Epistemology: Five Questions

Imagine n children playing together. The mother of these children has told them that if they get dirty there will be severe consequences. So, of course, each child wants to keep clean, but each would love to see the others get dirty. Now it happens during their play that some of the children, say k of them, get mud on their foreheads. Each can see the mud on others but not on his own forehead. So, of course, no one says a thing. Along comes the father, who says, “At least one of you has mud on your forehead,” thus expressing a fact known to each of them before he spoke (if k > 1). The father then asks the following question, over and over: “Does any of you know whether you have mud on your own forehead?” Assuming that all the children are perceptive,