Q. A young pair of chinchillas (one of each sex) is placed on an island. A pair of chinchillas does not breed until they are 2 months old. After they are 2 months old, each pair of chinchillas produces another pair each month (See pic below). Find a recurrence relation for the number of pairs of chinchillas on the island after n months, assuming that no chinchillas ever die.

**Solution:**

Let us denote the number of pairs of chinchillas after n months by f(n).

The chinchilla population can be modeled using a recurrence relation. At the end of the first month, the number of pairs of chinchillas on the island is f(1) = 1. Because this pair does not breed during the second month, f(2) = 1 also. To find the number of pairs after n months, add the number on the island the previous month, f(n−1) , and the number of newborn pairs, which equals f(n−2) , because each newborn pair comes from a pair at least 2 months old.

Consequently, the sequence {f(n)} satisfies the recurrence relation

f(n) = f(n−1) + f(n−2 )

for n ≥ 3 together with the initial conditions f(1) = 1 and f(2) = 1.This recurrence relation and the initial conditions uniquely determine this sequence.

If you closely observe, this is the recurrence relation for Fibonacci numbers. (See ‘Total Pairs’ column in the pic)

Hence, the number of pairs of chinchillas on the island after n months is given by the nth Fibonacci number.

**Reference:** http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

This puzzle is contributed by **Feroz Baig.** Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

## Recommended Posts:

- Puzzle 15 | (Camel and Banana Puzzle)
- Puzzle 24 | (10 Coins Puzzle)
- Puzzle 27 | (Hourglasses Puzzle)
- Puzzle 28 | (Newspaper Puzzle)
- Puzzle 29 | (Car Wheel Puzzle)
- Puzzle 31 | (Minimum cut Puzzle)
- Puzzle 33 | ( Rs 500 Note Puzzle )
- Puzzle 34 | (Prisoner and Policeman Puzzle)
- Puzzle 36 | (Matchstick Puzzle)
- Puzzle 38 | (Tic Tac Toe Puzzle)
- Puzzle 39 | (100 coins puzzle)
- Puzzle 51| Cheryl’s Birthday Puzzle and Solution
- Puzzle 81 | 100 people in a circle with gun puzzle
- Puzzle 85 | Chain Link Puzzle
- Puzzle | Elevator Puzzle
- Puzzle | 3 Priests and 3 devils Puzzle
- Puzzle | Man fell in well Puzzle
- Puzzle 1 | (How to Measure 45 minutes using two identical wires?)
- Puzzle 2 | (Find ages of daughters)
- Puzzle 3 | (Calculate total distance travelled by bee)