Two players, two hats, one of you must be right

Question

Alice and Bob each have a hat placed on their head, red or blue, chosen by a coin flip for each. Each can see the other's hat but not their own. At the same moment, without communicating, each must guess their own hat colour.

They win if at least one of them is right. They may agree a strategy in advance.

Is there a strategy that wins every time, no matter how the hats fall? If so, what is it, and why does it work?

Accepted Answer

Yes. Alice guesses that her hat is the same colour as Bob's. Bob guesses that his hat is the opposite colour to Alice's. Why it works. There are two kinds of world: the hats are the same colour, or they are different. - If the hats are the same, Alice, who guessed "same as Bob's", is right. - If the hats are different, Bob, who guessed "opposite to Alice's", is right. One of those two cases must hold, so at least one player is always right. In fact exactly one is right each time: the one whose "theory of the world" is correct. What is really going on. Neither player has any information about …

Two players, two hats, one of you must be right

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