For a finite irreducible Markov chain, what is the relationship between the invariant probability distribution and the mean recurrence times of states?

A particle moves on the $2^{n}$ vertices of the hypercube, $\{0,1\}^{n}$, in the following way: at each step the particle is equally likely to move to each of the $n$ adjacent vertices, independently of its past motion. (Two vertices are adjacent if the Euclidean distance between them is one.) The initial vertex occupied by the particle is $(0,0, \ldots, 0)$. Calculate the expected number of steps until the particle

(i) first returns to $(0,0, \ldots, 0)$,

(ii) first visits $(0,0, \ldots, 0,1)$,

(iii) first visits $(0,0, \ldots, 0,1,1)$.