Moser's circle problem (Solution)

This is a very pretty problem. It’s also fairly tricky. If you said the answer was \(2^{n-1}\), that’s not correct! If you want, stop reading and try a larger n. If you don’t beleive me, for n = 21, there are 6196 regions! See, just count them up:

I highly doubt I’ll do a better job of explaining the solution to this problem than 3blue1brown, so probably just go watch this video:

In case you don’t have 9 minutes to watch the video, but want to know whether you’re right, here’s the solution:

\[f(n) = 1 + {n \choose 2} + {n \choose 4}\]

\({n \choose 2}\) is the number of line segments connecting every pair of \(n\) points, and \({n \choose 4}\) is the number of intersections of those lines (not counting any intersection points on the circumference of the circle).

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Here’s an interactive version of the picture above. Move the slider to change the number of points on the circle. It’s backed by this Observable notebook.