# Leonardo of Pisa: Bunny Rabbits to Bull Markets - Fibonacci Numbers

Author(s):
Sandra Monteferrante

Practical advances aside, the mathematical legacy of Liber abaci is really the fascinating set of numbers introduced by one problem posed about rabbits.

How many rabbits can be produced from a single pair in a year if each pair begets a new pair every month, which from the second month on becomes productive, and deaths do not occur?

Click to see rabbits multiply

To illustrate the pattern of population growth, let r represent a newborn pair of rabbits and R a mature pair. A newborn pair, r, appears in the next month as a mature pair represented by R, that is r → R. An adult pair R appears in the next month with offspring, that is R → Rr. The table below illustrates the first 4 months of population growth.

Months Description Symbols Number
1 Newborn r 1
2 Mature R 1
3 Mature & offspring Rr 2
4 Mature, offspring & newly mature RrR 3

One way to look at the population in month 5 is to note that new offspring, r, come only from the rabbit pairs alive in month 3 and all rabbit pairs alive in month 4 continue as mature pairs, R.

Months Symbol Number
5 RrRRr 5
6 RrRRrRrR 8

Similarly, the population in month 6 is the sum of the populations of the two previous months, 8 = 5+3.

The pattern, that is, each month's population is the sum of the previous two months, is the key to this rabbit population growth. Thus the numbers of monthly populations is: 1, 1, 2, 3, 5, 8, 13 = 8+5, 21 = 13+8, 34 = 21+13, 55 = 34+21, 89 = 55+34, 144 = 89+55. Thus there are 144 pairs of rabbits in month 12.

The rabbit problem remained interesting over the years in mathematical circles. Henry E. Dudeney (1857 - 1930) thought it was too unrealistic, though. He changed Fibonacci's rabbits to cows and months to years before adding it to one of his books of puzzles, 536 Puzzles and Curious Problems:

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

Lesson Plan 2

Sandra Monteferrante, "Leonardo of Pisa: Bunny Rabbits to Bull Markets - Fibonacci Numbers," Convergence (August 2011)