Juxtaposition: Millionaire Triangles

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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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10 Responses to “Juxtaposition: Millionaire Triangles”

  1. David Petersen Says:

    I would draw a Pythagorean triangle as a right triangle. Maybe that one that Mu Alpha Theta uses as their logo with the squares coming off the sides to show that 3^2 + 4^2 = 5^2

  2. Ted Says:

    While working with Pythagorean Triples the other day, I ran across the Barning-Hall (Up-Across-Down, UAD) method to generate a complete tree of primitive Pythagorean triples, starting from the famous 3-4-5 triangle, on Ron Knott’s site.
    http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html
    The UAD Tree of Primitive Pythagorean Triangles (PPTs).

    In Wikipedia http://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples
    I see reference to “an entirely different ternary tree”
    [Price, H. Lee, The Pythagorean Tree: A New Species, (2008).] at
    http://arxiv.org/abs/0809.4324?context=math

  3. Ξ Says:

    That is so neat — thanks for the link!

  4. One Two Three « 360 Says:

    […] #2:  I learned of this problem via Ted’s comment here.  (Thanks Ted!)   He linked to Ron Knott’s site, which has all sorts of stuff about […]

  5. Ross Isenegger Says:

    I spent a bit of time thinking about arranging fractions (a la proof that the rationals are countable) in a triangle. It is well worth exploring. See http://mathfest.wikispaces.com/Pascal+Fractions+Triangle+Idea and http://mathfest.blogspot.com/2007/10/new-arrangement-on-old-idea.html.

  6. Ξ Says:

    Ross,

    I really like the visual nature of your triangle. In a similar vein, Have you seen the series Recounting the Rationals (Part I starts here) over on The Math Less Traveled? It’s a description of the paper Recounting the Rationals by Neil Calkin and Herbert Wilf, and it gives a completely different tree of fractions, but one in which all the fractions are in reduced form.

  7. Pascal’s Pyramid (part 1 of 3) « 360 Says:

    […] Pyramid (part 1 of 3) By TwoPi In an earlier post, Ξ had mentioned a question from Who Wants To Be A Millionaire that made reference to “a […]

  8. Ross Isenegger Says:

    The Recounting the Rationals paper and the blog posts about it contain the most mathematically rich ideas that I have explored in quite some time. Thanks so much for directing me to them. I believed (and may have been taught) that arranging the rationals in an organized list was impossible.

  9. kyzyl Says:

    Price’s method for generating a complete tree containing all primitive Pythagorean triples is nicely described in Part VII of this article: http://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples#VII.

  10. kyzyl Says:

    Ross Isenegger wonders about arranging fractions in a triangle. I think Price’s Pythagorean tree can also be used for this purpose. Simply read the fractions (q/p) and (q’/p’) from his tree of Fibonacci Boxes. Reduced fractions containing an even integer are found in the left-hand columns. All others are in the right-hand columns.

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