This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$:
Suppose there is a thief at the origin 0 and $N$ policemen at the point 2.
The thief and the policemen began their random walks independently at the same time following the same rule: move left or right both with probability 1/2.
Let $\tau_N$ denote the first time that some policeman meets the thief.
It's not hard to prove $E\tau_1=\infty$.
so what is the smallest $N$ such that $E\tau_N<\infty$?
I was shown this problem on my undergraduate course on Markov chains, but my teacher did not tell me the solution. Does anyone know the solution or references to the problem?