Is My Forehead Muddy?

Solution

A father tells his children, a boy and a girl, to play in the backyard without getting dirty, but they play in the mud anyway. Both children get mud on their foreheads. The father says, “at least one of you has mud on his or her forehead, which I asked you not to do. I’m going to take away desert tonight, but I’ll give you a chance to get your desert back.” He tells them he will ask the following question twice: do you know if you have mud on your forehead? Each child must answer honestly and simultaneously each time the question is asked. Each child can see the other’s forehead but not his or her own. If both children can correctly tell whether or not they have mud on their foreheads, they can have desert that night. What should the children do?

Solution

When the father asks the question the first time, “do you know if you have mud on your forehead,” both children will answer “no.” Each child sees the mud on the forehead of the other one. Since they know at least one of them has mud on his forehead, neither child can conclude whether he also has mud on his own forehead. They must answer “no” the first time the question is asked.

However, after the question is asked and answered, the girl will immediately conclude she must have mud on her forehead since the boy answered “no.” The boy will also conclude that he must have mud on his forehead for the same reason. So, when the father asks the same question again, they both answer “yes” and enjoy desert that night.