Archive for February 24th, 2009

Juxtaposition: Millionaire Triangles

February 24, 2009

There was a math question on Who Wants to Be a Millionaire yesterday!  According to this transcript of the February 23 game, this was the $50,000 question:

Named for the mathematician who designed it, a famous “pyramid” of numbers that starts with the number one on top is called what?
A. Fibonacci’s triangle
B. Pythagoras’s triangle
C. Pascal’s triangle
D. Fermat’s triangle

Fibonacci's TriangleThe correct answer is Pascal’s triangle.  But it got me thinking, what would the other triangles look like?  We’ve actually posted before on Fibonacci’s Triangle:  as envisioned by Doug Ensley, it’s a triangle with Fibonacci numbers down each side and with each interior number equal to the sum of the two numbers above it (as with Pascal’s triangle).  This is also called Hosoya’s triangle, according to wikipedia (named after Haruo Hosoya, who wrote about it in The Fibonacci Quarterly in 1976).

fermat-triangleSo what about Fermat’s triangle — what would that look like?  Since Fermat numbers are numbers of the form 2^{2^n} + 1, a natural adaptation to Fibonacci’s triangle would be to have the Fermat numbers on the outside, and to use sums of numbers to fill in the inside.  This looks like of funny, though — the numbers on the outside grow much faster than those on the inside.

What has me stumped is the notion of Pythagoras’s Triangle.  Would it have something to do with Pythagorean triples?  Maybe it would be a tetrahedron rather than a flat triangle, with a 1 on top, and then a Pythagorean triple beneath it — say the iconic 3, 4, 5 — and then…then I’m stumped.  What would come next?  Is there even an ordering for Pythagorean triples?  (Well, maybe:  TwoPi wrote earlier about how every Pythagorean triple can be written as (2uv, u^2-v^2, u^2+v^2), for positive integers u and v, so we could put an ordering on u and v and retrict ourselves to the primitive triples.)  But I’m not sure how to generate the entire tetrahedron.

Then again, for a Pythagorean triangle, perhaps we could just draw a right triangle and call it a day.

Incidentally, the link to the transcript also has commentary about whether Pascal’s triangle was the only reasonable answer.  The consensus seems to be YES, if only because the question referred to a “famous” triangle “pyramid”.


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