Problem of rabbits reproduction

Leonardo Pisano Fibonacci is considered rightfully as one of the most known mathematicians of the Middle Ages epoch. Later we will tell about Fibonacci and his role in development of the West-European mathematics in more detail. By irony of his fate Fibonacci introduced the outstanding contribution to development of mathematics, there was known in modern mathematics only as the author of interesting numeric sequence called Fibonacci numbers. This numeric sequence was obtained by Fibonacci at the solution of the famous "problem of rabbits reproduction". The formulation and solution of this problem is considered as the main Fibonacci's contribution to development of combinatorics. Just with the help of this problem Fibonacci anticipated the recurrences method, which is considered as one of the powerful methods of the combinatorial problems solution. The recursion formula obtained by Fibonacci at the solution of this problem is considered as the first recurrent formula in the mathematics history.

The essence of his "problem of rabbits reproduction" Fibonacci formulated extreme simply:

"Let's suppose that in the fenced place there is the couple of the rabbits (female male) in first day of January. This couple of the rabbits reproduces the new rabbits couple in the first day of February and then in the first day of each next month. Each newborn rabbits couple becomes mature in one month and then gives a life to the new rabbits couple each month after. There is a question: how much rabbits couples will be in the fenced place in one year, that is in 12 months from the beginning of reproduction?"

For the solution of this problem, which is demonstrated in figure above we will designate through A the couple of the mature rabbits, and through B the couple of the newborn rabbits. Then the process of "reproduction" can be described with the help of two "passages" described the monthly transformations of the rabbits.

 (1) (2)

Let's note, that the passage of (1) models the monthly transformation of each mature rabbits couple in two couples, namely in the same pair of the mature rabbits A and in the newborn couple Â. The passage of (2) models the process of rabbits "maturing", when the newborn couple is transformed into the mature couple of À. Then, if we will begin with the mature couple A, then the process of "rabbits reproduction" can be presented with the help of Table 1.

Table 1.

 Date Rabbits couples A B A + B January, 1 A 1 0 1 February, 1 AB 1 1 2 March, 1 ABA 2 1 3 April, 1 ABAAB 3 2 5 May, 1 ABAABABA 5 3 8 June, 1 ABAABABAABAAB 8 5 13

Note that in the columns A, B and A + B of Table 1 the numbers of the mature (A), newborn (B) and general (A + B) rabbits are indicated respectively.

Studying the A-, B- and (A + B)-sequences it is possible to establish the following regularity in these numeric sequences: each term of sequence is equal to the sum of two previous. If now to designate the n-th term of the sequence satisfying to this rule through Fn, then the mentioned above general rule can be written by the way of the following mathematical formula:

 Fn = Fn-1 + Fn-2. (3)

Such formula is called as the recurrent formula.

Note that the concrete values of the numeric sequence generated by the recurrent formula (3), depend on the initial values of the sequence F1 and F2. For example, we have F1 = F2 = 1 for A-numbers and for this case the recurrent formula (3) "generates" the following numeric sequence:

 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... . (4)

For the B-numbers we have: F1 = 0 and F2 = 1; then the corresponding numeric sequence for this case will look as the following:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... .

At last, for the (A + Â)-sequence we have: F1 = 1 and F2 = 2; then the conforming numeric sequence for this case will look as the following:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... .

In mathematics the numeric sequence of (4) is called, as a rule, Fibonacci numbers. They have a number of surprising mathematical properties, and we will tell about them at the next pages of our Museum. Follow us!