Random walk on a polygon

A particle moves among the vertices of an $(m+1)$-gon which are labeledclockwise as $0,1,...,m$. The particle starts at 0 and thereafter at each stepit moves to the adjacent vertex, going clockwise with a known probability $p$,or counterclockwise with probability $1-p$. The directions of successivemovements are independent. What is the expected number of moves needed to visitall vertices? This and other related questions are answered using recursiverelations.