Three ants are arranged on vertices of a triangle, one ant to a vertex. At some moment, all the ants begin crawiling along the sides of the triangel. Each one crawls along one of the two sides that connect to the vertex it is sitting on, with an equal probability of picking either.

Assuming that all the ants move with an equal speed, and that they keep crawling forever in the same direction along the triangle, what are the odds that no two will collide?

If two ants on adjacent vertices each head toward the vertex the other one is on, they will collide. If they both head toward the third vertex, one of them will collide with the ant who started on that vertex. Only if all three choose to go clockwise, or all three choose to go anti-clockwise will they avoid collisions.

Since each ant can choos either of two directions, and their decisions are independant, there are eight possible outcomes, of which two are collision-free. Therefore the chances of no collision are 2/8 or 1/4 and the odds are 1:3

Posted by TomM on 2002-11-20 17:33:30