Given a pair of concentric circles, chords $$AB, BC, CD, \ldots$$ of the outer circle are drawn such that they all touch the inner circle. If $$\angle ABC = 75^\circ$$, how many chords can be drawn before returning to the starting point?

Each chord is tangent to the inner circle, so at every vertex the direction of travel turns through the exterior angle $$180^\circ - 75^\circ = 105^\circ$$. Returning to the start after $$n$$ chords requires the total turning $$105n$$ to be a whole number of full turns, so $$105n = 360k$$ and $$n = \frac{24k}{7}$$. The smallest positive integer solution takes $$k = 7$$, giving $$n = 24$$.