I'm trying to do a project that requires me construct a circle with n chords. I need to find the maximum number of regions are obtained. I believe the way to do this is to have every new chord intersect every other chord without three chords intersecting at one point. In other words: is it possible to construct a circle such that all of the chords intersect with each other, there exists a new chord that will intersect each other chord, and no more than two chords are concurrent? After trying some small cases, it seems to me that when I do this, n regions are added to the previous region count. In other words, if f(n) is the maximum number of regions given n chords, then f(n) = n + f(n-1). The problem is I need to prove that n regions are added every time. Any hints are appreciated!