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 re: Solution -- another way of looking at it by Charlie)

... and the specific groupings have to do with the original directions of cars 1 through N.

In fact, it is possible to identify the original state of a particular bumper car track by a binary ID number:  one digit for each car, and, say, a zero for counterclockwise and a one for clockwise. 

It should be possible to determine how many periods it takes for the bumper car track to get back to its original state just by knowing the binary ID number.  This simple ID number won't tell you the exact starting positions, but I suspect all that matters is the order of the cars and the initial directions.

(where one period is the the length of the track divided by the cars' speed)

Posted by Larry on 2004-12-31 01:54:38