In The Hitchhiker’s Guide to the Galaxy by Douglas Adams, the number 42 is the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was!

Since today is Towel Day, let me reveal that now.

If you try to get several regular polygons to meet snugly at a point in the plane, what’s the most sides any of the polygons can have? The answer is 42.

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. If you do the math, you’ll see the reason this works is that

$latex \displaystyle{ \frac{1}{3} + \frac{1}{7} + \frac{1}{42} = \frac{1}{2} } $

There are actually 10 solutions of

$latex \displaystyle{ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2} } $

with $latex p \le q \le r,$ and each of them gives a way for…

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