Knights and Knaves
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Knights and Knaves is a type of logic puzzle devised by Raymond Smullyan.
On a fictional island, 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 what he needs to know.
An early example 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.
In some variations, inhabitants may also be alternators, who alternate between lying and telling the truth, or normals, who can say whatever they want (as in the case of Knight/Knave/Spy puzzles). 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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[edit] Some Examples of "Knights and Knaves" puzzles
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 is a prerequisite to understand the following examples. In particular, to solve Question 2 you must understand that the only way that an "if X then Y" statement can be false is for X to be true and Y to be false.
John and Bill are residents of the island of knights and knaves.
[edit] Question 1
John says: We are both knaves.
Who is who?
[edit] Question 2
John: If Bill is a knave then I am not a knight.
Bill: We are kind of different.
Who is who?
[edit] Question 3
Logician: Are you both knights?
John answers either Yes or No, but the Logician does not have enough information to solve the problem.
Logician: Are you both knaves?
John answers No, and then changes his mind and says Yes. The Logician can now solve the problem.
Who is who?
[edit] Question 4
Here is a rendition of perhaps the most famous of this type of puzzle:
John and Bill are standing at a fork in the road. You know that 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?
- By asking one yes/no question, can you determine whether John is a knight?
This version of the puzzle was further popularised by a scene in the 1980's fantasy film, Labyrinth, in which Sarah (Jennifer Connelly) finds herself faced with two doors each guarded by a two-headed knight. One door leads to the castle at the centre of the labyrinth, and one to certain doom.
It also showed up in the Doctor Who episode "Pyramids of Mars"
[edit] Solution to Question 1
This is what John is saying in a more extended form:
"John is a knave and Bill is a knave."
If John was a knight, he would not be able to say that he was a knave since he would be lying. Therefore the statement "John is a knave" must be true.
Since knaves lie, and one statement is true, the other statement must be false. Therefore the statement "Bill is a knave" must be false which leads to the conclusion that Bill is a knight.
The solution is that John is a knave and Bill is a knight.
[edit] Solution to Question 1, using Boolean algebra
We can use Boolean algebra to deduce who's who as follows:
Let J be true if John is a knight and let B be true if Bill is a knight. Now, either John is a knight and what he said was true, or John is not a knight and what he said was false. Translating that into Boolean algebra, we get:
(because 
)
(because 
)
(by de Morgan's law)
(by the law of distributivity)
Therefore John is a knave and Bill is a knight. Although most people can solve this puzzle without using Boolean algebra, the example still serves as a testament of the power of Boolean algebra in solving logic puzzles.
[edit] Solution for Question 2
John is a knave and Bill is a knight.
In this scenario, John is providing a misleading, conditional equation in which if Bill is a knave then John is also a knave. This is not possible: if Bill were a knave he would be lying when he says "We are kind of different". However, that would imply that John is both a knave and a truth-teller. Clearly they cannot both be knaves.
Rather, the solution is in the conditional statement. John says that if Bill is a knave then John is also a knave. He says nothing about the outcome if Bill were a knight. Therefore, if Bill is a knight, John's statement is irrelevant. Moreover, if Bill is a knight, he must be telling the truth when he says that he and John are different. As such, John must be a knave.
[edit] Solution to Question 4
For finding out which way leads to freedom the following question should be asked: "Will the other man tell me your path leads to freedom?"
If the man says "Yes", then the path does not lead to freedom, if he says "No", then it does. The following logic is used to solve the problem.
If the question is asked of the knight and the knight's path leads to freedom, he will say "No", truthfully answering that the knave would lie and say "No". If the knight's path does not lead to freedom he will say "Yes", since the knave would say that the path leads to freedom.
If the question is asked of the knave and the knave's path leads to freedom he will say "No" since the knight would say it does lead to freedom. If his path doesn't lead to freedom he would say Yes since the Knight would tell you it doesn't lead to freedom.
The reasoning behind this is that, whichever guardian the questioner asks, one would not know whether the guardian was telling the truth or not. Therefore one must create a situation where they receive both the truth and a lie applied one to the other. 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.
[edit] Knights and Knaves puzzles in popular culture
- In the role-playing console video game Final Fantasy VI (released as Final Fantasy III for the US market on the Super NES), the town of Zozo contains a knights and knaves puzzle that rewards the player for resolving it and applying the answer to an item in the town. (Though in this case, everyone in town is a "knave" except one person who says nothing related to the puzzle.)
- Near the end of the PC game The Elder Scrolls Adventures: Redguard, a spin-off of The Elder Scrolls series, the daedra Clavicus Vile poses a simple version of the knight and knaves puzzle for the hero, Cyrus, to solve.
- The PC game Dragonsphere has a variation of the knight and knaves where several faeries constantly alternate colors. A certain faerie may or may not change its state (always lie, always tell the truth) depending on its color. The puzzle is made even more difficult by the fact that they are always moving, making it difficult to identify an individual faerie.
- In the movie Labyrinth, Sarah must solve a knights-and-knaves puzzle to determine which door leads to the castle.
- In the cartoon Samurai Jack, Jack faces a huge two-headed serpent. The heads introduce themselves, claiming that one head always tells the truth and one always lies. They also claim that if Jack is swallowed by one of the heads, he will be transported further along his path, but if he is swallowed by the other, he will be digested in their shared belly. In neither case do they reveal which head is which. Jack solves the puzzle and allows himself to be swallowed by the helpful head. Immediately following, the heads gloat to each other that they are both liars.
- In the Doctor Who episode The Pyramids of Mars, Sarah Jane Smith is imprisoned in a glass cage and the Doctor must solve a knights-and-knaves puzzle (this one consisting of two robots) in order to set her free.
- In the animated series Yu-Gi-Oh!, the main characters were faced with a pair of brothers who claimed that one was a liar and the other a truth teller, but in the end it was revealed that both were liars.
- In the game Kingdom of Loathing, the Naughty Sorceress' lair is, among other things, guarded by four people: One who always lies, one who tells the truth, one who alternates between the two, and one who craves human flesh.
- The Videlectrix game Where's an Egg? is based on this puzzle.
- The point-and-click horror game 'Dark Seed II' has a castle guarded by two monsters who pose a knights-and-knaves puzzle to the protagonist.
- An issue of the webcomic Order of the Stick includes a 'test of the mind' where the characters have to decide which of two figures is giving them correct directions in the manner of the knights and knaves puzzle. This is solved by shooting one of them in the foot, the 'knight' is shocked at this treatment and the 'knave' denies that he has been shot.[1]
- In the webcomic xkcd, there is a comic strip entitled "Labyrinth Puzzle" which is a play on the standard knights-and-knaves puzzle.[2]
- In the webcomic girly, Otra has to face down a variation of the "Two Paths" riddle to lift a curse from Winter.[3]
