Three hats
Alice, Bob, and Charlie are playing a game with hats. Each is wearing a hat that is either red, yellow, or blue. Though there are three possible hat colors, they might not be distinct - perhaps there are three red hats, or two blue and one yellow.
Alice, Bob, and Charlie each look at the other two hats and simultaneously guess their own hat color. They must guess, and cannot convey any other information .
Before they don their hats, they can agree on a strategy. Is there a strategy that guarantees exactly one of them will guess correctly?
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Alice, Bob, and Charlie assign the numbers 0,1,2 to red, blue, yellow respectively.
Alice will guess as if the sum of the hat colors is divisible by 3. For example, if she sees a red hat (0) and a yellow hat (2), she will guess blue (1), since 0 + 2 + 1 = 3 .
Bob will guess as if the sum of the hat colors is one more than a multiple of 3. For example, if he sees a red hat (0) and a yellow hat (2), he will guess yellow (2), since 0 + 2 + 2 = 4 .
Charlie will guess as if the sum of the hat colors is one less than a multiple of 3. For example, if he sees a red hat (0) and a yellow hat (2), he will guess red (0), since 0 + 2 + 0 = 2 .
The actual sum of the hats must be equal to, one more than, or one less than a multiple of 3. Only the corresponding person will guess correctly, so there will be exactly one correct guess with this strategy. :)