Knights and Knaves
Knights and Knaves is a type of logic puzzle where some characters can only answer questions truthfully, and others only falsely. The name was coined by Raymond Smullyan in his 1978 work What Is the Name of This Book?
The puzzles are set on a fictional island where all inhabitants are either knights, who always tell the truth, or knaves, who always lie. The puzzles involve a visitor to the island who meets small groups of inhabitants. Usually the aim is for the visitor to deduce the inhabitants' type from their statements, but some puzzles of this type ask for other facts to be deduced. The puzzle may also be to determine a yes-no question which the visitor can ask in order to discover a particular piece of information.
One of Smullyan's examples of this type of puzzle involves three inhabitants referred to as A, B and C. The visitor asks A what type he is, but does not hear A's answer. B then says "A said that he is a knave" and C says "Don't believe B; he is lying!" To solve the puzzle, note that no inhabitant can say that he is a knave. Therefore, B's statement must be untrue, so he is a knave, making C's statement true, so he is a knight. Since A's answer invariably would be "I'm a knight", it is not possible to determine whether A is a knight or knave from the information provided.
Maurice Kraitchik presents the same puzzle in the 1953 book Mathematical Recreations, where two groups on a remote island – the Arbus and the Bosnins – either lie or tell the truth, and respond to the same question as above.
In some variations, inhabitants may also be alternators, who alternate between lying and telling the truth, or normals, who can say whatever they want. A further complication is that the inhabitants may answer yes/no questions in their own language, and the visitor knows that "bal" and "da" mean "yes" and "no" but does not know which is which. These types of puzzles were a major inspiration for what has become known as "the hardest logic puzzle ever".
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A large class of elementary logical puzzles can be solved using the laws of Boolean algebra and logic truth tables. Familiarity with boolean algebra and its simplification process will help with understanding the following examples.
John and Bill are residents of the island of knights and knaves.
John says "We are both knaves."
In this case, John is a knave and Bill is a knight. John's statement cannot be true because a knave admitting to being a knave would be the same as a liar telling the lie "I am a liar", which is known as the liar paradox. Since John is a knave this means he must have been lying about them both being knaves, and so Bill is a knight.
Same or different kinds
John says "We are the same kind.", but Bill says "We are of different kinds."
In this scenario they are making contradictory statements and so one must be a knight and one must be a knave. Since that is exactly what Bill said, Bill must be the knight, and John is the knave.
Fork in the road
"John and Bill are standing at a fork in the road. John is standing in front of the left road, and Bill is standing in front of the right road. One of them is a knight and the other a knave, but you don't know which. You also know that one road leads to Death, and the other leads to Freedom. By asking one yes–no question, can you determine the road to Freedom?"
This is perhaps the most famous rendition of this type of puzzle. This version of the puzzle was further popularised by a scene in the 1986 fantasy film, Labyrinth, in which the protagonist finds herself faced with two doors with guardians who follow the rules of the puzzle. One door leads to the castle at the centre of the labyrinth, and one to certain death. It had also appeared some ten years previously, in a very similar form, in the Doctor Who story Pyramids of Mars. It was also featured in The Enigma of Kaspar Hauser.
In Labyrinth, the protagonist's solution is to ask: "Answer yes or no. Would he tell me that this door leads to the castle?". As the question is about the other persons response, it will either result in a lie about the other truthfully saying which door leads to freedom or a truth about the other lying about which door leads to freedom. Either way, the response will be the opposite of what a truthful person would say had you asked them whether their door leads to freedom.
Another, perhaps simpler solution is to ask either man if they themselves would tell you that their door leads to freedom. If the man says "Yes", then their path leads to freedom, if he says "No", then it does not. This solution doesn't require that they know the behavior of the other person: only that they know their own.
These solutions are perhaps stated too simply and result in some ambiguity. For instance, if the knight/knave is only asked what they would respond, without presupposing that they would answer or that they were even asked a question, then they might validly respond to a different question to the intended one: "No" might mean that he would not respond because he was never asked.
To resolve this issue, the question could provide a more explicit description of the hypothetical being discussed: "If I asked you "Does your path lead to freedom?" and you responded, would the response be "Yes"?".
These solutions each use a known truth, specifically that the knight must tell the truth and the knave must lie, in the question to ensure the validity of the answer. The same technique can be used to find out any information either man knows. Notably, if all we want to know is whether a man is a knight or a knave, you can test this by simply asking "Is a truth, true?". As a truth is always true, this is a tautology and hence a known truth which we can test against the answer they give.