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muddy children problem

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A logical problem illustrating that all members of a group may know a proposition (1) to be true without that proposition being common knowledge in the group. Suppose that all children in a classroom can see one another's foreheads but not their own, and that exactly two of them have muddy foreheads. The teacher asks all those who know they have muddy foreheads to put up their hands. The two muddy children, who are assumed to be perceptive, intelligent, and honest, have no reason to own up. The teacher then repeats the request a second time, and the two muddy children still have no reason to own up. Now suppose that the teacher had first announced to the class as a whole that at least one child in the class is muddy, turning a proposition that every child already knows to be true—because every child can see at least one muddy child—into an item of common knowledge. When the teacher asked all those with muddy foreheads to put up their hands, the two muddy children would not own up the first time the question was asked, but they would both own up when it was repeated a second time, because at that point each would realize that the failure of the other muddy child to own up the first time implied that there must be two muddy children in the class. It can be proved that if n of the children are muddy, then they will all own up when the question is asked for the nth time. The problem is sometimes cast in terms of a group in which some members have cheating spouses that the others all know about, and it is also called the unfaithful wives problem or the cheating husbands problem. It is a classic problem in epistemic reasoning.

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