Sunday, January 15, 2012

Fibonacci's Rabbits

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.
Fluffy bunnies
Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...
How many pairs will there be in one year?
  1. At the end of the first month, they mate, but there is still one only 1 pair.
  2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
  3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
  4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
Fluffy bunnies family tree

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Can you see how the series is formed and how it continues? If not, look at the answer!
The first 300 Fibonacci numbers are here and some questions for you to answer.
Now can you see why this is the answer to our Rabbits problem? If not, here's why.
Another view of the Rabbit's Family Tree:
another view of same treeMultiples-of-Phi family tree
Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows:
  • All the rabbits born in the same month are of the same generation and are on the same level in the tree.
  • The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent's number. Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively.
  • The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree.
  • There are a Fibonacci number of new rabbits in each generation, marked with a dot.
  • There are a Fibonacci number of rabbits in total from the top down to any single generation.

Alexandra Brancovean, Candela Megido, Aida Zapico

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