Ants on a triangle

Question

Three ants sit at the three corners of an equilateral triangle. Simultaneously, each ant picks one of the two edges at its corner uniformly at random and walks along it at the same speed.

What is the probability that no two ants collide? Generalize to $n$ ants on an $n$ -gon.

Accepted Answer

Ants collide unless they all rotate the same way. If all three walk clockwise, or all three walk counterclockwise, each ant follows the ant ahead of it around the triangle and no two ever meet. In any other configuration, some pair of adjacent ants walks toward each other along a shared edge (or into the same corner) and collides: if the directions aren't unanimous, some ant walks counterclockwise while its clockwise neighbor walks toward it. Each ant chooses independently between 2 directions, so there are 2^3 = 8 equally likely configurations, of which exactly 2 (all-clockwise, all-counterc…

Ants on a triangle

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