Let $\{X_n\}$ be a Markov chain with transition matrix $P$, and $Y_n := X_{m-n}$, $m\ge n$.
Under what conditions is $\{Y_n\}_{n\ge 0}$ Markov chain?
I stared by proving that conditional probability of $Y_{n+1} = y$ depends only on $m, n, y$ and $Y_n$, but I wasn't able to obtain any additional conditions. What else should I consider?