Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 20H / c / Solution

Let $v_i$ be the vertex $i$ steps from $B$ toward $C$, and let $x_i$ be the mean time to reach $v_{i+1}$ from $v_i$. Part (b) gives $x_0=4n+1$. For $1\le i<n$, first-step analysis at $v_i$ givesbecause a step to $v_{i-1}$ must be followed by a fresh passage of mean $x_{i-1}$ back to $v_i$ before trying again. HenceBy the Strong Markov property, the expected time from $B$ to $C$ is the sum of these successive passage times:Adding the initial mean time $n^2$ from $A$ to $B$ gives