Consider the $k \times (k+1) \times (k+2)$ grid G.
Starting from a point $\vec x = (x_1,x_2,x_3)$, we perform the following random walk. We choose a coordinate $i \in \{1,2,3\}$ uniformly at random and increase that coordinate by 1, i.e. we move to the point $\vec x + \vec e_i$.
Let $P_0(s)$ be the probability of being in the grid G after $s$ random steps when starting from the point $(0,0,0)$. Moreover, let $P_1(s)$ be the probability of being in the grid G after $s$ random steps when starting from the point $(0,1,0)$.
Show that for all $s$, $P_1(s) \ge P_0(s+1)$.