This is from Reif's Statistical and Thermal Physics:
Two drunks start out together at the origin, each having equal probability of making a step to the left or right along the x-axis. Find the probability that they meet again after N steps. It is to be understood that the men make their steps simultaneously. (It may be helpful to consider their relative motion).
F. Reif has put you on the wrong foot by saying that "It is to be understood that the men make their steps simultaneously". It is not really necessary that the two steps are done simultaneously, you can just as well consider two random walkers where one takes his step just after the other. One complete step consists of both walkers making their moves. Suppose that you and I are these random walkers. You always start first and then I make my random move.
After your move, your relative position w.r.t. me changes by plus or minus one step. Then I make my move. After my move, the relative position again changes by plus or minus one step. As far as ...
A detailed solution is given.