King-move walk returning to the corner in two steps

A piece starts on a corner square of an $8\times8$ board. On each turn it moves to one of the squares adjacent to it horizontally, vertically or diagonally, picking uniformly among the moves that keep it on the board. What is the probability that it is back on its starting corner after exactly two moves?

$\tfrac{7}{40}$. From the corner there are 3 moves: two go to edge squares, each with 5 legal moves, and one goes to the diagonal neighbour, which has 8. Each of those squares has exactly one move back, so the probability is $\tfrac23\cdot\tfrac15+\tfrac13\cdot\tfrac18=\tfrac{7}{40}$.