smudged faces problem

The "smudged faces" problem is a logic puzzle that tests deductive reasoning. In this puzzle, three travelers emerge from a tunnel with soot on their foreheads. They can see each other's faces but not their own. A passenger instructs them to raise their hands if they see at least one smudged forehead on another traveler. All three raise their hands. The passenger then states that anyone who knows for sure if their own forehead is smudged should lower their hand. After a short pause, one traveler lowers their hand, indicating they know their forehead is smudged. This puzzle is solved through a process of elimination and logical inference, demonstrating how individuals can deduce information about themselves based on the actions and inactions of others.12