$\textbf{Two Doormen}$
Two doormen are guarding two rooms. One room contains tons of gold and the other is empty. Among these two doormen, one is honest who always tells the truth and the other is a liar who always gives false answers. While they know each other well, you do not know who is honest and who is not. If you are given just one chance to ask one question to one of them, what can you do in order to find out which room contains the gold?
$\textbf{Solution}$
We can ask any of them: "what the other man will say if I ask him which room has the gold?" Whichever the answer is, we shall open the other door. That room will have the gold.
$\textbf{Analysis}$
This is because if we happen to ask the truth teller, he will faithfully tell us that the other person will lie to us. Therefore, the answer we get is false. Meanwhile, if we happen to ask the liar, even though the other person tells the truth, the liar will alter the answer. As a result, the answer we get is false too. This means that, regardless of the scenarios, the answer we receive is always false.
$\textbf{Note}$
Conceptually, the problem is to extract the truth from two uncertain sources with just one question. As a result, we have to associate both sources. This is done by asking one person what the other person will say. Mathematically, the truth teller can be represented as TRUE and the liar can be represented as FALSE. While we do no know who is who, we can be sure that the expression (TRUE and FALSE) is always FALSE. This is the mathematical nature of this problem.