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?

Note that the fact that it's a triangle is irrelevant. The ants could move to random vertices on any $n$-gon and the result would be the same.
Put another way, if two people repeatedly choose random integers from $1$ to $n$, they will eventually choose the same number.
