Sunday, January 13, 2008

Leonardo Pisano -- The Greatest Mathematician of his time? No 'Fibbin'!

I didn't announce this week's Mystery Math Contest to see how that would play out. To be fair to those 200 or so subscribers to MathNotations who get a feed redirected through Feedburner, I guess I should mention it from now on, since the feed only shows new posts, not changes in the sidebar.

Anyway, we have two winners this week who correctly identified the one and only Fibonacci. BTW, this extraordinary gentleman had more names than George Foreman's sons! TC below mentioned one of his other names...

After visiting several websites, I found that he had devised some interesting word problems that seem to have survived to this day in one form or another. So now we know the rest of the story!

The following is from his most famous work, Liber abaci:

(1) A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
(2) A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
(3) Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.

Here are our winners...

Here's the info she supplied (the math symbols went a bit awry when I copied this from her email):

The mystery mathematician is Fibonacci. One interesting fact is that his real name was Leonardo Pisano.

Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the "Book of Squares" and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci also proves many interesting number theory results such as: there is no x, y such that x2 + y2 and x2 - y2 are both squares, and x4 - y4 = z2 has no non-trivial integral solutions.

He also defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x2 + c and x2 - c are both squares, then c is a congruum. He also proved that a square cannot be a congruum. The Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the seventeenth century French mathematician Pierre de Fermat.

And our 2nd winner is
who offered:

That's Leonardo Bigollo of Pisa.

This description of a famous sequence popularized by another famous Leonardo's code is interesting:

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?

It is almost a rite of passage for most math students to discover the closed form expression to this (featuring the Golden ratio, of course).

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