Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 19H / b / Solution

View each of the three level-one triangles in $G_2$ as a copy of $G_1$. Observe the walk only when it moves from one corner of such a copy to a different corner. By symmetry, the next of the two other corners is equally likely, so this embedded chain is the simple random walk on the coarse graph $G_1$. Part (a) says it makes an expected five coarse transitions before reaching $B$ or $C$.

Within each level-one copy, the mean time for one coarse transition is again five by part (a). The Strong Markov property at successive coarse-corner hitting times therefore gives