Question 16
Let $n \gt 1$ be an odd integer. Let $P_0$ and $P_1$ be two consecutive vertices of a regular polygon with sides $n$. For each $k \ge 2$, $P_{k}$ is defined as the vertex of the given polygon that lies on the bisector of $P_{k-1}$ and $P_{k-2}$. Determine for what values of $n$ the sequence $P_0, P_1, P_2, \dots$ runs through all the vertices of the polygon.