Some bumper cars are moving around a circular track at the same constant speed. However, they are not all going in the same direction. Collisions are perfectly elastic, so that two colliding cars instantaneously change directions (and continue at the same speed).

Show that at some point in the future, all the cars will be back to their starting positions and directions. Assume that each car has no length.

(In reply to Solution by Larry)

After imagining the cars to pass through each other, and allowing one revolution-time to pass after the start, the cars identities will be permuted in some fashion. 

Any permutation can be broken down into cycles, such as (1,2,3), (4,5) (6) just to take an example.  This means car 1 is in the position and direction previously (one turn before) occupied by car 2; car two is in the situation (state) previously occupied by car 3; car 3 is in the state previously occupied by car 1.  Also cars 4 and 5 have switched states, and car 6 is in the same position and direction as before.

After a number of revolution intervals has passed equal to the LCM of the cycle lengths, the identities of the cars will be the same as at the start, in the given positions and directions.

Posted by Charlie on 2004-12-30 18:58:58