= Solution
Invariant means $\pi P=\pi$. In a finite irreducible chain, $E_iT_i^+=1/\pi_i$ and the expected visits to $i$ between visits to $j$ are $\pi_i/\pi_j$. The first burglar has uniform invariant law on $64$ PINs, so (a) is $64$ and (b) is $16$. First-step equations by Hamming distance give the adjacent-state hitting time (c) as $63$. For the modified one-coordinate chain, destination weights are $1$ for digits $0$–$5$ and $2$ for $6,7$; its invariant weights are proportional to $w_a(10-w_a)$, namely $9$ and $16$. The product invariant law and the occupation formula give (d) $\pi_{(6,7)}/\pi_{(0,0)}=256/81$.
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