1. The problem statement, all variables and given/known data Suppose that five particles are traveling back and forth on the unit interval [0,1]. Initially, all five particles move to the right with the same speed. (The initial placement of the particles does not matter as long as they are not at the endpoints.) When a particle reaches 0 or 1, it reverses direction but maintains its speed. When two particles collide, they both reverse direction (and maintain speeds). How many particle-particle collisions occur before the particles once again occupy their original positions and are moving to the right? 3. The attempt at a solution I've drawn out the process if all particles are equal distances apart (1st one at 1\6, 2nd at 2\6, 3rd one at 3\6, 4th one at 4\6, and 5th one at 5\6), and counted the collisions. I've decided the answer is 20, but I don't know how to prove it. If I just write down the case I tried, then it only proves it for that one case, and that's no good. Does anybody have any suggestions as to where to start on this proof?