Told that the number of red hats is odd

Question

Three people sit in a circle. Each wears a hat that is either red or blue, and each can see the other two hats but not their own. A referee, who can see all three, announces truthfully: "The number of red hats is odd."

Who can now work out their own hat colour, and how? Does the answer change if the referee had instead said the number of red hats is even?

Accepted Answer

All three can name their own colour immediately. The reasoning. Suppose you see r red hats on the other two people, where r is 0, 1 or 2. The total number of red hats is r plus 1 if you are red, or r if you are blue. You are told the total is odd. - If you see an even number of red hats (0 or 2), the total can only be odd if your own hat is red. - If you see an odd number of red hats (exactly 1), the total can only be odd if your own hat is blue. Every person can do this, so every person knows their colour at once, and no waiting or listening is needed. If the referee had said "even". Exactly…

Told that the number of red hats is odd

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