Random walk on a polygon

Abstract

A particle moves among the vertices of an $(m+1)$-gon which are labeled clockwise as $0, 1, \ldots, m$. The particle starts at $0$ and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability $p$, or counterclockwise with probability $1-p$. The directions of successive movements are independent. What is the expected number of moves needed to visit all vertices? This and other related questions are answered using recursive relations.