Once upon a time a priest, a lawyer, and a math teacher ...



Once upon a time a priest, a lawyer, and a math teacher were about to be guillotined. The priest put his head on the block, they pulled the rope and nothing happened. He declared that he had been saved by divine intervention, so he was let go. Next the lawyer was put on the block, and again the rope didn’t release the blade. He exclaimed he can’t be executed twice for the same crime and is set free. They grab the math teacher and shove his head into the guillotine, he looks up at the release mechanism and says, “Wait a minute, I see your problem!”

Polya’s Problem

George Polya is the author of How To Solve It, the famous book on the heuristics of problem solving. In the 1960's Professor Polya made a movie for student teachers titled Let Us Teach Guessing. In the film he posed a problem that I feel is a classic for teaching problem solving. My presentation will start with a few warm-up problems that are classics themselves. We will then reenact the solving of Polya's Problem with a few new twists.

One of the classics of problem solving is the following puzzle.

The Nine Dot Puzzle

Draw no more than four straight lines (without lifting the pencil from the paper) which will cross through all nine dots.

A solution is difficult to find until one is willing to “think outside of the box.” But James Adams in his book Conceptual Blockbusting goes the original solution many steps further. He points out solutions that involve not four lines but three and then even fewer than three. The moral of the story? We often put our own unnecessary restrictions on problems we are trying to solve. It is an important tool of problem solvers to think outside the box, to see the problem from many perspectives.

Another classic of problem solving is the following story.

What are their ages?

From Overcoming Math Anxiety by Sheila Tobias

Two women meet after many years. One asks, “How old are your three daughters?

Answer: “The product of their ages is 36.”

Question: “But that’s not enough information.”

Answer: “Well, the sum of their ages is the same number as the post office box that we shared at college.”

Question: “But that’s still not enough information.”

Answer: “The oldest one looks like me.”

Question: “Oh, now I know their ages.”

I include this wonderful problem because it at first seems impossible to solve. There just doesn’t seem to be enough information, or we fear it is going to involve some silly trick. So telling students that there is a logical solution to the problem and applying a good dose of stick-to-it-ness, a student or group of students can indeed solve this one.

Another favorite of mine is H.E. Dudeney’s classic:

The Spider and the Fly

Henry E. Dudeney (1857 - 1930)

Inside a rectangular room, measuring 30’ in length and 12’ in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as shown at A; and a fly is on the opposite wall, 1 foot from the floor in the center, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.

The delight in this problem is that the typical first solution, straight up, over, and down, is not the shortest. Visual thinking plays a big role in most problem solving situations and this problem is an excellent one to point that out. Unfolding the box into nets in various ways yields a shorter route. In fact the shortest route traverses five of the six surfaces!

There are many other introductory problems that I feel should be included in a high school course that focuses on problem solving. The frog at the Bottom of a Well problem and the Two books Sitting on a Shelf problem are two more favorites but lets move on to Polya’s Problem.

In the film Let Us Teach Guessing, Polya introduces himself, explains the purpose of the day’s presentation, and then poses the problem: Into how many parts do five planes divide space? He starts by stating that good problem solving often begins by taking guesses. Guesses lead to better guesses. So he calls for guesses and a number of guesses are offered.

When a guess of six parts is suggested, a discussion as to how the planes are arranged is stimulated. Indeed different arrangements of the planes lead to different answers. If the planes are parallel then six is correct. After some discussion it is agreed that the planes are to be random and the meaning of random is agreed upon. “Into how many parts do five random planes divide space?” becomes the new statement of the problem. It is also agreed that it is equivalent to asking, “What is the maximum number of parts created by five planes dividing space?”

Eventually a guess of 32 is contributed and Polya’s eyebrows go up and he says that is a very good guess. But alas, Polya says with a twinkle in his eyes, that is not the solution. It is very difficult to visualize or even build a model of five random planes dividing space to verify this conjecture. Perhaps we need another approach?

One very powerful strategy in problem solving is looking at simpler cases. What is simpler than five random planes dividing space? Four planes, three, two, one, and zero planes. When a table of values is created it becomes clearer to the rest of the class where the “very good guess” came from. We seem however to be no closer to the solution if 32 is not correct.

Polya suggests that sometimes we need to explore not only simpler cases but analogous situations. What is analogous to planes dividing space? Lines dividing the plane. A table is created yet no connections are seen. Are there simpler analogous situations? Yes, points dividing the line. Another table is created.

I must confess, that I’ve presented this problem to high school students and teachers at conferences so many times that I’m unclear where my presentation begins to stray from Polya’s film presentation. So at this point I will present my latest development of the problem.

At this stage I suggest we look at the values in the tables. What do we notice? It appears that in the analogous patterns the values start off doubling then at some point it doubles less one.

|#points |0 |1 |2 |3 |4 |5 | |

|Parts of line |1 |2 |3 | | | | |

|# of lines |0 |1 |2 |3 |4 |5 | |

|Parts of Plane |1 |2 |4 |7 | | | |

Perhaps this is what happens in our table of Space/Planes. Is the next value 15 (double less one)? Do four random planes divide space into 15 parts?

|# of Planes |0 |1 |2 |3 |4 |5 | |

|Parts of Space |1 |2 |4 |8 | | | |

At this point I suggest we look more closely at those values in the tables, where the doubling stops. How does that case differ from previous cases in the table? What is the geometry, if you will, of that case that is different from earlier cases?

In the analogous situation of points dividing the line we see that when the second point is placed on the line we get two infinite or unbounded parts and one bounded.

In the analogous situation of lines dividing the plane we see that when the third line is placed in the plane we get all infinite or unbounded parts except one bounded region.

What would be the analogous situation in space? Can we model this situation?

At this point I guide my audience through the construction of a tetrahedron by folding a US dollar bill. We then count the regions of space.

An infinite region off each vertex. An infinite region off each edge.

An infinite region off each face. One bounded region enclosed within.

This totals 15, which confirms our conjecture.

Polya suggests we look at the relationships not only in each table of values but look for relationships between the tables. He reorganizes the three tables into one.

| |0 |1 |2 |3 |4 |5 |n |

|Space/planes |1 |2 |4 |8 |15 |? |? |

|Plane/lines |1 |2 |4 |7 |11 | |(n2 + n + 2)/2 |

|Line/points |1 |2 |3 |4 | | |n + 1 |

A pattern is detected by most at this stage. If you haven’t solved it yet, stop now and study the patterns in the table above.

This problem is wonderful in that it involves many familiar problem solving strategies. We look at easier problems to create patterns. We solve analogous problems. We build models to help visualize the problem. We look at special cases such as when the doubling patterns stop. Polya was quite content to end the presentation with the solution for n=5 without proof. His goal was to encourage good guessing and stimulate the problem solver in all of us.

A note.

The derivation of the formulas for the plane/lines and line/points patterns are quite accessible to high school students. Although the proofs are challenging they too are accessible. The formula for space/planes pattern is cubic and is best solved when students have become familiar with finite differences. The proof of the formula can be found in the literature (one source: Geometry by Discovery, David Gay, John Wiley and Sons, Inc.). I have not however had the courage to even attempt to present the proof before an audience of high school students or teachers.

The formula for the pattern line/points: n+1, can also be expressed by combinatorial notation: () + ().

The formula for the pattern plane/lines: (n2 + n + 2)/2 can also be expressed by combinatorial notation: () + () + ().

The formula for the pattern space/planes: (n3 + 5n + 6)/6 can also be expressed by combinatorial notation: () + () + () + ()

What about n 3D spaces intersecting in 4D space?

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download