An ant starts on a vertex of a cube. An ant-step consists of the ant choosing one of the vertices of the cube adjacent to its current vertex at random and moving there. (A vertex is adjacent to another vertex if there exists an edge between them.) After three ant-steps, what is the probability the ant ends up on the vertex opposite its starting position?
I am not sure what to do. I know the answer would be 1 over something and the denominator would a multiple of three but I can't figure out anything else. Any help would be appreciated.