Ants on a triangle

Question

Three ants sit at the three corners of a triangle. At the same instant, each ant picks one of the two edges leaving its corner uniformly at random and starts walking along it at the same speed.

What is the probability that none of the ants collide?

Follow-up: what about $n$ ants on the corners of an $n$ -gon?

Accepted Answer

Each ant independently chooses one of two directions, so there are 2^3 = 8 equally likely combinations of choices. Think of the directions as clockwise (CW) or counterclockwise (CCW) around the triangle. An ant walking CW along its edge won't crash into a neighbor as long as that neighbor is also going CW, they chase each other around without meeting. A collision happens whenever two adjacent ants walk toward each other, which occurs whenever the ants are not all circulating the same way. No collision happens in exactly two cases: - All three ants go clockwise, or - All three ants go counterc…

Ants on a triangle

Your answer

More Brainteasers questions