# A little counting problem

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 $n\geq 2$ consider $n$ 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 $a_n$) does this procedure split the circle into?

As a starting point, the following diagram shows the figures obtained for $n = 2, 3, 4, 5, 6, 7$

With this, we may see that $a_2 = 2, a_3 = 4, a_4 = 8, a_5 = 16, ...$ and so on and so forth.

1. I will just add that $a_6 = 30$ so all the excitement of $2^{(n-1)}$ is gone :(. Still don’t have a convincing proof for the temptative formula I have, though.