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

ImageWith 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.

  2. I am wondering…. is a_4 correct? I am only able to count 6 not 8…

    • Oh, sorry about that, somehow the left and bottom segments got erased when uploading the picture (which would create the two regions you’re missing). But, as the problem states, all the segments should be included when drawing out the picture.

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