Two ants on a triangle puzzle Last Saturday's Guardian newspaper contained the following puzzle: > Two soldier ants start on different vertices of an equilateral triangle. With each move, each ant moves independently and randomly to one of the other two vertices. If they meet, they eliminate each other. Prove that mutual annihilation is eventually assured. What are the chances they survive... exactly N moves. I understand that the probability of a collision on any one move is 1/4, but I don't understand the quoted proof of eventual annihilation: > If the chances of eventual mutual annihilation are are P, then P = 1/4 + 3/4 P, so P = 1. I scratched my head for a while but I still couldn't follow it. Do they mean the probability in the limit of an infinite number of moves? Or is there something crucial in that calculation of P that I'm not getting?

It's a simple geometric distribution (worth a google/wiki), which in and of itself is a justification why eventual annihilation occurs with probability 1. It's worth noting that the question assumes they can pass by each other without dying - i.e. they only die at vertices.

However their argument is simpler; it simply says that at any given stage of the scenario P(eventually die) = P(die on this move) + P(don't die this move)*P(eventually die), which can be solved algebraically as they did. The argument is simply additivity and multiplicity of probabilities.