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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?*

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?*

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