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? He changes months into years and rabbits into bulls (male) and cows (females) in problem 175 in his book 536 puzzles and Curious Problems (1967, Souvenir press): He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females! In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee) wrote several excellent books of puzzles (see after this section). We can get round this by saying that the female of each pair mates with any male and produces another pair.Īnother problem which again is not true to life, is that each birth is of exactly two rabbits, one male and one female. It seems to imply that brother and sisters mate, which, genetically, leads to problems. The Rabbits problem is not very realistic, is it? There are many other interesting mathematical properties of this tree that are explored in later pages at this site.Ġ, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987. There are a Fibonacci number of rabbits in total from the top down to any single generation.There are a Fibonacci number of new rabbits in each generation, marked with a dot.The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree.Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively. The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent's number.All the rabbits born in the same month are of the same generation and are on the same level in the tree.Rabbits have been numbered to enable comparisons and to count them, as follows: Now can you see why this is the answer to our Rabbits problem? If not, here's why.Īnother view of the Rabbit's Family Tree:īoth diagrams above represent the same information. The first 300 Fibonacci numbers are here and some questions for you to answer. Ĭan you see how the series is formed and how it continues? If not, look at the answer! 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. 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.At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.At the end of the first month, they mate, but there is still one only 1 pair.How many pairs will there be in one year? 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. 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 a newly-born pair of rabbits, one male, one female, are put in a field. The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series. Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This, the first, looks at the Fibonacci numbers and why they appear in various "family trees" and patterns of spirals of leaves and seeds.The second page then examines why the golden section is used by nature in some detail, including animations of growing plants. Fibonacci Numbers and Nature This page has been split into TWO PARTS.
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