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 in 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".
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 a character finds herself faced with two doors each guarded by a knight. One door leads to the castle at the centre of the labyrinth, and one to certain doom. It had also appeared some ten years previously, in a very similar form, in the Doctor Who story Pyramids of Mars.
One alternative is asking the following question: "Will the other man tell me that your path leads to freedom?" The following logic is used to solve the problem:
- If the question is asked of the knight and the knight's path does lead to freedom, he will say "No", truthfully answering that the knave (the other man) would lie and say "No." If the knight's path does not lead to freedom, he will say "Yes," truthfully answering that the knave will lie and tell us that the path does lead to freedom (when in fact it does not).
- If the question is asked of the knave and the knave's path does lead to freedom, because the knight would say, "Yes, it does lead to freedom," the knave is forced to lie about that fact and will say "No." If the Knave's path does not lead to freedom, since the knight would say, "No," the knave is forced to respond that the knight would say, "Yes."
In either case, if the man says "No", then the path does lead to freedom, if he says "Yes", then it does not. Because one does not know whether they are speaking to a knight or a knave, one must phrase the question in such a way that the answers of "Yes" or "No" ultimately reach the same conclusion despite the fact that one "Yes" (from the Knave) is a lie and the other "Yes" (from the Knight) is a truth. Therefore if they ask the Knight, they will receive the truth about a lie; if they ask the Knave then they will receive a lie about the truth.
Note that the above solution requires that each of them know that the other is a knight/knave. An alternate solution is to ask of either man, "What would your answer be if I asked you if your path leads to freedom?' If the man says "Yes", then the path leads to freedom, if he says "No", then it does not. The reason is fairly easy to understand, and is as follows:
- If the knight is asked if their path leads to freedom, they will answer truthfully, with "yes" if it does, and "no" if it does not. They will also answer this question truthfully, again stating correctly if the path led to freedom or not.
- If the knave is asked if their path leads to freedom, they will answer falsely about their answer, with "no" if it does, and "yes" if it does not. However, when asked this question, they will lie about what their false answer would be. In a sense, they are lying about their lie. They would answer correctly, with their first lie canceling out the second.
This question forces the knight to say a truth about a truth, and the knave to say a lie about a lie, resulting, in either case, with the truth.