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Leonardo Fibonacci was an Italian mathematician in the late 12th century and early 13th century. He published a puzzle which produced a number sequence that had far reaching consequences and applications. He first published this in his book Liber Abaci in 1202. The puzzle has been a source of fascination to mathematicians ever since as it is not the answer that is so fascinating but rather the sequence of numbers that occurs when you try and find the answer. The greatest fascination comes from the way that the sequence discovered appears in the most unexpected places!

The puzzle

In January you are given a pair of new born rabbits. After two months they produce another pair of rabbits and they continue to produce a new pair every month after that. Each new pair of rabbits, after two months, produces another new pair and goes on to produce a new pair each month thereafter. How many pairs will there be in December?

|January |

| |

| |

|February |

| |

| |

|March |

| |

| |

|April |

| |

| |

|May |

| |

| |

| |

|Continue the pattern: |

| |

|JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC |

|1 1 2 3 |

Now you have found Fibonacci Series. This is the series where the next term is found by adding together the previous two terms. You have seen this before. The most fascinating link with Fibonacci is with plants. It is found in plants which spiral:

Celery – Each new stalk on celery overlaps the last in a spiral. There are three spirals altogether, 1 is anti-clockwise and 2 clockwise.

Pineapple – The outside of a pineapple is formed using spirals. In this case there are 8 clockwise spirals and 13 anti-clockwise.

Pinecones – This again is formed by spirals. This uses the Fibonacci numbers of 5 and 8 spirals.

The unusual thing is that they all use adjacent Fibonacci numbers. Other examples are sunflowers, daisies, cacti, palm trees, etc. Look around your kitchen or garden can you find any?

Problems:

The following are Fibonacci-like sequences. The next term is found by adding the previous two terms together. Can you fill in the missing numbers?

1. 2, 3, 5, , , , 34. 13. 0, , , 12, 18, 30.

2. 4, 5, 9, , , . 14. 8, , , 8, 16.

3. 9, 21, , 51, . 15. 2, , , 18, 28.

4. 9, 3, , 15, . 16. -2, 4, , , 8, .

5. 8, 3, , , . 17. -1, 6, , , 16, .

6. 3, , 9, , . 18. 5, -3, , , 1, .

7. , 8, 12, , . 19. 6, -2, , , 6, .

8. , , 18, 30, . 20. 10, , 8, , .

9. , , 16, , 42 . 21.* 2, , , , 22.

10. , , 20, , 60. 22.* 4, , , , 50.

11. 2, , , 18, 28. 23.* -2, , , , 20.

12. 20, , , 32, 58. 24.* 1, , , , , 43.

ANSWERS

1. 2, 3, 5, 8 , 13, 21 , 34. 13. 0, 6, 6, 12, 18, 30.

2. 4, 5, 9, 14 , 23 , 37 . 14. 8, 0 , 8 , 8, 16.

3. 9, 21, 30, 51, 81 . 15. 2, 8 , 10, 18, 28.

4. 9, 3, 12, 15, 27 . 16. -2, 4, 2, 6, 8, 14 .

5. 8, 3, 11, 14, 25 . 17. -1, 6, 5 , 11, 16, 27 .

6. 3, 6, 9, 15, 24. 18. 5, -3, 2, -1, 1, 0 .

7. 4, 8, 12, 20 , 32. 19. 6, -2, 4, 2, 6, 8 .

8. 6, 12, 18, 30,48 . 20. 10, -2, 8, 6 , 14 .

9. 6,10, 16,26 , 42 . 21.* 2, 6, 8, 14, 22.

10. 0, 20, 20, 40, 60. 22.* 4, 14, 18, 32, 50.

11. 2, 8, 10, 18, 28. 23.* -2, 8, 6, 14 , 20.

12. 20, 6, 26 , 32, 58. 24.* 1, 8, 9, 17, 26, 43.

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Original pair

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Original pair

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New pair

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New pair

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Original pair

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New pair

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New pair

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Original pair

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Original pair

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New pair

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New pair

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New pair

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