Car and Key Hide-and-Seek


I got this problem from Rustan Leino, who was inspired to create it by a puzzle Aistis Simaitis gave him.

I solved it and wrote up my solution.


In a room are three boxes that on the outside look identical. One of the boxes contains a car, one contains a key, and one contains nothing. You and a partner get to decide amongst yourselves to each point to two boxes. When you've made your decision, the boxes are opened and their contents revealed. If one of the boxes your partner is pointing to contains the car and one of the boxes you're pointing to contains the key, then you both win. What strategy maximizes the probability of winning, and what's the probability that you'll win?

Solution     Reveal