Prisoners in red hats

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Here is different version of the Green eyed dragons puzzle.

Puzzle

Twenty-eight tongueless prisoners awake to see that they are all wearing hats (although they cannot see their own). Although they do not know it, it turns out that all the hats are red. The guard comes in and announces that from now on, once a day, every prisoner will write on a piece of paper either that he does not know what color hat he has, or the color of his hat. If the prisoner correctly writes down his own hat color, he goes free. If he guesses wrong, he is executed.

This continues for many months with no prisoner ever guessing his hat color (or even attempting a guess, for fear of his life). Eventually the guard decides to give the prisoners a hint. He announces, “at least one of you is wearing a red hat.” Twenty-eight days later, the prisoners all go free. How did this happen?

See also