As I was working away at an assignment today, a group of freshmen were discussing a counting problem nearby which I ended up hearing about. Since I found it interesting, I thought I might just share it with you all.
So we start by considering a circle and, for each consider evenly spaced points along the circle and all the possible segments between them (i. e. as in a complete graph). Now, the question is, how many regions (say ) does this procedure split the circle into?
As a starting point, the following diagram shows the figures obtained for