Ants On A Circle I

Question

Nine ants are placed at equal spacing around a circle. Each ant independently chooses clockwise or counterclockwise and then moves at constant speed so that, if uninterrupted, each would make exactly one full revolution in one minute. When two ants meet they instantly reverse direction and continue at the same speed. The ants are indistinguishable, so we only care about which geometric points around the circle are occupied, not which ant is which. What is the probability that after one minute the set of occupied points is exactly the same set of nine starting points?

QuantSoc · Accepted Answer

Instead of thinking that ants reverse direction at collisions, imagine they pass through one another without changing direction. Since the ants are indistinguishable, this makes no difference. Under this pass-through picture each ant follows a continuous path at the same constant speed and therefore makes exactly one full circuit in one minute, so every ant ends up at the same geometric point where it started. Note that this does not necessarily mean each individual ant returns to its original position; it only means that the set of points occupied by the ants initially and finally is the same. Therefore the required probability is 1.