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Problem of the Week 1088

Star of David

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Prove that a Star of David in Pascal's Triangle has the property that the product of the entries in each triangle is the same.

By a Star of David is meant the two small triangles surrounding the binomial coefficient [n, k]. In this diagram, 15 × 35 × 56 = 20 × 70 × 21.

The result, known as the Star of David Theorem, is a variation on this due to H. W. Gould and says that

gcd([n, k+1], [n-1, k-1], [n+1, k]) = gcd([n, k-1], [n+1, k+1], [n-1, k])

© Copyright 2007 Stan Wagon. Reproduced with permission.

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7 December 2007