I have this question which I am not sure how to solve:
One hundred indistinguishable ants are dropped on a hoop of diameter 1. Each ant is traveling either clockwise or counterclockwise with a constant speed of 1 meter per minute. When two ants meet, they bounce off each other and reverse directions. Will the ants ever return to their original configuration? After how many minutes?
There is a well-known trick with these ant problems: instead of them bouncing off each other, just pretend that they pass by each other. So it looks like each ant moves with a constant speed without changing direction. Now it's easy to see that the original configuration will recur every $\pi$ minutes.