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Muddy Children and Common Knowledge

Interesting article on common knowledge and a simple form of the red hat problem.

But does he know that I know that he knows?
From No Right to Believe by Ezra Resnick


Five children have been playing together, and three of them have gotten mud on their foreheads. Each child can see mud on others but not on himself. When the children come home, their father says that at least one of them has mud on his forehead; he then asks if anyone can deduce whether there is mud on his own forehead. The children look around, but no one answers. So the father asks again: Does anyone know whether he has mud on his own forehead? Silence. The father then repeats his question a third time, at which point all three dirty children immediately step forward and proclaim that their foreheads must be muddy.