In lieu of an abstract I offer a problem.

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 Jayadev Athreya and Mauricio Duarte.