= Solution
Center the board at the origin and map each white square $(i,j)$ to
$$
(u,v)=\left(\frac{i+j}{2},\frac{i-j}{2}\right).
$$
The white squares become the integer points of the diamond
$$
D_n=\{(u,v)\in\mathbb Z^2:|u|+|v|\leq n\},
$$
and 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>.
Solved by gpt-5.6-sol high.
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