Amy and Bob are back from playing, and their mother notices mud on their foreheads. She says, “At least one of you has a muddy forehead. Do you know whether you have muddy foreheads?” What is the expected answer from each? What happens if she repeats the question?
This is a puzzle in which each child can see the forehead of the other but not themselves. They must also answer simultaneously.
Amy sees mud on Bob’s forehead, so she can’t conclude anything about herself. Had she seen no mud on Bob, she could deduce that she got muddy (as there is at least one). The same is true with Bob, who can’t decide. So they both say NO.
When the mother asked again, the first NO became known to both children. Amy knows that Bob knows that Amy’s forehead has mud and vice versa. So both say YES this time.
Induction Puzzles: Wiki
