# Fibonacci’s Rabbits

*By toni On 20 August, 2020 · Leave a Comment*

Fibonacci (in the year 1202) investigated a problem about how fast a population of rabbits would grow in the following circumstances, starting with just one pair of rabbits:

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.

**Use the diagram to find out how many pairs of rabbits there will be in one year and so to solve the problem?**

The diagram shows the population of rabbits at times t = 0, 1, 2, 3, 4 and 5 months.

At time t = 0 the original pair of new-born rabbits are shown by a yellow disc.

At time t = 1 month the same pair are shown in pink, as young rabbits.

At time t = 2 months the diagram shows the same rabbits (now mature) in red together with a pair of new-borns (in yellow).

At time t = 3 months the original pair have produced a 2^{nd} pair of new-borns and their 1^{st} pair are now young and shown in pink.

The population is also shown at times t = 4 and 5 months.

Extend the tree diagram to show the population at t = 6 months.

Fill in the table for times t = 0 to t = 6 months.

Write down formulae for *N _{t}, Y_{t}, M_{t}* and

*F*and explain why the pattern will continue month after month.

_{t }In 1202 Leonardo of Pisa, who is now known as Fibonacci, published a ground-breaking book called ‘The Book of Calculation’ (Liber Abaci in Latin) which included this problem. Fibonacci had travelled in Asia as a merchant and he brought from India to Europe a new way of writing numbers – the Hindu-Arabic Numeral system. His methods influenced the development of mathematics in Europe and provided merchants with efficient ways to record commercial transactions.

Henry E Dudeney (1857 – 1930), an Englishman famous for his puzzles, adapted Fibonacci’s Rabbits problem. He changed months into years, and rabbits into bulls (male) and cows (females) and stated the problem as follows:

**If cows produce their first she-calf at age 2 years, and after that another she-calf every year, how many she-calves are there after 12 years, assuming that none die? **

Explain the solution to this problem.

Click here for the FIBONACCI’S RABBITS worksheet

Click here for the FIBONACCI’S RABBITS Notes for teachers

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