Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 1 / a / Solution

Center the board at the origin and map each white square $(i,j)$ toThe white squares become the integer points of the diamondand a bishop move changes exactly one coordinate. From $(u,v)$ the chain can move first to $(u,0)$ and then to $(0,0)$, since both points lie in $D_n$. Reversing such paths connects any two states, so the bishop random walk is an irreducible Markov chain.