Knights and Knaves is a basic puzzle used in the fields of mathematics, analysis, or any area of study that utilizes logic. The problem was developed and popularized by mathematician Raymond Smullyan, who is known (well, maybe not) for being prolific in recreational mathematics. The premise of the Knights and Knaves problem is as follows: you’re trying to make your way through an elaborate maze when you come across two ominous, identical doors. A nearby sign informs you that one door leads to a double-headed dragon that will most certainly devour you, while the other leads to riches beyond your wildest dreams and the path to freedom. Complicating the matter are two guards stationed nearby who know where the doors lead; one guard can only tell the truth (the knight) while the other can only lie (the knave). If you’re allowed a single yes/no question between the two of them, how can you figure out which door to pick?
Clearly, you can’t just ask one of the guards whether his door will lead to freedom; the answer will either be yes or no, and there’s no way to tell whether he’s the honest knight or the lying knave. The solution to the dilemma is to ask either man the following question: “What answer will you give me if I asked you whether your door led to freedom?” Phrasing the inquiry this way forces the guard to reveal the truth, regardless of whether he is a knight or a knave. Consider the following:
- If you ask the knight and his door leads to freedom, he’ll say yes and you’ll have the correct answer. If his door leads to doom, he’ll say no and you’ll also have the correct answer.
- If you ask the knave and his door leads to freedom, his response will be “yes”— he’d say “no” if you asked him straight out where his door went, but he has to lie in response to the way you phrased your question. The “double lie” ends up revealing the truth. Similar reasoning applies if his door leads to doom.
The Knights and Knaves puzzle has become a common trope in literature and entertainment, especially in heroic fantasy when a protagonist is required to complete some sort of test of mental acuity. The fantastic movie Labyrinth is one such example. Savvy writers will try to subvert this trope in some manner— a character in the Order of the Stick webcomic bypasses the test by shooting one of the guards in the foot and confirming the validity of his protests. Xkcd presents a variation of the problem with three guards— “one always lies, one always tells the truth, and one stabs people who ask tricky questions.” At its core, the Knights and Knaves puzzle is a great exercise in basic logic and reasoning, and a good opportunity for beginners with mathematical proofs to practice using truth tables and deductive reasoning.