= Solution
A <discrete-time Markov chain> with initial distribution $\widehat\pi$ is <reversible> if $(Y_0,\ldots,Y_n)$ and $(Y_n,\ldots,Y_0)$ have the same law for every $n$. Its <detailed balance> equations are $\widehat\pi_i\widehat P_{ij}=\widehat\pi_j\widehat P_{ji}$. These equations are equivalent to reversibility. Indeed, reversing every factor in the <probability> $\widehat\pi_{i_0}\prod_{r=1}^n\widehat P_{i_{r-1}i_r}$ gives the reverse path <probability>; the converse follows by taking $n=1$.
Summing <detailed balance> over $i$ gives
$$
(\widehat\pi\widehat P)_j=\sum_i\widehat\pi_i\widehat P_{ij}=\widehat\pi_j\sum_i\widehat P_{ji}=\widehat\pi_j.
$$
Hence $\widehat\pi$ is an <stationary distribution>.
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