= Solution
Use coordinates $(i,j)\in\{0,\ldots,7\}^2$, with the starting corner $(0,0)$. The bishop stays on the 32 squares where $i+j$ is an <even number>. Its moves form a <random walk on a graph>, with an edge between distinct squares whenever $|i-i'|=|j-j'|$. If $d(i,j)$ is the number of reachable squares, the transition probability to each neighbor is $1/d(i,j)$.
The graph is connected: from $(i,j)$ move along its antidiagonal to $((i+j)/2,(i+j)/2)$, then along the main diagonal to the corner. The midpoint coordinates are <integers> because $i+j$ is an <even number>, and this square always lies on the board. Thus this is the full communicating state space of the <bishop random walk>.
The <invariant distribution> is proportional to degree. Its normalizing constant can be counted by summing the ordered pairs along each diagonal. The diagonals of one orientation have lengths $8,6,6,4,4,2,2$, contributing $144$; the other have lengths $1,3,5,7,7,5,3,1$, contributing $136$. Therefore
$$
\boxed{\pi(i,j)=\frac{d(i,j)}{280}.}
$$
For any neighboring states $s,t$, $\pi(s)P(s,t)=1/280=\pi(t)P(t,s)$, proving the <detailed balance> equations and hence that the chain is a <reversible Markov chain>.
The corner has seven neighbors. <Kac's lemma> gives its expected first positive return time:
$$
\boxed{\mathbb E_{(0,0)}T_{(0,0)}^+=\frac1{\pi(0,0)}=\frac{280}{7}=40\text{ moves}.}
$$
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