Consider two particles $p_1$, $p_2$ performing a uniform random walk on the integers $[0,n]$ (the integers $0, 1, ..., n$), under the following modification: at every time-step, $p_1$ selects uniformly at random whether to step left or right, and then attempts to step to that location. If $p_2$ was already in that location, then $p_1$ instead stays put. Then, $p_2$ performs the same procedure, with respect to $p_1$.
The particles start at $0$ and $1$ respectively, and $n$ is an absorbing state in the sense that when a particle steps on it, it is eliminated.
The question is: how long will it take for both $p_1$ and $p_2$ to get eliminated in such a system?