A particle performs a random walk on the vertices of a cube. At each step it remains where it is with probability 1/4, and moves to each of its neighbouring vertices with probability 1/4. Let v and w be diametrically opposite vertices. If the walk starts at v, find:
1) the mean number of steps until its first return to v.
2) the mean number of steps until its first visit to w.
The only way I can think of doing this would be to solve some system of 8 simultaneous equations. This seems pretty nasty. How could you do this?