Everyone shouts the number of red hats, one must be right

Question

Ten people each wear a red or blue hat. Everyone can see the other nine hats but not their own. At a signal, all ten simultaneously shout out a guess for the total number of red hats in the room (a number from 0 to 10). They may agree on a plan beforehand but cannot communicate once the hats are on. Guarantee that at least one person shouts the exactly correct total.

Accepted Answer

Each person is almost sure of the answer. If you see r red hats on the others, then the true total is either r (if your own hat is blue) or r+1 (if your own hat is red). So every person is torn between just two consecutive numbers, and consecutive numbers have opposite parity (one even, one odd). The plan. Split the ten people into two teams by their pre-agreed labels: people 1 to 5 assume the true total is even, and people 6 to 10 assume the true total is odd. Each person then shouts whichever of their two candidates, r or r+1, matches their assumed parity. Since one of any two consecutive n…

Everyone shouts the number of red hats, one must be right

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