Fibonaccis Rabbits

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.

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.

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 100 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:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The Rabbits problem is not very realistic, is it?

It seems to imply that brother and sisters mate, which, genetically, leads to problems. We can get round this by saying that the female of each pair mates with any male and produces another pair. Another problem which again is not true to life, is that each birth is of exactly two rabbits, one male and one female.

Dudeney's Cows

The English puzzlist, Henry E Dudeney (1857 - 1930) wrote several excellent books of puzzles. In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females!

He changes months into years and rabbits into bulls (male) and cows (females) in problem 175 in his book 536 puzzles and Curious Problems (1967, Souvenir press):

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

This is a better simplification of the problem and quite realistic now.

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But Fibonacci does what mathematicians often do at first, simplify the problem and see what happens - and the series bearing his name does have lots of other interesting and practical applications as we see later. So let's look at another real-life situation that is exactly modelled by Fibonacci's series - honeybees.