An ant returning to its corner

An ant sits on a corner of a cube. Each step it moves along an edge to one of the 33 adjacent corners, chosen uniformly at random. Find the expected number of steps until the ant first returns to its starting corner.

Show hints (2)+
  1. The corners form a 33-regular graph on 88 vertices; the stationary distribution is uniform.
  2. Mean return time =1/π(v)=1/\pi(v), and π(v)=18\pi(v)=\tfrac18 for every corner.

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8

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Asked at: Jane Street, Two Sigma

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