  We consider irreducible Markov chains on a finite state space. We show that
the mixing time of any such chain is equivalent to the maximum, over initial
states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$
starting from $x$. We prove that in the case of the $d$-dimensional torus the
maximum hitting time of moving targets is equal to the maximum hitting time of
stationary targets. Nevertheless, we construct a transitive graph where these
two quantities are not equal, resolving an open question of Aldous and Fill on
a "cat and mouse" game.
