Mean return time for a random walk on cube corners

An ant sits on a corner of a cube. At each step it walks along one of the three edges at its current corner, choosing each with probability $\tfrac13$. Starting from its initial corner, what is the expected number of steps until it is back at that corner for the first time?

$8$. The walk is a random walk on a regular graph with 8 vertices, so its stationary distribution is uniform, $\tfrac18$ per corner. For an irreducible Markov chain the mean return time to a state is the reciprocal of its stationary probability, i.e. $8$. (A distance-from-start chain gives the same.)