Proof Without Words: Square Triangular Numbers and Almost Isosceles Pythagorean Triples Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR Theorem (1, §4.9). Let Tn = 1 + 2 + · · · + n = n(n + 1)/2 denote the nth triangular number. Then Tn = k 2 if and only if (2n + 2k + 1)2 = (n + 2k)2 + (n + 2k + 1)2 . Proof. (Using inclusion-exclusion, shown for (n, k) = (8, 6).) n

n + 2k + 1

n+1

n + 2k + 1

n + 2k

n

n + 2k

n+1

(2n + 2k + 1)2 = (n + 2k)2 + (n + 2k + 1)2 − (2k)2 + 2n(n + 1), (2n + 2k + 1)2 = (n + 2k)2 + (n + 2k + 1)2 ⇐⇒ 4k 2 = 4Tn . Summary. We illustrate wordlessly a one-to-one correspondence between square triangular numbers and almost isosceles Pythagorean triples.

References 1. W. Sierpinski, Pythagorean Triangles. Trans. A. Sharma. Yeshiva Univ., New York, 1962. Republished by Dover, Mineola, NY, 2003. http://dx.doi.org/10.4169/college.math.j.47.3.179 MSC: 05A15, 11D09

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