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, ...Month #0 - At the beginning of the experiment, there is one pair of rabbits (condition #1). Month #1 - After one month, the two rabbits have mated but have not given birth. Therefore, there is still only one pair of rabbits. Month #2 - After two months, the first pair of rabbits gives birth to another pair, making two pair in all. Month #3 - After three months, the original pair gives birth again, and the second pair mate, but do not give birth. This makes three pair. Month #4 - After four months, the original pair give birth, and the pair born in month #2 give birth. The pair born in month #3 mate, but do not give birth. This makes two new pair, for a total of five pair. Month #5 - After five months, every pair that was alive two months ago gives birth. This makes three new pair, for a total of eight. Can we find a way to predict each number without going through each individual pair? Of course we can, we're mathematicians. The way you find the total number of rabbits for each month is to find out how many pairs of rabbits were newly born that month and add that to the number of rabbits you had before the new ones were born. So how many pairs of rabbits are newly born every month? Well, since it takes two months for each new pair to give birth, each pair of rabbits that was alive two months ago will give birth to a new pair. In other words, the number of new pairs in each month is equal to the number of pairs alive two months ago. A Rabbit Next we need to find the number of rabbit pairs that were alive before the new ones were born. It should be obvious that this is simply the number of pairs alive the month before. In other words, to find the total number of pairs of rabbits, you simply add together the number of pairs that were alive in the preceding two months. Now, do you know of any series of numbers which begins with one and one and continues by adding the preceding two numbers to get the next? Of course you do. It's the Fibonacci series. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .