A simple model is presented for the motion of a grain down a rough inclined surface with a staircase profile. The model is an extension of an earlier model of ours where we now allow for bouncing, i.e., we consider a non-vanishing normal coefficient of restitution. It is shown that in parameter space there are three regions of interest: (i) a region of smaller inclinations where the orbits are always bounded (and we argue that the particle always stops); (ii) a transition region where halting, periodic and unbounded orbits co-exist; and (iii) a region of large inclinations where no halting orbit exists (and we conjecture that the motion is always unbounded). Fixed points are also found at precisely the inclination separating regions (i) and (ii).