On number of collisions of billiard balls

Consider three billiard balls of the same radius and mass, undergoing totally elastic reflections on a billiard table with no walls (the whole plane).  All three balls can be given non-zero initial velocities.  What is the maximum (supremum) possible number of collisions among the three balls?  The supremum is taken over all initial positions and initial velocities.  I will discuss this problem and its generalization to any finite family of balls in one, two and higher dimensions.

Joint work with Mauricio Duarte.