Knights and Knaves Puzzles

What Is The Name of This Book?These puzzles have to do with a strange island inhabited by two types of people: people who only tell the truth (knights) and people who only tell lies (knaves). Raymond Smullyan collected dozens of puzzles like this in his book, What is the Name of This Book?. The following list progresses quickly from easy to very difficult. When you have mastered these, try the Hardest Logic Puzzle Ever.

1. Two people, Red and Blue, stand before you. Red says, “We are both knaves.” What are they really?

See Answer
Red cannot be a knight, because then he would be lying by saying he is a knave. Therefore he is a knave, and his statement is a lie, meaning that Blue must be a knight.

 

Knights or Knaves2. Two people again. Red says, “We are both the same type of people,” and Blue says, “We are each different types of people.” What are their identities?

See Answer
As their statements contradict, it must be that one is a knight and one a knave. Therefore Blue is the knight.

 

3. You come to a fork in a road and need to learn which path leads to your destination. Two individuals are standing there. You know that one is a knight and one is a knave, but you don’t know which is which. What single yes-or-no question could you ask of one of the islanders to learn the correct path?

See Answer
Point to a path and ask of one, “Would the other one tell me that this is the correct path?” If the answer is No, that is the correct path; if Yes, it is the incorrect path.

 

4. Now you encounter a fork in the road but there is only one inhabitant standing there, and you do not know whether he is a knight or a knave. What one yes-or-no question could you ask to learn the correct path?

See Answer
Point to a path and ask, “If I asked you if this is the correct path, would you say yes?” You may treat a “Yes” answer as indicating that the path is correct and “No” meaning that it is the incorrect path. Here’s why:

First of all, suppose that the person is a knight and the path is correct. The knight answers yes. If the person is a knight and the path is not correct, then the knight will answer no. If the person is a knave and the path is correct, then the knave would lie and say no if asked. However, you are asking if he would say yes if asked, so he must lie and say yes. If the person is a knave and the path is incorrect, then the knave would lie and say yes if asked. But you are asking if he would say yes if asked, so he must lie and say no.

 

Three Knights or Knaves5. To complicate matters, a new island is discovered, with a third type of inhabitant: spies, who can either tell the truth or lie. You encounter three people, one of each, but you don’t know which is which. Red says, “I am a knight.” Blue says, “Red is telling the truth.” Green says, “I am the spy.” Who is what?

See Answer
Could Blue be lying? That would mean that Red is also lying, so Green would have to be the knight. But Green cannot be the knight, because the knight would not lie about who he is. Therefore Blue is telling the truth, so Red is the knight. Since Blue is a non-knight who told the truth, he must be the spy, making Green the knave.

 

6. Now add the wrinkle of language difficulties. On yet another island, the inhabitants are all either knights or knaves, and though they understand English, they will not utter the words Yes and No. They use “Ja” and “Da,” but we do not know which means Yes and which means No. Upon encountering an inhabitant of this island, you ask, “Does Ja mean Yes?” He answers, “Ja.” Have you learned what Ja means? Have you learned whether the person is a knight or a knave?

See Answer
You have not learned the meaning of Ja and Da, but you do know that the person is a knight. If Ja means yes, then the knight would answer Ja (yes). If Ja does not mean yes, then the knight would answer Ja (no). If the person were a knave and Ja meant yes, he would lie and say Da (no). If Ja did not mean Yes then he would lie and say Da (yes).

 

7. You encounter another inhabitant of the same island and are determined to discover the meanings of Ja and Da. What one yes-or-no question can you ask in order to learn this?

See Answer
Simply ask, “Are you a knight?” No matter whether the inhabitant is a knight or a knave, he will answer with the word for Yes.

 

8. Suppose that you have not determined the meanings of Ja and Da, but you encounter one inhabitant of the same island at a fork in the road and you must find out the correct path to take. Can you do it with one yes-or-no question?

See Answer
You can force the inhabitant to indicate the correct path, even without learning the person’s identity or the meaning of Ja and Da. Point to a path and ask, “If I asked you if this was the correct path, would you say Ja?” If the answer is Ja, that is the correct path. If the answer is Da, it is not the correct path. Here’s why:

If the person is a knight and it is the correct path and Ja means Yes, then he will answer Ja. If the person is a knight and it is the correct path and Ja means No, then he will answer Ja (No I would not answer no).

If the person is a knight and it is the wrong path and Ja means yes, then he will answer Da (no). If the person is a knight and it is the wrong path and Ja means no, then he will answer Da (yes I would say no).

If the person is a knave and it is the correct path and Ja means yes, then he will lie and answer Ja (yes I would say yes). If the person is a knave and it is the correct path and Ja means no, then he will lie and answer Ja (no I would not say no).

If the person is a knave and it is the wrong path and Ja means yes, then he will lie and answer Da (no I would not say yes). If the person is a knave and it is the wrong path and Ja means no, then he will lie and answer Da (yes I would say no).

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