One hundred people stand in a line, all facing in the same direction. Each is wearing a red or a blue hat, assigned at random. Each person can see all the hats before him in the line, but not his own or those of the people behind him. Starting at the back of the line, each person in turn must guess the color of his own hat. Each person can hear all the prior guesses. If the group are allowed to discuss strategy beforehand, how many can be sure of guessing correctly?
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The answer, surprisingly, is 99 — the first guess may be wrong, but everyone else can guess correctly.
Before any guesses are made, each person in line counts the red hats before him. If the total is odd, he prepares the answer “red”; if it’s even (including 0), he prepares the answer “blue.”
Now, while awaiting his turn to speak, each person listens to the guesses behind him. Each time he hears “red,” he switches the color of his own prepared guess from red to blue or back again. When his turn comes to speak, he gives his prepared guess.
(Thanks, Jeff.)
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