We prove a conjecture of Gérard Rauzy related to the structure of billiard trajectories in the cube : let us associate to any such trajectory the sequence with values in $\{\mathbf {1},\mathbf {2},\mathbf {3}\}$ given by coding $\mathbf {1}$ (resp. $\mathbf {2}$, $\mathbf {3}$) any time the particle rebounds on a frontal (resp. lateral, horizontal) side of the cube. We show that, if the direction is totally irrational, the number of distinct finite words of length $n$ appearing in this sequence is exactly $n^2+n+1$.