Three balls in the plane

Hello everyone,

I'm trying to find the exact velocities of three balls in the plane after they collide elastically. I'm assuming arbitrary positive masses and arbitrary positive radii. Of course, three balls can collide in many ways:

1. in an equilateral triangle: one coming from north, one coming from roughly east south east, one coming from roughly west south west
2. in a "Newton's cradle": two balls stationary, touching; third ball comes in and hits one of them
3. two balls "sandwiching" the third

In such a collision, I know that the sum of the initial momenta will equal the sum of the final momenta, and I know that the initial kinetic energy will equal the final kinetic energy. This gives me two equations to solve. A third equation is needed. This one usually comes from positional considerations and our intuition about the collision: in (1), the north ball will go north, etc; in (2), if L is the line passing through the stationary balls' centers, and the third ball is traveling along L, one ball will stand still (or maybe not), but if the third ball is perpendicular to L, the blow might be glancing or direct, and who knows what happens; in (3), the sandwiched ball will act like a buffer and the momentum equations will pretend it's not there (unless it's moving and happens to be sandwiched at exactly the right time).

The details of each particular case are extensive. My intuition fails in more complicated cases. What are some general principles that I can use to come up with a recipe for solving this in general? For example, is it true that if a few balls are touching, they act like one solid object except they break off "at the extremities"?

I am not satisfied with approximations (e.g. to avoid a 3-at-once collision, nudge one ball very slightly to turn the problem into two 2-at-once collisions). However, I am curious to see whether nudging a ball's position by a small vector ε, solving for the final velocities after the two collisions, and letting ||ε|| → 0⁺, (whether) the final velocities "make sense", as if they were the final velocities correctly calculated after one 3-at-once collision. For this, though, I need to know what the correct final velocities are, in order to actually check that these limiting velocities equal them.

Thank you,
-unit