Möbius prism

This week’s Fiddler is a problem about a multi-sided Möbius prism:

Consider an three-dimensional prism whose bases are regular N-gons. I twist it and stretch it into a loop, before finally connecting the two bases. Suppose that my twist is by a random angle, such that the two bases are aligned when they are coonected. Among all whole number values of N less than or equal to 1,000, for which value of N will a randomly twisted regular N-gon prism have the most distinct faces, on average?

My solution:
[Show Solution]

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