Fibonacci Rabbits

How did Fibonacci discover his famous numbers?
Originally in the year 1202, Fibonacci was presented with a problem of how quickly the rabbit population will grow in ideal conditions:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

This problem states several important factors:

  • rabbits take 1 month to grow up
  • after they have matured (for 1 month) it takes a pair of rabbits 1 more month to produce another pair of newly born rabbits.
  • we assume that rabbits never die
  • we assume that whenever a new pair of rabbits is produced, it is always a male and a female
  • we assume that these rabbits live in ideal conditions
  • the problem begins with just 1 pair of newly born rabbits (1 male, 1 female) Given all this information, how many pairs of rabbits will there be in 1 year (i.e. 12 months)?

    The diagram below shows the number of rabbits which will result after 4 months:



    Number of pairs : Explanation

    1 Pair : we start with 1 pair of newly born rabbits

    1 Pair : our rabbits take 1 month to mature

    2 Pair : At the end of 2nd month, our rabbits produce 1 newly born pair, so that now we have 2 pairs of rabbits.

    3 Pair : At the end of 3rd month, we have 3 pairs of rabbits (our original pair + 2 pairs of babies)

    5 Pair : At the end of 4th month, 1 pair of babies of the original rabbits produce a newly born pair, while the other pair of babies mature, and our original rabbits produce another new pair. This gives us 5 pairs of rabbits.

    The rabbit population creates the sequence: 1, 1, 2, 3, 5, ...

      Questions:
    • Can you complete the diagram above for the next 2 generations of rabbits, i.e. 2 months?
    • How many rabbits will there be at the end of the 5th month? At the end of the 6th month?
    • Do you recognize the sequence being formed? Without drawing the diagram, what are the numbers of rabbits at the end of the 7th and 8th month? How did you find that?
    What was Fibonacci's answer to the original question? How many rabbits will there be in 1 year? I.e. at the end of 12 months? (Do not draw the diagram)

    Reference:The Number Devil: a mathematical adventure by Hans Magnus Enzensberger. ISBN 0-8050-5770-6
    http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits
    http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Fibonacci.html