36 has long been one of my favorite numbers, but faced with this carnival, it was hard to figure out why. It’s a square number that’s a product of two squares, but that’s not too rare. (Why?) It’s the 6th perfect square and the sum of the first six odds, but that’s not too remarkable. (Why?) It’s the 8th triangular number, but not a Sierpinski step or anything… wait! It’s a square triangular number? How common is that? 1, 36, then…?

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Great, a question I can answer without thought (perhaps no one else would even bother)! The fact that 16 is the sum of the first six odd positive integers and is also the sixth square number is not unusual because the nth perfect square is always the sum of the first n odd positive integers. There are a couple of proofs, but the one I like best is an informal visual demonstration in which odd numbers are displayed as Ls, and the squares are build by successively fitting the previous square into the next L (odd number) to make the next square (keeping in mind that 1 is both the first odd number and first perfect square, and that as a single dot, it’s a ‘degenerate’ L and square).

The other question, about numbers that are both triangular and square, is not something I knew off the top of my head. There is plenty of information available on-line about that should people be curious. A quick glance suggests that it leads to some interesting places. Thanks for raising the issue.