The d$-$dimensional hypercube graph Qd is the graph with vertices corresponding to all subsets of $[$d$]$ and two vertices u and v adjacent if and only if their symmetric difference has exactly one element. Equivalently, we can regard vertices as bit strings of length d and vertices are adjacent if and only if they differ in only one coordinate. Consider a random walk on Qd starting at a vertex v$.$ Determine the expected return time to v and the expected number of visits to v$^$c before returning to v$.$