Closed Polygonal Chains within a circle The circumference of a circle is divided into N equal parts by N points. How many different closed polygonal chains can be constructed, if they must be made of N segments of the same length, and use all of the N given points as vertices? (A closed polygonal chain is a connected series of line segments that ends at the point where it begins. The line segments may intersect. Two closed polygonal chains are considered identical if one will coincide with the other after rotation.) how many such unique polygonal chains can be formed for any N? (AKA create a general formula) Hint (Given to me by teacher): The formula is directly related to Euler's Totient Function

I think you want the regular star polygons. There's one such $n$-gon for each $k < n/2$ that's relatively prime to $n$, so $\varphi(n)/2$ of them. You build them by stepping through the points by jumps of $k$.

I probably should have just given the last sentence above as a hint. Try drawing them for all possible values of $k$ for various values of $n$.