Consider an $n$ times $n$ grid. Assume there are two agents $A$ and $B$.
At time $t_0$, agent $A$ is at square $(n,n)$ and $B$ is at random square. At time $t+1$, agent $B$ has moved one square from his current position to an adjacent square, one left, right, above or below, provided he isn't at the edge. At the same time, $A$ can also move one square in similar fashion or stay put.
Assume that $A$ tries to find $B$, ie. be at time $t$ at the same square $B$ is.
What is the optimal way for $A$ to move?
To me, two strategies strategies come to mind. One is for $A$ to stay in the same square all the time, for the random walked will certainly intercept him at some point. Other would be for $A$ to go on a random walk too. (If this would be human "game", psychologically the second sounds more appealing)