, B = 20, 12.

Leonardo Fibonacci. La Practica di Geometria. Volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano. Dal Codice 2186 della Biblioteca Riccardiana di Firenze, 1448. Ed. by Gino Arrighi, Domus Galilaeana, Pisa, 1966. P. 37 and fig. 18. Same as Fibonacci 1202.

Zhu Shijie. Siyuan Yujian (Precious Mirror of the Four Elements). 1303. ??NYS -- English given in Li & Du, p. 179. Questions in verse, no. 1. Two reeds, 14 apart, which reach 2½ and 1 out of the water. When they lean together, they just touch at the water surface. The water is assumed to have the same depth at both reeds, which reduces the problem from two variables to one variable.

Gherardi?. Liber habaci. c1310. Pp. 139-140. H, B = 20, 12

Gherardi. Libro di ragioni. 1328. Pp. 77-78. Ship's mast with H, D = 131, 4. B&W picture on p. 77, from f. 46v.

Columbia Algorism. c1350.

No. 135, pp. 138-139. Sliding ladder: H, B = 10, 6.

No. 140, pp. 149-150. Leaning tree: H, B = 20, 10.

(I have colour slides of the illustrations to these problems.)

Pseudo-dell'Abbaco. c1440. No. 167, pp. 138-140, with B&W picture on p. 139. Sliding spear: H, D = 30, 4. I have a colour slide of this.

della Francesca. Trattato. c1480. F. 44r (108). B, D = 6, 2. English in Jayawardene.

Pacioli. Summa. 1494. Part II, ff. 54v-55v.

Prob. 25 (misprinted 52). B, D = 6, 2. = della Francesca.

Prob. 26. H, H - D = 10, 8.

Prob. 27. H, D = 10, 4.

Prob. 28. H, D = 10, B/3.

Prob. 29. D + H = 12, DH = 12.

Prob. 30. H = D + 4, DH = 12.

Prob. 41. Tree of height 40 with a rope of length 50 tied to the top which reaches to the ground at a point 30 away. Length 10 of the rope is pulled, causing the tree to lean. How high is the top of the tree now? We now have a triangle of sides 40, 40, 30 and want the altitude to the side of length 30.

PART II.

F. 68r, prob. 98. Tree of height 30 has a rope of length 50 tied to the top, so it reaches the ground 40 away. How much rope has to be pulled to move the top of the tree to being 8 away from the vertical? He neglects to give the value 50 in the problem statement and the ground distances of 8 and 32 are interchanged in the diagram.

van Etten. 1624. Prob. 89 (86), part V (4), p. 135 (214). Sliding ladder: H, B = 10, 6.

Ozanam. 1694. Prob. 42 & fig. 48, plate 10, 1696: 123-124; 1708: 129; 1725: 320-321 & plate 10 (11). Ladder 25 long with foot 7 from wall. Foot is pulled out 8 more -- how much does the top come down?

Vyse. Tutor's Guide. 1771? Prob. 16 (in verse), 1793: p. 179, 1799: p. 190 & Key p. 228. H, B = 100, 10.

Mittenzwey. 1880. Prob. 294, pp. 53-54 & 104; 1895?: 324, pp. 57 & 106; 1917: 324, pp. 52 & 100. Leaning reed, B, D = 5, 1. [I have included this and the next entry as this problem is not so common in the 19C and 20C as in earlier times.]

Clark. Mental Nuts. 1904, no. 69; 1916, no. 95. The boatman's puzzle. Leaning pole, B, D = 12, 6. Find H - D.

6.BF.3. WELL BETWEEN TWO TOWERS

The towers have heights A, B and are D apart. A well or fountain is between them and equidistant from the tops of the towers. I denote this by (A, B, D). Vogel, in his DSB article on Fibonacci, says the problem is Indian, and Dold pointed me to Mahavira. Pseudo-dell'Abbaco introduces the question of a sliding weight or pulley -- see Pseudo-dell'Abbaco, Muscarello, Ozanam-Montucla, Tate, Palmaccio, Singmaster.

I have just found that Bhaskara I gives several unusual variations on this.

See Tropfke, p. 622. See also 10.U.

INDEX of A, B, D problems, with A ( B.

0 4 8 Chaturveda

0 9 27 Bhaskara II

0 12 24 Bhaskara I

0 18 81 Bhaskara I

5 6 12 Bhaskara I

10 10 12 Bhaskara I

13 15 14 Mahavira

18 22 20 Mahavira

20 24 22 Mahavira

20 30 30 Gherardi?

20 30 50 Perelman

30 40 50 Fibonacci, Muscarello, Cardan

30 50 100 Bartoli

30 70 100 Tate

40 50 30 Muscarello

40 50 70 Pseudo-dell'Abbaco

40 50 100 della Francesca

40 60 50 Lucca 1754

60 80 100 Calandri c1485

70 100 150 Columbia Algorism, Pacioli

80 90 100 Calandri 1491

Bhaskara I. 629. Commentary to Aryabhata, chap. II, v. 17, part 2. Sanskrit is on pp. 97-103; English version of the examples is on pp. 296-300. The material of interest is examples 2 and 3.

These are 'hawk and rat problems'. A hawk is sitting on a wall of height A and a rat is distance D from the base of the wall. The rat tries to get to its hole, in the wall directly under the hawk. The hawk swoops, at the same speed as the rat runs, and catches the rat when it hits the ground. Hence this is the same as our two tower problem, but with B = 0, so I will denote this version by (A, 0, D). Bhaskara I attributes this type of problem to unspecified previous writers. Shukla adds that later writers have it, including Chaturveda and Bhaskara II, qqv.

Ex. 2: (12, 0, 24).

Ex. 3: (18, 0, 81). Bhaskara I explains the solution in detail and Shukla gives an English precis of it. Let ABOC be the horizontal diameter of a circle and let LBM be a vertical chord. LB is our pole, with the hawk at L, and the rat is at C and wants to get to B. The point of capture is O, because LO = OC. From LB2 = AB x BC, we can determine AB and hence the other values.

Looking at Chaturveda (below), I now see that turning this sideways gives the same diagram as the broken bamboo problem -- the tree was BC and breaks at O to touch the ground at L. So the broken bamboo problem (H, D) is the same as the two towers or hawk and rat problem (D, 0, H).

Bhaskara I. 629. Ibid. Examples 8 and 9 are 'crane and fish problems'. A fish is at the NE corner of a rectangular pool and a crane is at the NW corner and they move at the same speeds. The fish swims obliquely to the south side, but the crane has to walk along the edge of the pool. The fish unfortunately gets to the side just as the crane reaches the same point and gets eaten. This again like our two tower problem, but with one pigeon unable to fly, so it has to walk down the tower and across. Because the pool is rectangular, the two values A and B are equal.

Ex. 8: (6, 6, 12).

Ex. 9: (10, 10, 12). The meeting point is 3 3/11 from the SW corner.

Chaturveda. 860. Commentary to the Brahma-sphuta-siddhanta, chap. XII, section IV, v. 41, example 4. In Colebrooke, p. 310. Cat and rat, where the cat behaves like the hawk of Bhaskara I: (4, 0, 8).

Mahavira. 850. Chap. VII, v. 201-208, pp. 249-251.

He gives several problems, but he usually also asks for the equal distance from the top

of each tower to the fountain.

v. 204. Two pillars, with a rope between them which touches the ground but with equal lengths to the tops. (13, 15, 14).

v. 206. Two hills with mendicants who are able to fly along the hypotenuses. (22, 18, 20)

v. 208. Same context. (20, 24, 22).

Bhaskara II. Lilavati. 1150. Chap. VI, v. 149-150. In Colebrooke, pp. 65-66. Peacock and snake version of the hawk and rat problem: (9, 0, 27).

Fibonacci. 1202. De duabus avibis [On two birds], pp. 331-332 (S: 462-463). (40, 30, 50). He does the same problem differently on pp. 398-399 (S: 544-545).

Gherardi?. Liber habaci. c1310. P. 139. (20, 30, 30).

Lucca 1754. c1330. F. 54v, pp. 120-121. (60, 40, 50).

Columbia Algorism. c1350. No. 136, pp. 139-140. (70, 100, 150).

Bartoli. Memoriale. c1420. Prob. 10, f. 76r (= Sesiano 138-139 & 148-149, with reproduction of the relevant part of f. 76r on p. 139). (50, 30, 100).

Pseudo-dell'Abbaco. c1440.

Prob. 80, p. 72, with picture on p. 71. (50, 40, 70).

Prob. 158-159, pp. 129-133, with illustrations on pp. 130 & 132, deal with the related problem where a rope with a sliding weight hangs between two towers, and the diagram clearly shows the weight in the air, not reaching the ground, so that the resulting triangles are similar. [I found it an interesting question to determine when the rope was long enough to reach the ground, and if not, how much above the ground the weight was -- see Muscarello, Ozanam-Montucla, Singmaster below.]

Prob. 158 has A, B, D = 40, 60, 40 and a rope of length L = 110, so the rope is more than long enough for the weight to reach the ground, but all he does is show that the two parts of the rope are 66 and 44, which is a bit dubious as there is slack in the rope. The diagram clearly shows the weight in the air. I have a colour slide of this.

Prob. 159 has A, B = 40, 60 with L = 120 such that the weight just touches the ground -- find the distances of the weight to the towers.

Prob. 160, p. 133. (40, 30, 50).

Muscarello. 1478.

Ff. 95r-95v, pp. 222-223. A, B, D = 50, 40, 30. Place a rope between the towers just long enough to touch the ground.

Ff. 95v-96r, pp. 223-224. A, B, D = 30, 20, 40. A rope of length L = 60 with a sliding weight is stretched between them -- where does the weight settle?

F. 99r, pp. 227-228. Fountain between towers for doves: (40, 30, 50).

della Francesca. Trattato. c1480. F. 22r (72). (40, 50, 100). English in Jayawardene.

Calandri. Aritmetica. c1485. Ff. 89r-89v, pp. 178-179. (60, 80, 100). (Tropfke, p. 599, shows the illustration in B&W.)

Calandri. Arimethrica. 1491. F. 100v. Well between two towers. (80, 90, 100). Nice double size woodcut picture.

Pacioli. Summa. 1494. Part II.

F. 59v, prob. 62. (70, 100, 150) = Columbia Algorism.

Ff. 59v-60r, prob. 63. A, B, D = 30, 40, 20 with rope of length 25 between the towers with a sliding lead weight on it. How high is the weight from the ground.

Ff. 61r-61v, prob. 66. Three towers of heights A, B, C = 125, 135, 125, with distances AB, AC, BC = 150, 130, 140. Find the point on the ground equidistant from the tops of the towers.

Cardan. Practica Arithmetice. 1539. Chap. 67.

Section 9, f. NN.vi.r (p. 197). (40, 30, 50).

Section 10, ff. NN.vi.r - NN.vii.v (pp. 197-198). Three towers of heights A, B, C = 40, 30, 70, with distances AB, AC, BC = 50, 60, 20. Find the point on the ground equidistant from the tops of the towers. Same idea as Pacioli, prob. 66.

Ozanam-Montucla. 1778. Vol. II, prob. 7 & fig. 5, plate 1. 1778: 11; 1803: 11-12; 1814: 9-10; 1840: 199. Rope between two towers with a pulley on it. Locate the equilibrium position. Uses reflection.

Carlile. Collection. 1793. Prob. XLV, p. 25. Find the position for a ladder on the ground between two towers so that leaning it each way reaches the top of each tower. (80, 91, 100). He simply states how to do the calculation, x = (D2 + A2 - B2)/2D for the distance from the base of tower B.

T. Tate. Algebra Made Easy. Chiefly Intended for the Use of Schools. New edition. Longman, Brown, Green, and Longman, London, 1848. P. 111.

No. 36. A, B, D = 30, 70, 100. Locate P such that the towers subtend the same angle, i.e. the two triangles are similar. Clearly P divides D as 30 to 70.

No. 37. Same data. Locate P so the distance to the tops is the same. This gives 70 to 30 easily because A + B = D.

No. 38. Same data. Locate P so the difference of the squares of the distances is 400. Answer is 68 from the base of the shorter tower.

Perelman. MCBF. 1937. Two birds by the riverside. Prob. 136, pp. 224-225. (30, 20, 50). "A problem of an Arabic mathematician of the 11th century."

Richard J. Palmaccio. Problems in Calculus and Analytic Geometry. J. Weston Walch, Portland, Maine, 1977. Maximum-Minimum Problems, No. 3, pp. 9 & 70-71. Cable supported pulley device over a factory. A, B, D = 24, 27, 108 with cable of length L = 117. Find the lowest point. He sets up the algebraic equations corresponding to the the two right triangles, assumes the distance from one post and the height above ground are implicit functions of the length of the cable from the pulley to the same post and differentiates both equations and sets equal to zero, but it takes half a page to get to the answer and he doesn't notice that the two triangles are similar at the lowest point.

David Singmaster, proposer; Dag Jonsson & Hayo Ahlburg, solvers. Problem 1748: [The two towers]. CM 18:5 (1992) 140 & 19:4 (1993) 125-127. Based on Pseudo-dell'Abbaco 158-159, but the solution by reflection was later discovered to be essentially Ozanam-Montucla.

David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics Teacher; Corwin Press, NY, 1996, pp. 273-286. Gives the reflection solution.

Yvonne Dold-Samplonius. Problem of the two towers. IN: Itinera mathematica; ed. by R. Franci, P. Pagli & L. Toti Rigatelli. Siena, 1996. Pp. 45-69. ??NYR. The earliest example she has found is Mahavira and it was an email from her in about 1995 that directed me to Mahavira.

6.BF.4. RAIL BUCKLING.

A railway rail of length L and ends fixed expands to length L + ΔL. Assuming the rail makes two hypotenuses, the middle rises by a height, H, satisfying H2 = {(L+ΔL)/2}2 - (L/2)2, hence H ( ((LΔL/2).

However, one might assume the rail buckled into an arc of a circle of radius r. If we let the angle of the arc be 2θ, then we have to solve rθ = (L + ΔL)/2; r sin θ = L/2. Taking sin θ ( θ - θ3/6, we get r2 ( (L + ΔL)3/ 24 ΔL. We have H = r (1 - cos θ) ( rθ2/2 and combining this with earlier equations leads to H ( ({3(L+ΔL)ΔL/8} which is about (3 / 2 = .866... as big as the estimate in the linear case.

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 149, prob. 12. L = 1 mile, ΔL = 1 ft or 2 ft -- text is not clear. "Answer: More than 54 ft." However, in the linear case, ΔL = 1 ft gives H = 51.38 ft and ΔL = 2 ft gives H = 72.67 ft, while the exact answers in the circular case are 44.50 ft and 62.95 ft.

Sullivan. Unusual. 1943. Prob. 15: Workin' on the railroad. L = 1 mile, ΔL = 2 ft. Answer: about 73 ft.

Robert Ripley. Mammoth Believe It or Not. Stanley Paul, London, 1956. If a railroad rail a mile long is raised 200 feet in the centre, how much closer would it bring the two ends? I.e. L = 1 mile, H = 200 ft. Answer is: "less than 6 inches". I am unable to figure out what Ripley intended.

Jonathan Always. More Puzzles to Puzzle You. Tandem, London, 1967. Gives the same question as Ripley with answer "approximately 15 feet". The exact answer is 15.1733.. feet or 15 feet 2.08 inches.

David Singmaster, submitter. Gleaning: Diverging lines. MG 69 (No. 448) (Jun 1985) 126. Quotes from Ripley and Always.

David Singmaster. Off the rails. The Weekend Telegraph (18 Feb 1989) xxiii & (25 Feb 1989) xxiii. Gives the Ripley and Always results and asks which is correct and whether the wrong one can be corrected -- cf Ripley above.

Phiip Cheung. Bowed rail problem. M500 161 (?? 1998) 9. ??NYS. Paul Terry, Martin S. Evans, Peter Fletcher, solvers and commentators. M500 163 (Aug 1998) 10-11. L = 1 mile, ΔL = 1 ft. Terry treats the bowed rail as circular and gets H = 44.49845 ft. Evans takes L = 1 nautical mile of 6000 ft and gets almost exactly H = 50 ft. Fletcher says it took 15 people to lift a 60ft length of rail, so if someone lifted the 1 mile rail to insert the extra foot, it would need about 1320 people to do the lifting.

6.BF.5. TRAVELLING ON SIDES OF A RIGHT TRIANGLE.

New section. See also the Mittenzwey example in 10.A.6.

Brahmagupta. Brahma-sphuta-siddhanta. 628. Chap. XII, sect. IV, v. 39. In Colebrooke, p. 308. Rule for the problem illustrated by Chaturveda.

Mahavira. 850. Chap. VII, v. 210-211, pp. 251-252. A slower traveller goes due east at rate v. A faster traveller goes at rate V and starts going north. After time t, he decides to meet the other traveller and turns so as to go directly to their meeting point. How long, T, do they travel? This gives us a right triangle with sides vT, Vt, V(T-t) leading to a quadratic in T whose constant term drops out, yielding T = 2t V2/(V2-v2). If we set r = v/V, then T = 2t/(1-r2), so we can determine T from t and r without actually knowing V or v. Indeed, if we let ρ = d/D, we get 2ρ = 1 - r2. v, V, t = 2, 3, 5.

Chaturveda. 860. Commentary to the Brahma-sphuta-siddhanta, chap. XII, section IV, v. 39. In Colebrooke, p. 308. Two ascetics are at the top of a (vertical!) mountain of height A. One, being a wizard, ascends a distance X and then flies directly to a town which is distance D from the foot of the mountain. The other walks straight down the mountain and to the town. They travel at the same speeds and reach the town at the same time. Example with A, D = 12, 48.

Bhaskara II. Lilavati. 1150. Chap. VI, v. 154-155. In Colebrooke, pp. 66-67. Similar to Chaturveda. Two apes on top of a tower of height A and they move to a point D away. A, D = 100, 200.

Bhaskara II. Bijaganita. 1150. Chap. IV, v. 126. In Colebrooke, pp. 204-205. Same as Lilavati.

6.BG. QUADRISECT A PAPER SQUARE WITH ONE CUT

This involves careful folding. One can also make mn rectangles with a single cut.

Mittenzwey. 1880. Prob. 193 & 194, pp. 36 & 89; 1895?: 218 & 219, pp. 41 & 91; 1917: 218 & 219, pp. 37 & 87. Make four squares. Make four isosceles right triangles.

Walter Gibson. Big Book of Magic for All Ages. Kaye & Ward, Kingswood, Surrey, 1982. Tick-tack-toe, pp. 68-69. Take a 4 x 4 array and mark alternate squares with Xs and Os. By careful folding and cutting, one produces eight free squares and a connected lattice of the other eight squares, with the free squares being either all the Os or all the Xs, depending on how the final part of the cut is made.

David Singmaster. Square cutting. Used in my puzzle columns.

Weekend Telegraph (18 & 25 Mar 1989) both p. xxiii.

G&P, No. 16 (Jul 1995) 26. (Publication ceased with No. 16.)

6.BH. MOIRÉ PATTERNS

I have not yet found any real history of this topic. One popular book says moirés were first made in 15C China. The OED has several entries for Moire and Moiré. It originally refers to a type of cloth and may be a French adaptation of the English word mohair -- Pepys refers to 'greene-waterd moyre' and this is the earliest citation. In the early 19C, the term began to be used for the 'watered' effect on cloth and metal. At some point, the term was transferred to the optical phenomena, but the OED does not have this meaning.

Journal of Science and Arts 5 (1818) 368. On the Moiré Metallique, or Fer blanc moiré. ??NYS -- cited in OED as their first citation for the noun use of the term.

John Badcock. Domestic Amusements, or Philosophical Recreations, ... Being a Sequel Volume to Philosophical Recreations, or Winter Amusements. T. Hughes, London, nd [Preface dated Feb 1823]. [BCB 16-17; OCB, pp. 180 & 196. Heyl 21. Toole Stott 78-80. Wallis 34 BAD. HPL [Badcock]. These give dates of 1823, 1825, 1828.] Pp. 139-141, no. 169: Moiré Metal, or Crystallised Tin & no. 170: Moiré Watering, by other Methods. "Quite new and splendid as this art is, .... M. Baget, a Frenchman, however, claims the honour of a discovery of this process, attributing the same to accident, ...." Cited in the OED as the first adjectival use of the term, though the previous entry seems to also have an adjectival usage.

Rational Recreations. 1824. Experiment 16, p. 15: Metallic watering, or, fer blanc moire. Says it is of Parisian invention and gives the method of applying sulphuric acid to tin.

Endless Amusement II. 1826? Pp. 24-25: Application of the moiré métallique to tin-foil. This deals with obtaining a moiré effect in tin-foil and is quite different than Badcock.

Young Man's Book. 1839. Pp. 312-314. Identical to Endless Amusement II.

Tom Tit, vol. 2. 1892. Le papier-canevas et les figures changeantes, pp. 137-138. Uses perforated card.

Hans Giger. Moirés. Comp. & Maths. with Appls. 12B:1/2 (1986) [= I. Hargittai, ed., Symmetry -- Unifying Human Understanding, as noted in 6.G.] 329-361. Giger says the technique of moiré fabrics derives from China and was first introduced into France in 1754 by the English manufacturer Badger (or Badjer). He also says Lord Rayleigh was the first to study the phenomenon, but gives no references.

6.BI. VENN DIAGRAMS FOR N SETS

New topic. I think I have seen more papers on this and Anthony Edwards has recently sent several more papers.

Martin Gardner. Logic diagrams. IN: Logic Machines and Diagrams; McGraw-Hill, NY, 1958, pp. 28-59. Slightly amended in the 2nd ed., Univ. of Chicago Press, 1982, and Harvester Press, Brighton, 1983, pp. 28-59. This surveys the history of all types of diagrams. John Venn [Symbolic Logic, 2nd ed., ??NYS] already gave Venn diagrams with 4 ovals and with 4 ovals and a disconnected set. Gardner describes various binary diagrams from 1881 onward, but generalised Venn diagrams seem to first occur in 1909 and then in 1938-1939, before a surge of interest from 1959. His references are much expanded in the 2nd ed. and he cites most of the following items.

John Venn. On the diagrammatic and mechanical representation of propositions and reasonings. London, Edinburgh and Dublin Philos. Mag. 10 (1880) 1-18. ??NYS -- cited by Henderson.

John Venn. Symbolic Logic. 2nd ed., Macmillan, 1894. ??NYS. Gardner, p. 105, reproduces a four ellipse diagram.

Lewis Carroll. Symbolic Logic, Part I. 4th ed., Macmillan, 1897; reprinted by Dover, 1958. Appendix -- Addressed to Teachers, sections 5 - 7: Euler's method of diagrams; Venn's method of diagrams; My method of diagrams, pp. 173-179. Describes Euler's simple approach and Venn's thorough approach. Reproduces Venn's four-ellipse diagram and his diagram for five sets using four ellipses and a disconnected region. He notes that Venn suggests using two five-set diagrams to deal with six sets and does not go further. He then describes his own method, which easily does up to eight sets. The diagram for four sets is the same as the common Karnaugh diagram used by electrical engineers. For more than four sets, the regions become disconnected with the cells of the four-set case being subdivided, using a simple diagonal, then his 2-set, 3-set and 4-set diagrams within each cell of the 4-set case. ?? -- is this in the 1st ed. -- ??NYS date??

Carroll-Gardner, p. 61, says this is in the first ed. of 1896.

William E. Hocking. Two extensions of the use of graphs in elementary logic. University of California Publications in Philosophy 2:2 (1909) 31(-??). ??NYS -- cited by Gardner who says Hocking uses nonconvex regions to get a solution for any n.

Edmund C. Berkeley. Boolean algebra and applications to insurance. Record of the Amer. Inst. of Actuaries 26:2 (Oct 1937) & 27:1 (Jun 1938). Reprinted as a booklet by Berkeley and Associates, 1952. ??NYS -- cited by Gardner. Uses nonconvex sets.

Trenchard More Jr. On the construction of Venn diagrams. J. Symbolic Logic 24 (Dec 1959) 303-304. ??NYS -- cited by Gardner. Uses nonconvex sets.

David W. Henderson. Venn diagrams for more than four classes. AMM 70:4 (1963) 424-426. Gives diagrams with 5 congruent irregular pentagons and with 5 congruent quadrilaterals. Considers problem of finding diagrams that have n-fold rotational symmetry and shows that then n must be a prime. Says he has found an example for n = 7, but doesn't know if examples can be found for all prime n.

Margaret E. Baron. A note on the historical development of logic diagrams: Leibniz, Euler and Venn. MG 53 (No. 384) (May 1969) 113-125. She notes Venn's solutions for n = 4, 5. She gives toothed rectangles for n = 5, 6.

K. M. Caldwell. Multiple-set Venn diagrams. MTg 53 (1970) 29. Does n = 4 with rectangles and then uses indents.

A. K. Austin, proposer; Heiko Harborth, solver. Problem E2314 -- Venn again. AMM 78:8 (Oct 1971) 904 & 79:8 (Oct 1972) 907-908. Shows that a diagram for 4 or more sets cannot be formed with translates of a convex set, using simple counting and Euler's formula. (The case of circles is in Yaglom & Yaglom I, pp. 103-104.) Editor gives a solution of G. A. Heuer with 4 congruent rectangles and more complex examples yielding disconnected subsets.

Lynette J. Bowles. Logic diagrams for up to n classes. MG 55 (No. 394) (Dec 1971) 370-373. Following Baron's note, she gives a binary tooth-like structure with examples for n = 7, 8.

Vern S. Poythress & Hugo S. Sun. A method to construct convex connected Venn diagrams for any finite number of sets. Pentagon (Spring 1972) 80-83. ??NYS -- cited by Gardner.

S. N. Collings. Further logic diagrams in various dimensions. MG 56 (No. 398) (Dec 1972) 309-310. Extends Bowles.

Branko Grünbaum. Venn diagrams and independent families of sets. MM 48 (1975) 12-22. Considers general case. Substantial survey of different ways to consider the problem. References to earlier literature. Shows one can use 5 identical ellipses, but one cannot use ellipses for n > 5.

B. Grünbaum. The construction of Venn diagrams. CMJ 15 (1984) 238-247. ??NYS.

Allen J. Schwenk. Venn diagram for five sets. MM 57 (1984) 297. Five ovals in a pentagram shape.

A. V. Boyd. Letter: Venn diagram of rectangles. MM 58 (1985) 251. Does n = 5 with rectangles.

W. O. J. Moser & J. Pach. Research Problems in Discrete Geometry. Op. cit. in 6.T. 1986. Prob. 27: On the extension of Venn diagrams. Considers whether a diagram for n classes can be extended to one for n+1 classes.

Mike Humphries. Note 71.11: Venn diagrams using convex sets. MG 71 (No. 455) (Mar 1987) 59. His fourth set is a square; fifth is an octagon.

J. Chris Fisher, E. L. Koh & Branko Grünbaum. Diagrams Venn and how. MM 61 (1988) 36-40. General case done with zig-zag lines. References.

Anthony W. F. Edwards. Venn diagrams for many sets. New Scientist 121 (No. 1646) (7 Jan 1989) 51-56. Discusses history, particularly Venn and Carroll, the four set version with ovals and Carroll's four set version where the third and fourth sets are rectangles. Edwards' diagram starts with a square divided into quadrants, then a circle. Fourth set is a two-tooth 'cogwheel' which he relates to a Hamiltonian circuit on the 3-cube. The fifth set is a four-tooth cogwheel, etc. The result is rather pretty. Edwards notes that the circle in the n set diagram meets the 2n-1 subsets of the n-1 sets other than that given by the circle, hence travelling around the circle gives a sequence of the subsets of n-1 objects and this is the Gray code (though he attributes this to Elisha Gray, the 19C American telephone engineer -- cf 7.M.3). The relationship with the n-cube leads to a partial connection between Edwards' diagram and the lattice of subsets of a set of n things.

New Scientist (11 Feb 1989) 77 has: Drawing the lines -- letters from Michael Lockwood -- describing a version with indented rectangles -- and from Anthony Edwards -- noting some errors in the article.

Ian Stewart. Visions mathématiques: Les dentelures de l'esprit. Pour la Science No. 138 (Apr 1989) 104-109. c= Cogwheels of the mind, IN: Ian Stewart; Another Fine Math You've Got Me Into; Freeman, NY, 1992, chap. 4, pp. 51-64. Exposits Edwards' work with a little more detail about the connection with the Gray code.

A. W. F. Edwards & C. A. B. Smith. New 3-set Venn diagram. Nature 339 (25 May 1989) 263. Notes connection with the family of cosine curves, y = 2-n cos 2nx on [0, π] and Gray codes and with the family of sine curves, y = 2-n sin 2nx on [0, 2π] and ordinary binary codes. Applying a similar phase shift to Edwards' diagram leads to diagrams where more than two set boundaries are allowed to meet at a point.

A. W. F. Edwards. Venn diagrams for many sets. Bull. Intern. Statistical Inst., 47th Session, Paris, 1989. Contrib. Papers, Book 1, pp. 311-312.

A. W. F. Edwards. To make a rotatable Edwards map of a Venn diagram. 4pp of instructions and cut-out figures. The author, Gonville and Caius College, Cambridge, CB2 1TA, 21 Feb 1991.

A. W. F. Edwards. Note 75.39: How to iron a hypercube. MG 75 (No. 474) (1991) 433-436. Discusses his diagram and its connection with the n-cube.

Anthony Edwards. Rotatable Venn diagrams. Mathematics Review 2:3 (Feb 1992) 19-21. +  Letter: Venn revisited. Ibid. 3:2 (Nov 1992) 29.

6.BJ. 3D DISSECTION PUZZLES

This will cover a number of cases which are not very mathematical. I will record just some early examples. See also 6.G (esp. 6.G.1), 6.N, 6.W (esp. 6.W.7), 6.AP for special cases. The predecessors of these puzzles seem to be the binomial and trinomial cubes showing (a+b)3 and (a+b+c)3, which I have placed in 6.G.1. Cube dissections with cuts at angles to the faces were common in the 19C Chinese puzzle chests, often in ivory. I have only just started to notice these. It is hard to distinguish items in this section from other burr puzzles, 6.W.7, and I have tried to avoid repetition, so one must also look at that section when looking at this section.

Catel. Kunst-Cabinet. 1790. Der Vexierwürfel, p. 11 & fig. 32 on plate II. Figure shows that there are some cuts at angles to the faces, so this is not an ordinary cube dissection, but is more like the 19C Chinese dissected cubes.

C. Baudenbecher. Sample book or catalogue from c1850s. Op. cit. in 6.W.7. One whole folio page shows about 20 types of wooden interlocking puzzles, including most of the types mentioned in this section and in 6.W.5 and 6.W.7. Until I get a picture, I can't be more specific.

Slocum. Compendium. Shows: Wonderful "Coffee Pot"; Magic "Apple"; Magic "Pear"; Extraordinary "Cube"; Magic "Tub" from Mr. Bland's Illustrated Catalogue of Extraordinary and Superior Conjuring Tricks, etc.; Joseph Bland, London, c1890. He shows further examples from 1915 onward.

Hoffmann. 1893. Chap. II, pp. 107-108 & 141-142 = Hoffmann-Hordern, pp. 106-107, with photo.

No. 37: The Fairy Tea-Table. Photo on p. 107 shows a German example, 1870-1895.

No. 38: The Mystery. Photo on p. 107 shows a German example, 1870-1895, with instructions.

Western Puzzle Works, 1926 Catalogue.

No. 5075. Unnamed -- Fairy Tea-Table.

Last page shows 20 Chinese Wood Block Puzzles, High Grade. These are unnamed, but the shapes include various burr-like objects, cube, spheres, egg, barrel, tankard, pear and apple.

P. M. Grundy. A three-dimensional jig-saw. Eureka 7 (Mar 1942) 8-10. Consider a 2 x 2 x 2 array of unit cubes. He suggests removing and/or adding lumps on the interior faces to make a jig-saw. He then considers lumps in the form of a isosceles right triangular prism with largest face being a unit square. He finds there are 25 such pieces, subject to the conditions that there are most two removals, and that when there are two, they must be parallel. He gives a graphical view of a 2 x 2 x 2 formed from such pieces which gives some necessary, but not sufficient, conditions for a set of eight such pieces to be able to make a cube. One solution is shown. [If one assumes there is just one removal and one addition, I find just four pieces, which form a 1 x 2 x 2 block. One could view this as a 3-dimensional matching puzzle, where the internal faces have to match both like a head to tail matching, but also with correct orientation. MacMahon thought of notching pieces as an alternative to colouring edges, but his pieces were two-dimensional.]

6.BK SUPERELLIPSE

New section.

Gardner. SA (Sep 1965) = Carnival, chap. 18. Describes how the problem arose in the design of Sergel's Square, Stockholm, in 1959. Addendum in Carnival gives the results given by Gridgeman, below. Also says road engineers used such curves with n = 2.2, called '2.2 ellipses', from the 1930s for bridge arches.

N. T. Gridgeman. Lamé ovals. MG 54 (No. 387) (Feb 1970) 31-37. Lamé (c1818) seems to be the first to consider (x/a)n + (y/b)n = 1. Hein's design in Stockholm uses a/b = 6/5 and n = 5/2. Gerald Robinson used a/b = 9/7 and n = 2.71828..., which he determined by a survey asking people which shape they liked most. Gridgeman studies curvature, area, perimeter, evolutes, etc.

6.BL. TAN-1 ⅓ + TAN-1 ½ = TAN-1 1, ETC.

This problem is usually presented with three squares in a row with lines drawn from one corner to the opposite corners of the squares. New section. Similar formulae occur in finding series for π. See 6.A and my Chronology of π.

L. Euler. Introductio in Analysin Infinitorum. Bousquet, Lausanne, 1748. Vol. 1, chap. VIII, esp. § 142 (??NYS). = Introduction to the Analysis of the Infinite; trans. by John D. Blanton; Springer, NY, 1988-1990; Book I, chap. VIII: On transcendental quantities which arise from the circle, pp. 101-115, esp. § 142, pp. 114-115. Developing series to calculate π, he considers angles a, b such that a + b = π/4, then examines the formula for tan (a + b) and says: "If we let tan a = ½, then tan b = ⅓ .... In this way we calculate ... π, with much more ease than ... before." Conway & Guy give some more details.

Carroll. ?? -- see Lowry (1972) and Conway & Guy (1996).

Størmer. 1896. See Conway & Guy.

Gardner. SA (Feb 1970) = Circus, chap. 11, prob. 3. Says he received the geometric problem from Lyber Katz who had been given it when he was in 4th grade in Moscow.

C. W. Trigg. A three-square geometry problem. JRM 4:2 (Apr 1971) 90-99. Quotes a letter from Katz, dating his 4th year as 1931-32. Trigg sketches 54 proofs of the result, some of which generalize.

H. V. Lowry. Note 3331: Formula for π/4. MG 56 (No. 397) (Oct 1972) 224-225. tan-1 1/a  = tan-1 1/b + tan-1 1/c implies a(b+c) = bc - 1, hence (b-a)(c-a)  =  a2 + 1, whence all integral solutions can be determined. Conway & Guy say this was known to Lewis Carroll.

J. R. Goggins & G. B. Gordon. Note 3346: Formula for π/4 (see Note 3331, Oct 1972). MG 57 (No. 400) (Jun 1973) 134. Goggins gets π/4 = Σn=1 tan-1 1/F2n+1, where Fn is the n-th Fibonacci number. [I think this formula was found by Lehmer some years before??] Gordon also mentions Eureka No. 35, p. 22, ??NYS, and finds recurrences giving tan-1 1/pn + tan-1 1/qn = tan-1 1/rn.

Douglas A. Quadling. Classroom note 304: The story of the three squares (continued). MG 58 (No. 405) (Oct 1974) 212-215. The problem was given in Classroom note 295 and many answers were received, including four proofs published by Roger North. Quadling cites Trigg and determines which proofs are new. Trigg writes that tan-1 1/F2n+2 + tan-1 1/F2n+1 = tan-1 1/F2n, which is the basis of Goggins' formula.

Alan Fearnehough. On formulas for π involving inverse tangent functions and Prob. 23.7. MS 23:3 (1990/91) 65-67 & 95. Gives four basic theorems about inverse tangents leading to many different formulae for π/4. The problem gives a series using inverse cotangents.

John H. Conway & Richard K. Guy. The Book of Numbers. Copernicus (Springer-Verlag), NY, 1996. Pp. 241-248 discusses relationships among values of tan-1 1/n which they denote as tn and call Gregory's numbers. Euler knew t = t2 + t3, t1 = 2 t3 + t7 and t1 = 5 t7 + 2 t18 - 2 t57 and used them to compute π to 20 places in an hour. They say Lewis Carroll noted that tn = tn+c + tn+d if and only if cd = n2 + 1. In 1896, Størmer related Gaussian integers to Gregory numbers and showed how to obtain a Gregory number as a sum of other Gregory numbers. From this it follows that the only two-term expressions for π/4 are t2 + t3, 2 t2 - 4 t7, 2 t3 + t7 and 4 t5 - t239. This is described in Conway & Guy, but they have a misprint of 8 for 18 at the bottom of p. 246.

6.BM. DISSECT CIRCLE INTO TWO HOLLOW OVALS

Consider a circle of radius 2. Cut it by two perpendicular diameters and by the circle of radius 1 about the centre. Two of the outer pieces (quarters of the annulus) and two of the inner pieces (quadrants) make an oval shape, with a hollow in the middle. The problem often refers to making two oval stools and the hollows are handholds! After the references below, the problem appears in many later books.

Jackson. Rational Amusement. 1821. Geometrical Puzzles, no. 9, pp. 25 & 84 & plate I, fig. 6. Mentions handholes. Solution is well drawn.

Endless Amusement II. 1826? Mentions handholes. Solution is well drawn.

Crambrook. 1843. P. 4, no. 6: A Circle to form two Ovals. Check??

Magician's Own Book. 1857. Prob. 36: The cabinet maker's puzzle, pp. 277 & 300. Solution is a bit crudely drawn. = Book of 500 Puzzles, 1859, pp. 91 & 114. = Boy's Own Conjuring Book, 1860, prob. 35, pp. 240 & 265.

Family Friend (Dec 1858) 359. Practical puzzle -- 4. I don't have the answer.

The Secret Out. 1859. The Oval Puzzle, pp. 380-381. Asks to 'produce two perfect ovals.' Solution is a bit crudely drawn, as in Magician's Own Book, but the text and numbering of pieces is different.

Illustrated Boy's Own Treasury. 1860. Prob. 34, pp. 401 & 441. Same crude solution as Magician's Own Book, but with different text, neglecting to state that the stools have handholes in their centres.

Magician's Own Book (UK version). 1871. On p. 282, in the middle of an unrelated problem, is the solution diagram, very poorly drawn -- the pieces of the oval stools are shown as having curved edges almost as though they were circular arcs. There is no associated text.

Hanky Panky. 1872. The oval puzzle, p. 123. Same crude solution as Magician's Own Book, but different text, mentioning handholes.

Mittenzwey. 1880. Prob. 256, pp. 46 & 97; 1895?: 285, pp. 50 & 99; 1917: 285, pp. 45 & 94. The stools are very poorly drawn, with distinct wiggles in what should be straight lines.

Hoffmann. 1893. Chap. II, no. 32: The cabinet maker's puzzle, pp. 104 & 137-138 = Hoffmann-Hordern, p. 102. Mentions hand holes. Well drawn solution.

Benson. 1904. The cabinet-maker's puzzle, p. 201. Mentions hand holes. Poor drawing.

6.BN. ROUND PEG IN SQUARE HOLE OR VICE VERSA

Wang Tao-K'un. How to get on. Late 16C. Excerpted and translated in: Herbert A. Giles; Gems of Chinese Literature; 2nd ed. (in two vols., Kelly & Walsh, 1923), in one vol., Dover, 1965, p. 226. "... like square handles which you would thrust into the round sockets ..."

Sydney Smith. Sketches of Moral Philosophy. Lecture IX. 1824. "If you choose to represent the various parts in life by holes upon a table, of different shapes, -- some circular, some triangular, some square, some oblong, -- and the persons acting these parts by bits of wood of similar shapes, we shall generally find that the triangular person has got into the square hole, the oblong into the triangular, and a square person has squeezed himself into the round hole. The officer and the office, the doer and the thing done, seldom fit so exactly that we can say they were almost made for each other." Quoted in: John Bartlett; Familiar Quotations; 9th ed., Macmillan, London, 1902, p. 461 (without specifying the Lecture or date). Irving Wallace; The Square Pegs; (Hutchinson, 1958); New English Library, 1968; p. 11, gives the above quote and says it was given in a lecture by Smith at the Royal Institution in 1824. Bartlett gives a footnote reference: The right man to fill the right place -- Layard: Speech, Jan. 15, 1855. It is not clear to me whether Layard quoted Smith or simply expressed the same idea in prosaic terms . Partially quoted, from 'we shall ...' in The Oxford Dictionary of Quotations; 2nd ed. revised, 1970, p. 505, item 24. Similarly quoted in some other dictionaries of quotations.

I have located other quotations from 1837, 1867 and 1901.

William A. Bagley. Paradox Pie. Vawser & Wiles, London, nd [BMC gives 1944]. No. 17: Misfits, p. 18. "Which is the worst misfit, a square peg in a round hole or a round peg in a square hole?" Shows the round peg fits better. He notes that square holes are hard to make.

David Singmaster. On round pegs in square holes and square pegs in round holes. MM 37 (1964) 335-337. Reinvents the problem and considers it in n dimensions. The round peg fits better for n < 9. John L. Kelley pointed out that there must be a dimension between 8 and 9 where the two fit equally well. Herman P. Robinson kindly calculated this dimension for me in 1979, getting 8.13795....

David Singmaster. Letter: The problem of square pegs and round holes. ILEA Contact [London] (12 Sep 1980) 12. The two dimensional problem appears as a SMILE card which was attacked as 'daft' in an earlier letter. Here I defend the problem and indicate some extensions -- e.g. a circle fits better in a regular n-gon than vice-versa for all n.

6.BO. BUTTERFLY PROBLEM

I have generally avoided classical geometry of this sort, but Bankoff's paper deserves inclusion.

Leon Bankoff. The metamorphosis of the butterfly problem. MM 60 (1987) 195-210. Includes historical survey of different proofs. The name first appears in the title of the solution of Problem E571, AMM 51 (1944) 91 (??NYS). The problem first occurs in The Gentlemen's Diary (1815) 39-40 (??NYS).

6.BP. EARLY MATCHSTICK PUZZLES

There are too many matchstick puzzles to try to catalogue. Some of them occur in other sections, e.g. 6.AO.1. Here I only include a few very early other examples. At first I thought these would date from mid to late 19C when matches first started to become available, but the earliest examples refer to slips of paper or wood. The earliest mention of matches is in 1858.

Rational Recreations. 1824. Exer. 23, p. 132. Double a sheep pen by adding just two hurdles. I have just realised this is a kind of matchstick puzzle and I suspect there are other early examples of this. It is a little different than most matchstick puzzles in that one is usually not given an initial pattern, but must figure it out. (A hurdle is a kind of panel woven from sticks or reeds used by shepherds to make temporary pens.)

Family Friend 2 (1850) 148 & 179. Practical Puzzle -- No. V. = Illustrated Boy's Own Treasury, 1860, Prob. 46, pp. 404 & 443. "Cut seventeen slips of paper or wood of equal lengths, and place them on a table, to form six squares, as in the diagram. ...."

Magician's Own Book. 1857. Prob. 20: Three square puzzle, pp. 273 & 296. (I had 87 & 110 ??) Almost identical to Family Friend, with a few changes in wording and a different drawing, e.g. "Cut seventeen slips of cardboard of equal lengths, and place them on a table to form six squares, as in the diagram. ...."

The Sociable. 1858. Prob. 2: The magic square, pp. 286 & 301. "With seventeen pieces of wood (lucifer matches will answer the purpose, but be careful to remove the combustible ends, and see that they are all of the same length) make the following figure: [a 2 x 3 array of squares]", then remove 5 matches to leave three squares.

Book of 500 Puzzles. 1859.

Prob. 2: The magic square, pp. 4 & 19. As in The Sociable.

Prob. 20: Three square puzzle, pp. 87 & 110. Identical to Magician's Own Book.

Boy's Own Conjuring Book. 1860. Three-square puzzle, pp. 235 & 259. Identical to Magician's Own Book.

Elliott. Within-Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 5: The three squares, pp. 28 & 31. Almost identical to Magician's Own Book, prob. 20, with a slightly different diagram.

Mittenzwey. 1880. Prob. 156-171, 202-212, 240, 242, pp. 32-33, 37-38, 44 & 83-85, 90, 94; 1895?: 179-196, 227-237, 269, 271, pp. 37-38, 41-42, 48 & 85-87, 92, 96; 1917: 179-196, 227-237, 269, 271, pp. 33-34, 38-39, 44 & 82-84, 88, 92. This is the first puzzle book to have lots of matchstick problems, though he doesn't yet use the name, calling them 'Hölzchen' (= sticks). Several problems occur elsewhere, e.g. in 6.AO.1. The last problem is the same as in Jackson.

Sophus Tromholt. Streichholzspiele. Otto Spamer, Leipzig, 1889; 5th ed., 1892; 14th ed., Leipzig, 1909; slightly revised and with a Preface by Rüdiger Thiele, Zentralantiquariat der DDR, Leipzig, 1986; Hugendubel, 1986. [Christopher 1017 is Spamer, 1890. C&B give 1890. There is an edition by Ullstein, Frankfurt, 1990.] This is the earliest book I know which is entirely devoted to matchstick puzzles.

Gaston Tissandier. Jeux et Jouets du jeune age Choix de récréations amusantes & instructives. Ill. by Albert Tissandier. G. Masson, Paris, nd [c1890]. P. 40, no. 4-5: Le problème des allumettes. Make five squares with nine matches. Solution has four small squares and one large one.

Clark. Mental Nuts. 1897, no. 56. The toothpicks. Use 12 toothpicks to make a 2 x 2 array of squares. Move three picks to form three squares.

H. D. Northrop. Popular Pastimes. 1901. No. 1: The magic square, pp. 65 & 71. = The Sociable.

6.BQ. COVERING A DISC WITH DISCS

The general problem is too complex to be considered recreational. Here I will mainly deal with the carnival version where one tries to cover a circular spot with five discs. In practice, this is usually rigged by stretching the cloth.

Eric H. Neville. On the solution of numerical functional equations, illustrated by an account of a popular puzzle and of its solution. Proc. London Math. Soc. (2) 14 (1915) 308-326. Obtains several possible configurations, but says "actual trial is sufficient to convince" that one is clearly the best, namely the elongated pentagon with 2-fold symmetry. This leads to four trigonometric equations in four unknown angles which theoretically could be solved, but are difficult to solve even numerically. He develops a modification of Newton's method and applies it to the problem, obtaining the maximal ratio of spot radius to disc radius as 1.64091. Described by Gardner and Singleton.

Ball. MRE. 10th ed., 1922. Pp. 253-255: The five disc problem. Sketches Neville's results.

Will Blyth. More Paper Magic. C. Arthur Pearson, London, 1923. Cover the spot, pp. 66-67. "This old "fun of the fair" game has been the means of drawing many pennies from the pockets of frequenters of fairs." Says the best approach is an elongated pentagon which has only 2-fold symmetry.

William Fitch Cheney Jr, proposer; editorial comment. Problem E14. AMM 39 (1932) 606 & 42 (1935) 622. Poses the problem. Editor says no solution of this, or its equivalent, prob. 3574, was received, but cites Neville.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Cover the spot, pp. 132-134. Uses discs of diameter 1 5/8 in to cover a circle of diameter 2 1/2 in. This is a ratio of 20/13 = 1.538..., which should be fairly easy to cover?? His first disc has its edge passing through the centre of the circle. His covering pattern has bilateral symmetry, though the order of placing the last two discs seems backward.

Walter B. Gibson. The Bunco Book. (1946); reprinted by Citadel Press (Lyle Stuart Inc.), Secaucus, New Jersey, 1986. Spotting the spot, pp. 24-25. Also repeated in summary form, with some extra observations in: Open season on chumps, pp. 97-106, esp. pp. 102-103. The circles have diameter 5" and the discs "are slightly more than three inches in diameter." He assumes the covering works exactly when the discs have five-fold symmetry -- which implies the discs are 3.090... inches in diameter -- but that the operator stretches the cloth so the spot is unsymmetric and the player can hardly ever cover it -- though it still can be done if one plays a disc to cover the bulge. "There is scarcely one chance in a hundred that the spectator will start correctly ...." On p. 103, he adds that "in these progressive times" the bulge can be made in any direction and that shills are often employed to show that it can be done, though it is still difficult and the operator generally ignores small uncovered bits in the shills' play in order to make the game seem easy.

Martin Gardner. SA (Apr 1959)?? c= 2nd Book, chap. 13. Describes the five disc version as Spot-the-Spot. Cites Neville.

Colin R. J. Singleton. Letter: A carnival game -- covering disks with smaller disks. JRM 24:3 (1992) 185-186. Responding to a comment in JRM 24:1, he points out that the optimum placing of five discs does not have pentagonal symmetry but only bilateral. Five discs of radius 1 can then cover a disc of radius 1.642.., rather than 1.618..., which occurs when there is pentagonal symmetry. He cites Gardner and E. H. Neville. His 1.642 arises because Gardner had truncated the reciprocal ratio to three places.

6.BR. WHAT IS A GENERAL TRIANGLE?

David & Geralda Singmaster, proposers; Norman Miller, solver. Problem E1705 -- Skewness of a triangle. AMM 71:6 (1964) 680 & 72:6 (1965) 669. Assume a ( b ( c. Define the skewness of the triangle as S = max {a/b, b/c, c/a} x min {a/b, b/c, c/a}. What triangles have maximum and minimum skewness? Minimum is S = 1 for any isosceles triangle. Maximum occurs for the degenerate triangle with sides 1, φ, 1+φ, where 1 + φ = φ2, so φ = (1 + (5)/2 is the golden mean.

Baruch Schwarz & Maxim Bruckheimer. Let ABC be any triangle. MTr 81 (Nov 1988) 640-642. Assume AB < AC < BC and (A < 90o. Drawing BC and putting A above it leads to a small curvilinear triangular region where A can be. Making A equidistant from the three boundaries leads to a triangle with sides proportional to (33, 7, 8 and with angles 44.5o, 58.5o, 77o. The sides are roughly in the proportion 6 : 7 : 8.

Gontran Ervynck. Drawing a 'general' triangle. Mathematics Review (Nov 1991). ??NYS -- cited by Anon., below. Notes that if we take an acute triangle with angles as different as possible, then we get the triangle with angles 45o, 60o, 75o.

Anon. [possibly the editor, Tom Butts]. What is a 'general' triangle? Mathematical Log 37:3 (Oct 1993) 1 & 6. Describes above two results and mentions Guy's article in 8.C. Gives an argument which would show the probability of an acute triangle is 0.

Anon. [probably the editor, Arthur Dodd]. A very scalene triangle. Plus 30 (Summer 1995) 18-19 & 23. (Content says this is repeated from a 1987 issue -- ??NYS.) Uses the same region as Schwarz & Bruckheimer, below. Looks for a point as far away from the boundaries as possible and takes the point which gives the angles 45o, 60o, 75o.

In 1995?, I experimented with variations on the definition of skewness given in the first item above, but have not gotten much. However, taking a = 1, we have 1 ( b ( c ( b + 1. Plotting this in the b, c plane gives us a narrow strip extending to infinity. For generality, it would seem that we want c = b + ½, but there is no other obvious condition to select a central point in this region. As fairly random points, I have looked at the case where c = b2, which gives b = (1 + (3)/2 = 1.366.., c = 1.866.. -- this triangle has angles about 31.47o, 45.50o, 103.03o -- and at the case where b = 3/2, which gives a triangle with sides proportional to 2, 3, 4 with angles about 28.96o, 46.57o, 104.46o. In Mar 1996, I realised that the portion of the strip corresponding to an acute triangle tends to 0 !!

I have now (Mar 1996) realised that the situation is not very symmetric. Taking c = 1, we have 0 ( a ( b ( 1 ( a + b and plotting this in the a, b plane gives us a bounded triangle with vertices at (0, 1), (½, ½), (1, 1). There are various possible central points of this triangle. The centroid is at (1/2, 5/6), giving a triangle with sides 1/2, 5/6, 1 which is similar to 3, 5, 6, with angles 29.93o, 56.25o, 93.82o. An alternative point in this region is the incentre, which is at (½, ½{-1 + 2(2}), giving a triangle similar to 1, -1 + 2(2, 2 with angles 29.85o, 65.53o, 84.62o. The probability of an acute triangle in this context is 2 - π/2 = .429.

6.BS. FORM SIX COINS INTO A HEXAGON

O O O O O

Transform O O O into O O in three moves.

O O

New section -- there must be older versions.

Young World. c1960. P. 13: Ringing the change. He starts with the mirror image of the first array.

Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. The ring of coins, p. 31. Says it is described by Gardner, but gives no details. Notes that if you show the trick to someone

O O O and then give the coins in the mirror image pattern shown at the left, he will

O O O not be able to do it.

Putnam. Puzzle Fun. 1978.

No. 12: Create a space, pp. 4 & 27. With four coins, create the pattern OO  

on the right. [Takes two moves from a rhombic starting pattern.] O O

No. 13: Create a space again, pp. 4 & 38. Standard hexagon problem.

6.BT. PLACING OBJECTS IN CONTACT

New section. The objects involved are usually common objects such as coins or cigarettes, etc. The standard recreation is to have them all touching one another. However, the more basic question of how many spheres can touch a sphere goes back to Kepler and perhaps the Greeks. Similar questions have been asked about cubes, etc.

Endless Amusement II. 1826? Problem II, p. 189. "Five shillings or sixpences may be so placed over each other, as to be all visible and all be in contact." Two solutions. The first has two coins on the table, then two coins on top moved far enough onto one of the lower coins that a vertical coin can touch both of them and the two lower coins at once. The second solution has one coin with two coins on top and two slanted coins sitting on the bottom coin and touching both coins in the second layer and then touching each other up in the air. [I have recently read an article analysing this last solution and showing that it doesn't work if the coin is too thick and that the US nickel is too thick.] = New Sphinx, c1840, p. 131.

Will Baffel. Easy Conjuring without apparatus. Routledge & Dutton, nd, 4th ptg [c1910], pp. 103-104. Six matches, each touching all others. Make a V with two matches and place a third match in the notch to make a short arrow. Lie one of these on top of another.

Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Five in contact, pp. 98-101. Same as Endless Amusement II.

Rohrbough. Puzzle Craft. 1932. Six Nails, p. 22 (= p. 22 of 1940??). As in Baffel.

Meyer. Big Fun Book. 1940. Five coins, p. 543. Same as the first version in Endless Amusement II.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 29, pp. 35 & 92. Second of the forms given in Endless Amusement II.

Ripley's Puzzles and Games. 1966. P. 36. Six cigarettes, as in Baffel.

I recall this is in Gardner and that a solution with 6 cigarettes was improved to 7.

6.BU. CONSTRUCTION OF N-GONS

New Section. This is really a proper geometric topic, but there is some recreational interest in two aspects.

A. Attempts to construct regular n-gons for impossible values of n, e.g. n = 7, either by ruler and compass or by origami or by introducing new instruments -- see 6.BV.

B. Attempts to construct possible cases, e.g. n = 5, by approximate methods.

Aspect A is closely related to the classic impossible problems of trisecting an angle and duplicating a cube and hence some of the material occurs in books on mathematical cranks -- see Dudley. Further, there were serious attempts on both aspects from classic times onward.

Abu’l-Jūd. 11C. He "devised a geometrical method to divide the circle into nine equal parts." [Seyyed Hossein Nasr; Islamic Science -- an Illustrated Study; World of Islam Festival Publishing Co., London?, 1976, p. 82. Q. Mushtaq & A. L. Tan; Mathematics: The Islamic Legacy; Noor Publishing House, Farashkhan, Delhi, 1993, p. 70.]

Pacioli. De Viribus. c1500. These problems are discussed by Mackinnon, op. cit. in 6.AT.3, pp. 167, 169, citing Agostini, p. 5. Let ln be the side of a regular n-gon inscribed in a unit circle.

Ff. 146r-147r, XXIII afare la 7a fia dicta nonangolo. cioe de .9. lati difficile (XXIII to make the 7th figure called nonagon, that is of 9 sides, difficult) = Peirani 198-199. Asserts l9 = (l3 + l6)/4. Mackinnon computes this gives .6830 instead of the correct .6840.

Ff. 148r-148v, XXV. Documento della 9 fia recti detta undecagono (XXV. on the 9th rectilinear figure called undecagon) = Peirani 200. Asserts l11 = φ (l3 + l6)/3, where φ is the golden mean: (1 - (5)/2. Mackinnon computes this gives .5628 instead of the correct .5635.

F. 148v, XXVI. Do. de' .13. (XXVI. on the 13th) = Peirani 200. Asserts l13 = (1 φ)·5/4. Mackinnon computes this gives .4775 instead of the correct .4786.

Ff. 149r-149v, XXVIII. Documento del .17. angolo cioe fia de .17. lati (XXVIII on the 17-angle, that is the figure of 17 sides) = (Peirani 201-202). Peirani says some words are missing in the second sentence of the problem and Agostini says the text is too corrupt to be reconstructed. MacKinnon suggests l17 = (l3 - l6)/2 which gives .3660 instead of the correct .3675.

Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice, (1569); facsimile by Arnaldo Forni, 1980, HB. [The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.] Pp. 26-27 includes discussion of constructing a regular heptagon, but it just seems to say to divide the circumference of a circle into seven equal parts -- ??

Christian Huygens. Oeuvres Complètes. Vol. 14, 1920, pp. 498-500: problem dated 1662, "To inscribe a regular heptagon in a circle." ??NYS -- discussed by Archibald.

R. C. Archibald. Notes (to Problems and Solutions section) 24: Problems discussed by Huygens. AMM 28 (1921) 468-479 (+??). The third of the problems discussed is the construction of the heptagon quoted above. Archibald gives an extensive survey of the topic on pp. 470-479. A relevant cubic equation was already found by an unknown Arab writer, c980, and occurs in Vieta and in Kepler's Harmonices Mundi, book I, Prop. 45, where Kepler doubts that the heptagon can be constructed with ruler and compass.

An approximate construction was already given by Heron of Alexandria and may be due to Archimedes -- this says the side of the regular heptagon is approximately half the side of the equilateral triangle inscribed in the same circle. Jordanus Nemorarius (c1230) called this the Indian method. Leonardo da Vinci claimed it was exact. For the central angle, this approximation gives a result that is about 6.5' too small.

Archibald then goes on to consider constructions which claim to work or be good approximations for all n-gons. The earliest seems to be due to Antoine de Ville (1628), revised by A. Bosse (1665). In 1891, A. A. Robb noted that a linkage could be made to construct the heptagon and J. D. Everett (1894) gave a linkage for n-gons.

Italo Ghersi; Matematica Dilettevoli e Curiosa; 2nd ed., Hoepli, 1921; pp. 425-430: Costruzioni approssimate.

He says the following construction is given by Housel; Nouvelles Annales de Mathématiques 12 (1853) 77-?? with no indication of its source. Ghersi says it also occurs in Catalan's Trattato di Geometria, p. 277, where it is attributed to Bion. However, Ghersi says it is due to Rinaldini (probably Carlo Renaldini (1615-1698)). Let AOB be a horizontal diameter of a circle of radius 1 and form the equilateral triangle ABC with C below the diameter. Divide AB into n equal parts and draw the line through C and the point 4/n in from B. Where this line hits the circle, say P, is claimed to be 1/n of the way around the circumference from B. Ghersi obtains the coordinates of P and the angle BOP and computes a table of these values compared to the real values. The method works for n = 2, 3, 4, 6. For n = 17, the error is 36'37".

On pp. 428-430, he discusses a method due to Bardin. Take AOB as above and draw the perpendicular diameter COD. Divide the diameter into n equal parts and extend both diameters at one end by this amount to points M, N. Draw the line MN and let it meet the circle near B at a point P. Now the line joining P to the third division point in from B is claimed to be an edge of the regular n-gon inscribed in the circle. Ghersi computes this length, finding the method only works for n ( 5, and gives a table of values compared to the real values. This is exact for n = 6 and is substantially more accurate than Renaldini's method. For n = 17, the error is 1'10.32".

The "New" School of Art Geometry, Thoroughly Remodelled so as to Satisfy all the Requirements of the Science and Art Department for Science Subject I. Sections I. and II, Practical Plane and Solid Geometry, (Cover says: Gill's New School of Art Geometry Science Subject I.) George Gill and Sons, London, 1890.

Pp. 26-27, prob. 66 -- To describe any regular Polygon on a given straight line, AB. He constructs the centre of a regular n-gon with AB as one edge. Taking the side AB as 1, the height hn of the centre is given by hn  =  (n-4) (3/4  -  (n-6)/4, while the correct answer is ½ cot π/n. For large n, the relative error approaches 14.99%. He gives no indication that the method is only approximate and doesn't even work for n = 5.

Pp. 74-75, prob. 188 -- To inscribe any Regular Polygon in a given circle. He gives three methods. The first is to do it by trial! The second requires being able to construct the regular 2n-gon! The third construction is Renaldini's, which he does indicate is approximate.

R. C. Archibald, proposer; H. S. Uhler, solver. Problem 2932. AMM 28 (1921) 467 (??NX) & 30 (1923) 146-147. Archibald gives De Ville's construction and asks for the error. Uhler gives values of the error for n = 5, 6, ..., 20, and the central angles are about 1o too large, even for n = 6, though the error seems to be slowly decreasing.

T. R. Running. An approximate construction of the side of a regular inscribed heptagon. AMM 30 (1923) 195-197. His central angle is .000061" too small.

W. R. Ransom, proposer; E. P. Starke, solver. Problem E6. AMM 39 (1932) 547 (??NX) & 40 (1933) 175-176. Gives Dürer's method for the pentagon and asks if it is correct. Starke shows the central angle is about 22' too large.

C. A. Murray, proposer; J. H. Cross, E. D. Schell, Elmer Latshaw, solvers. Problem E697 -- Approximate construction of regular pentagon. AMM 52 (1945) 578 (??NX) & 53 (1946) 336-337. Describes a method similar to that of de Ville - Bosse and asks if it works for a pentagon. Latshaw considers the general case. The formula is exact for n = 3, 4, 6. For n = 5, the central angle is 2.82' too small. For n > 6, the central angle is too large and the error is increasing with n.

J. C. Oldroyd. Approximate constructions for 7, 9, 11, 13-sided polygons. Eureka 18 (Oct 1955) 20. Gives fairly simple constructions which are accurate to a few seconds.

Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. Méthode dite d'Albert Dürer, pp. 45-47. Describes Dürer's approximate method for the pentagon and says it fails by 22'.

Underwood Dudley. Mathematical Cranks. MAA Spectrum, 1992. This book discusses many related problems, e.g. duplication of the cube, trisection of the angle. The chapter: Nonagons, Regular, pp. 231-234 notes that there seem to be few crank constructors of the heptagon but that a nonagoner exists -- Dudley does not identify him. Actually he constructs 10o with an error of about .0001', so he is an excellent approximater, but he claims his construction is exact.

Robert Geretschläger. Euclidean constructions and the geometry of origami. MM 68:5 (Dec 1995) 357-371. ??NYS -- cited in next article, where he states that this shows that all cubic equations can be solved by origami methods.

Robert Geretschläger. Folding the regular heptagon. CM 23:2 (Mar 1997) 81-88. Shows how to do it exactly, using the result of his previous paper.

Dirk Bouwens, proposer; Alan Slomson & Mick Bromilow, independent solvers. An early protractor. M500 171 (Dec 1999) 18 & 173 (Apr 2000) 16-17. Draw a semicircle on diameter AOB. Draw the perpendicular through O and extend it to C so that BC = BA (the diameter) [this makes c = OC = (3] . If P divides AO in the ratio λ : 1-λ, then draw CP to meet the semicircle at D and OD divides the arc ADE in approximately the same ratio. He finds the exact value of the angle AOD and finds that the maximum error in the process is only .637o (when λ ( .18). Second author provides a graph of the error and says the maximum error is at about 18o. [I get .637375o at .181625o.]

Ken Greatrix. A better protractor. M500 175 (Aug 2000) 14-15. Taking c = 1.67721 in the previous construction gives a more accurate construction, with maximum error about .32o at about 13o. [I get .324020o at .130164o.]

6.BV. GEOMETRIC CONSTRUCTIONS

New Section. This is really a proper geometric topic, but there is some recreational interest in it, so I will cite some general references.

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 82-101 & 168-191. Excellent survey. After considering use of straightedge and compasses, he considers: compasses only; folds and creases; straightedge only; straight line linkages; straightedge with fixed figure (circle, square or parallelogram); straightedge with restricted compasses (collapsible compass, rigid (or rusty) compass or rigid dividers); parallel and angle rulers; higher order devices (marked ruler, carpenter's square, tomahawk, compasses of Hermes, two right angle rulers, straightedge with compasses and fixed conic); plane linkages in general. Each section has numerous references.

6.BW. DISTANCES TO CORNERS OF A SQUARE

New section. If ABCD is a rectangle, P is a point and a, b, c, d are the distances of P from the corners of the rectangle, then the basic relation a2 + c2 = b2 + d2 is easily shown. This leads to a number of problems. A little research has found references back to 1896, but the idea might be considerably older.

AMM 3 (1896) 155. ??NYS -- this is the earliest reference given by Trigg, cf below.

SSM 15 (1915) 632. ??NYS -- cited by Trigg, below.

AMM 35 (1928) 94. ??NYS -- cited by Trigg, below.

SSM 32 (1932) 788. ??NYS -- cited by Trigg, below.

NMM 12 (1937) 141. ??NYS -- cited by Trigg, below.

AMM 47 (1940) 396. ??NYS -- cited by Trigg, below.

NMM 16 (1941) 106. ??NYS -- cited by Trigg, below.

NMM 17 (1942) 39. ??NYS -- cited by Trigg, below.

AMM 50 (1943) 392. ??NYS -- cited by Trigg, below.

SSM 46 (1946) 89, 783. ??NYS -- cited by Trigg, below.

SSM 50 (1950) 324. ??NYS -- cited by Trigg, below.

SSM 59 (1959) 500. ??NYS -- cited by Trigg, below.

"A. Polter Geist", proposer; Joseph V. Michalowicz, Mannis Charosh, solvers, with historical note by Charles W. Trigg. Problem 865 -- Locating the barn. MM 46:2 (Mar 1973) 104 & 47:1 (Jan 1974) 56-59. For a square with an interior point, a, b, c = 13, 8, 5. How far is P from the nearest side? First solver determines the side, s, by applying the law of cosines to triangles BPA and BPC and using that angles ABP and BPC are complementary. This gives a fourth order equation which is a quadratic in s2. Second solver uses a more geometric approach to determine s. The distances to the sides are then easily determined. Trigg gives 13 references to earlier versions of the problem -- see above.

Anonymous. Puzzle number 35 -- Eccentric lighting. Bull. Inst. Math. Appl. 14:4 (Apr 1978) 110 & 13:5/6 (May/Jun 1978) 155. Light bulb in a room, with distances measured from the corners. a, b, c = 9, 6, 2. Find d. Solution uses the theorem of Apollonius to obtain the basic equation.

David Singmaster, proposer and solver. Puzzle number 40 -- In the beginning was the light. Bull. Inst. Math. Appl. 14:11/12 (Nov/Dec 1978) 281 & 15:1 (Jan 1979) 28. Assuming P is inside the rectangle, what are the conditions on a, b, c for there to be a rectangle with these distances? When is the rectangle unique? When can P be on a diagonal? Solution first obtains the basic relation, which does not depend on P being in the rectangle. Reordering the vertices if necessary, assume b is the greatest of the distances. Then a2 + c2 ( b2 is necessary and sufficient for a rectangle to exist with these distances. This is unique if and only if equality holds, when P = D. If the distances are all equal, then P is at the centre of the rectangle, which can have a range of sizes. If the distances are not all equal, there is a unique rectangle having P on a diagonal and it is on the diagonal containing the largest and smallest distances.

Marion Walter. Exploring a rectangle problem. MM 54:3 (1981) 131-134. Takes P inside the rectangle with a, b, c = 3, 4, 5. Finds the basic relation, noting P can be anywhere, and determines d. Then observes that the basic relation holds even if P is not in the plane of ABCD. Ivan Niven pointed out that the problem extends to a rectangular box. Mentions the possibility of using other metrics.

James S. Robertson. Problem 1147 -- Re-exploring a rectangle problem. MM 55:3 (May 1982) 177 & 56:3 (May 1983) 180-181. With P inside the rectangle and a, b, c given, what is the largest rectangle that can occur? Observes that a2 + c2 > b2 is necessary and sufficient for a rectangle to exist with P interior to it. He then gives a geometric argument which seems to have a gap in it and finds the maximal area is ac + bd.

I. D. Berg, R. L. Bishop & H. G. Diamond, proposers. Problem E 3208. AMM 94:5 (May 1987) 456-457. Given a, b, c, d satisfying the basic relation, show that a rectangle containing P can have any area from zero up to some maximum value and determine this maximum.

Problem 168.2. M500 168 ??? Seven solvers, M500 170 (Oct 1999) 15-20. Given P inside a square and a, b, c = 5, 3, 4, find the side of the square.

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