Solution
= Solution
For a finite irreducible <Markov chain> with <invariant distribution> $\pi$, the <mean recurrence time> of state $i$ is
$$
\mathbb E_iT_i^+=\frac1{\pi_i}.
$$
The <simple random walk on the hypercube> has the uniform invariant distribution because the <hypercube graph> is regular, so every vertex has mass $2^{-n}$. Therefore the expected first positive return time to the initial vertex is
$$
\boxed{2^n.}
$$