Each of three people is wearing either a red hat or a blue hat. Each can see the color of the others’ hats but not her own. Each is told to raise her hand if she sees a red hat on another player. The first to guess the color of her own hat correctly wins.
All three raise their hands. A few minutes pass in which no guesses are made, and then one player says “Red” and wins. How did she know the color of her hat?
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All three players raised their hands, so each can see at least one red hat. This means that at least two of the hats are red; if two or more were blue then there’d be at least one player who didn’t raise her hand. But any player who can see a blue hat can immediately infer that her own hat must be red, because she can see a red-wearing player whose hand is raised. In the puzzle this doesn’t happen: All three players raise their hands and yet none of them makes this inference. That allows one of the players to conclude that none of them is wearing a blue hat; all three hats must be red.
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