Three muddy children and a father who keeps asking

Question

Three children come in from playing. All three have mud on their foreheads. Each can see the others' foreheads but not their own, and there are no mirrors. Their father says: "At least one of you has a muddy forehead. If you know you are muddy, step forward." Nobody moves. He repeats the exact same request. Nobody moves. He repeats it again. On which asking do the muddy children step forward, and why did they wait? The children are perfect logicians and all know that of each other.

Accepted Answer

All three step forward on the third asking. One muddy child. She sees two clean foreheads. Father says at least one is muddy, so it must be her. She steps forward at the first asking. Two muddy children. Each sees one muddy forehead. At the first asking, each thinks: "Perhaps I am clean and the other child is the only muddy one." So neither moves. But then each reasons: "If I were clean, the other child would have seen only clean foreheads and would have stepped forward at the first asking. She did not. So I am muddy." Both step forward at the second asking. Three muddy children. Each sees tw…

Three muddy children and a father who keeps asking

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