Several years ago, I think in 2002, there was a Mandelbrot contest problem that asked for the largest number of regions in the plane that could be generated by two triangles. The answer turns out to be 8, and it’s interesting to think about why there can’t be more.

Naturally, this got me wondering about the integer sequence: what is the largest number of regions in the plane you can generate using two n-gons? At first this problem seems very simple: 8, 10, 12, 14, …

But after a little while, I realized it went 8, 18, ?, 38, ?, …, where the number of regions for two 2n-gons is easy to compute. You can probably figure out what I was assuming that wasn’t actually a requirement.

For odd numbers of sides, things are much more complicated. In fact, to this day I’m still not sure of the answer. I have a drawing that I think gives the maximum number of regions, but I haven’t been able to prove that there isn’t a better drawing out there somewhere. Anyone have any ideas for me?

Problems like this one are appealing to me because they are easy to play with but hard to finish. There are more elementary ones (like regions created by lines), classics (like regions in space determined by planes, as in Polya’s famous video), interesting side notes (ever wonder why we only draw Venn diagrams with 3 circles and not with 4? Have you seen diagrams with 4 sets — and noticed that they don’t use circles?), and difficult ones like the case of two odd-sided polygons.

Maybe this would make a good topic for a future Numberplay column.

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