Spider crawling on a tetrahedron
Question
Suppose a spider is crawling along the edges of a tetrahedron with sides ABCD and edges of length 1. You can assume the spider starts at point A on the tetrahedron and at each vertex chooses its next edge at random (so it has a 1/3 chance of going back along the edge it came on, and a 1/3 chance of going along each of the other two).
Given this information, find the probability that after it has crawled a distance of 5, it is again at point A.