Coxeter groups are groups generated by involutions $s_i$, with the relations of form $(s_is_j)^m=$ id. For each Coxeter group, we will be discussing a particular system of “voracious” paths between any two vertices of the Cayley graph. This system turns out to have the “fellow traveller” property and is generated by a finite state automaton. This is joint work with Damian Osajda.