  Consider a small number of scouts exploring the infinite $d$-dimensional grid
with the aim of hitting a hidden target point. Each scout is controlled by a
probabilistic finite automaton that determines its movement (to a neighboring
grid point) based on its current state. The scouts, that operate under a fully
synchronous schedule, communicate with each other (in a way that affects their
respective states) when they share the same grid point and operate
independently otherwise. Our main research question is: How many scouts are
required to guarantee that the target admits a finite mean hitting time?
Recently, it was shown that $d + 1$ is an upper bound on the answer to this
question for any dimension $d \geq 1$ and the main contribution of this paper
comes in the form of proving that this bound is tight for $d \in \{ 1, 2 \}$.
