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**Amanda1990** 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?