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Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:

  • the product of non-adjacent vertices is constant.

  • the greatest common divisor of non-adjacent vertices is constant.

Below is one such hexagon. As an example, here we have that $4 \cdot 10 \cdot 15 = 6 \cdot 20 \cdot 5$, as well as $\gcd(4, 10, 15) = \gcd(6,20,5)$.

$$ 1 \\ 1 \qquad 1\\ 1\qquad 2\qquad 1\\ 1\qquad3\qquad3\qquad1\\ 1\qquad\mathbf{4}\qquad\mathbf{6}\qquad4\qquad1\\ 1\qquad\mathbf{5}\qquad10\qquad\mathbf{10}\qquad5\qquad1 \\ 1\qquad6\qquad\mathbf{15}\qquad\mathbf{20}\qquad15\qquad6\qquad1$$

There is a quick proof here (pdf). The original proof should be in V. E. Hoggatt, Jr., & W. Hansell. "The Hidden Hexagon Squares." The Fibonacci Quarterly 9(1971):120, 133. but I cannot access it.

I am, however, intereseted in a purely combinatorial proof. I do not know how to approach this at all: I cannot see what the non-adjacent vertices represent and/or I do not know how to remodel their meaning. Can anyone help?

EDIT: To specify my question more closely, what I am looking for is some natural bijection between the two sets of triads that create the hexagon.

Thanks.

  • 0
    The Hoggatt and Hansell article has now been brought online: [page 1](http://www.fq.math.ca/Scanned/9-2/hoggatt1-a.pdf), [page 2](http://www.fq.math.ca/Scanned/9-2/hoggatt1-b.pdf). I don't think, however, that it helps with your question.2014-01-28
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    About the BOUNTY: Mitch's answer provides a combinatorial interpretation of each side of the identity, showing that each side counts a certain collection of sets. His answer does not, however, show these two collections to be equinumerous, and therefore is NOT A PROOF. His answer asserts that the two collections are, in fact, the same, which would immediately establish that they are equinumerous, but this assertion is INCORRECT, as demonstrated in the comments. I am offering the bounty in hopes that someone will find a bijection between the two collections.2014-03-16

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