Expected return time of a random walk on a cube

Question

An ant starts at a corner of a cube and at each step crawls along an edge to one of the three neighbouring corners, chosen with equal probability. What is the expected number of steps until it first returns to its starting corner?

Short answer

$8$ . The walk is uniform over the $8$ corners in the long run, and the mean return time to a state is the reciprocal of its stationary probability, $1/(1/8)=8$ . Equivalently, solve the distance-layer equations for distances $1,2,3$ .

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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