The Medieval Mathematician Leonardo Fibonacci's Problem About Rabbits

2024 Author: Malcolm Clapton | [email protected]. Last modified: 2023-12-17 03:44

Calculate what offspring a pair of animals will give by the beginning of next year.

Leonardo Fibonacci was an outstanding medieval mathematician. It is believed that it was he who introduced Arabic numerals into use. In The Book of the Abacus, a work that expounds and promotes decimal arithmetic, Fibonacci gives his famous problem on rabbits. Try to solve it.

In early January, a pair of newborn rabbits (male and female) were placed in a pen, fenced on all sides. How many pairs of rabbits will they produce by early next year? It is necessary to take into account the following conditions:

• Rabbits reach sexual maturity two months after their birth, that is, by the beginning of the third month of life.
• At the beginning of each month, each sexually mature couple gives birth to only one couple.
• Animals are always born in pairs "one female + one male".
• Rabbits are immortal, predators cannot eat them.

Let's see how the number of rabbits grows in the first six months:

Month 1. One pair of young rabbits.

Month 2. There is still one original pair. Rabbits have not yet reached childbearing age.

Month 3. Two pairs: the original one, having reached childbearing age + a pair of young rabbits that she gave birth to.

Month 4. Three pairs: one original pair + one pair of rabbits that she gave birth to at the beginning of the month + one pair of rabbits that were born in the third month, but have not yet reached puberty.

Month 5. Five couples: one original couple + one couple born in the third month and reached childbearing age + two new couples that they gave birth to + one couple that was born in the fourth month, but has not yet reached maturity.

Month 6. Eight couples: five couples from last month + three newborn couples. Etc.

To make it clearer, let's write the received data into the table:

If you carefully examine the table, you can identify the following pattern. Each time the number of rabbits present in the n-th month is equal to the number of rabbits in the (n - 1) -th, previous month, summed up with the number of newly born rabbits. Their number, in turn, is equal to the total number of animals as of the (n - 2) month (which was two months ago). From here you can derive the formula:

F = F_{n ‑ 1}+ F_{n ‑ 2}, where F - the total number of pairs of rabbits in the n-th month, F_{n ‑ 1} is the total number of pairs of rabbits in the previous month, and F_{n ‑ 2} - the total number of pairs of rabbits two months ago.

Let's count the number of animals in the following months using it:

Month 7. 8 + 5 = 13.

Month 8. 13 + 8 = 21.

Month 9. 21 + 13 = 34.

Month 10. 34 +21 = 55.

Month 11. 55 + 34 = 89.

Month 12. 89 + 55 = 144.

Month 13 (beginning of next year). 144 + 89 = 233.

At the beginning of the 13th month, that is, at the end of the year, we will have 233 pairs of rabbits. Of these, 144 will be adults and 89 will be young. The resulting sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 is called Fibonacci numbers. In it, each new final number is equal to the sum of the two previous ones.

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