Ants on a Triangle

Question

Three ants sit on the corners of a triangle. Each simultaneously 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?

Accepted Answer

Collisions are avoided only if all three ants walk the same way around the triangle: all clockwise or all counterclockwise. Any mixed choice puts two ants on the same edge walking toward each other (head-on) or makes two ants converge on a shared corner... in fact any deviation from unanimity creates one edge traversed in both directions. Each ant chooses independently between 2 directions: 2^3 = 8 equally likely configurations, of which 2 are safe: P = frac28 = frac14. For n ants on an n-gon, the same logic gives frac22^n = 2^1-n. This is the gentlest member of the "find the global structure behind local choices" family, the answer is one observation, and interviewers use it as a warm-up to see if you look for invariants before enumerating.

Ants on a Triangle

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