Ants on a ring
Suppose you have a circular ring. On the ring are ants, randomly placed and moving clockwise or counterclockwise randomly. Each ant moves at the same constant speed. Once two ants collide, they reverse directions instantaneously keeping the same speed (there is no loss of kinetic energy in the collision). Stating from an initial configuration, which we define by the position of each ant and the direction of its motion (clockwise or counterclockwise), we assume that at a later time each ant regains its initial position, but at this time its direction of motion is inverted compared to the initial configuration. What can we say about the number of ants?
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1 solution
Assuming that the ants' movement is filmed starting from the initial configuration and ending at the moment when the conditions of the problem are satisfied. If the video is played backward, the result should be exactly the same as if it is played forward. Hence, starting from the mid-point of the video, whether the video is played forward or backward, the result should also be the same. To be convinced of this, one should simply imagine that the video is played forward and backward simultaneously. The mid-point of the video is reached at the same time in either the forward played version and the backward one, this is obvious, and the two played versions should look exactly the same. It follows that at the mid-point all the ants are in collision, hence there is an even number of ants.