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
The 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).
So what about Fermat’s triangle — what would that look like? Since Fermat numbers are numbers of the form 
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 
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”.
In a comment on 
Many people are familiar with the Rubik’s Cube, the 3x3x3 cube with colored faces that can be moved out of position and then, ideally, twisted back into place. This toy, originally called the Magic Cube, was invented by Ernő Rubik in 1974 while he was a lecturer at the Academy of Applied Arts and Crafts in Budapest, Hungary. In 1980 the cube made its way to other countries and spawned, among other things, a one-season cartoon series 
One such juxtaposition is the concept of Fibonacci’s Triangle: a combination of the Fibonacci Sequence (1, 1, 2, 3, 5, 8, … where each number is the sum of the previous two) and Pascal’s Triangle*, shown to the left, which is created by placing “1”s along the outside and then filling the inside by adding two adjacent numbers and placing the sum between them on the next row.
There are many questions that can be asked about Fibonacci’s triangle: Is there an explicit formula for the entries in each row the way there is for Pascal’s triangle? Does a Sierpinski-like pattern develop when you shade in the odd numbers the way it does for Pascal’s triangle? The answer to the first question is yes: see Doug Ensley’s article, “Fibonacci’s Triangle and Other Abominations” in the September 2003 issue of Math Horizons. For the second question, the best solution is to draw it out and see for yourself!
