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Knights and knaves

Logic and puzzles · level 3, lesson 3 of 6

A question to start

You land on a strange island. Everyone who lives there is either a knight or a knave.

  • Knights always tell the truth. Every sentence a knight says is true.
  • Knaves always lie. Every sentence a knave says is false.

You cannot tell them apart by looking. You meet an islander, and you wonder what they might say about themselves.

Could an islander say "I am a knight"? Yes, anyone could. A knight would be telling the truth. A knave would be lying, which is what knaves do.

Could an islander say "I am a knave"? Try it. If a knight said it, the knight would be saying something false, and knights never do. If a knave said it, the knave would be saying something true, and knaves never do. So nobody on the island can ever say "I am a knave".

Puzzles like these were made famous by the puzzle writer and mathematician Raymond Smullyan. They look like riddles, but they train the most important skill in logic: trying every case and crossing off the impossible ones.

The idea

The two rules, said carefully

For any sentence an islander says:

  • If the speaker is a knight, the sentence is true.
  • If the speaker is a knave, the sentence is false.

And it works backwards too. If you find out a sentence is true, the speaker must be a knight (a knave could not have said it). If a sentence is false, the speaker must be a knave.

A B "We are both knaves."

The method: try both cases

Each islander is one of two things, so you can always try both. Here is the plan.

  1. Pick a person (usually the first speaker).
  2. Case 1: suppose they are a knight. Then their sentence is true. Follow where that leads.
  3. Case 2: suppose they are a knave. Then their sentence is false. Follow where that leads.
  4. If a case leads to something impossible (a knight who lies, a person who is both, a sentence that is both true and false), cross it off.
  5. Whatever survives is the answer. If both cases survive, see what they have in common: that part is certain.

Why does this work? Because the person really is one of the two kinds. If one case is impossible, the real answer must be the other one. It is like a two-door room: if one door is locked, you know which door you came through.

Solving the picture

A says: "We are both knaves."

Case 1: A is a knight. Then the sentence is true, so A and B are both knaves. But that makes A a knave, and we supposed A is a knight. A cannot be both. Impossible. Cross it off.

Case 2: A is a knave. Then the sentence is false. "Both knaves" is false means at least one of them is a knight (remember from lesson 1: the opposite of an and is an or). A is a knave, so the knight must be B.

Answer: A is a knave, B is a knight. Check: A, a knave, said "we are both knaves", which is false because B is a knight. A knave saying something false. Everything fits.

Writing a case table

When there are more people, a table keeps you tidy. Put the cases as columns and follow each one down.

A says: "B is a knave." B says: "A is a knave."

Case 1: A is a knight Case 2: A is a knave
A's sentence "B is a knave" is ... true false
So B is ... a knave a knight
B's sentence "A is a knave" is ... false true
Does B fit? Yes: a knave says something false. Yes: a knight says something true.

Both cases survive! So we cannot tell who is who. But in both cases one is a knight and the other is a knave. So we can be sure: exactly one of them is a knight. Even when a puzzle does not tell you everything, it may tell you something for certain.

Sentences with "or" and "if"

Islanders can use any of the little words from lessons 1 and 2. You just have to use the math meanings.

A says: "I am a knave or B is a knight."

Case 2 first: A is a knave. Then the first part "I am a knave" is true, so the whole or sentence is true (an or needs only one true part). But a knave cannot say something true. Impossible.

Case 1: A is a knight. Then the sentence is true. Its first part, "I am a knave", is false, so the second part must be true: B is a knight.

Answer: both are knights.

Worked examples

1. A three-person puzzle. A says: "All three of us are knaves." B says: "Exactly one of us is a knight." What is each person?

  • Could A be a knight? Then "all three are knaves" is true, so A is a knave. Impossible. So A is a knave, and "all three are knaves" is false: at least one of them is a knight.
  • Now B. Case: B is a knave. Then "exactly one of us is a knight" is false. A and B are both knaves, so C would have to be a knight (someone is). But then exactly one is a knight (C), which makes B's sentence true. A knave saying something true: impossible.
  • Case: B is a knight. Then exactly one is a knight, and that is B. So C is a knave. Check A's sentence: "all three are knaves" is false (B is a knight). Good, A is a knave.

Answer: A knave, B knight, C knave.

2. Certain about the silent one. A says: "B is a knave." B says: "A and C are the same kind." C says nothing.

Case 1: A is a knight Case 2: A is a knave
So B is ... a knave a knight
B's sentence is ... false: A and C differ true: A and C are the same
So C is ... a knave (A is a knight) a knave (A is a knave)

We cannot tell A and B apart, but in both cases C is a knave, so that is certain, even though C never spoke.

3. An island that cannot exist. A says: "B is a knight." B says: "A is a knave."

  • Case 1: A is a knight. Then B is a knight, so B's sentence is true, so A is a knave. Impossible.
  • Case 2: A is a knave. Then B is a knave, so B's sentence is false, so A is a knight. Impossible.

Both cases are crossed off. That means these two sentences cannot both be said on the island. A puzzle like this has no answer, and proving that is an answer too.

4. The two guards. You reach two doors. One leads out, one leads to a pit. Each door has a guard: one is a knight, one is a knave, and you do not know which. You may ask one guard one question.

Ask either guard: "Which door would the other guard tell me leads out?" Then take the other door.

  • If you asked the knight: the knave would point to the pit, and the knight truthfully reports that. You hear "the pit door".
  • If you asked the knave: the knight would point to the exit, and the knave lies about that. You hear "the pit door".

In both cases you hear the pit door, so you take the other one and walk free.

Try it

  1. An islander says: "2 plus 2 is 4." What are they?
  2. An islander says: "2 plus 2 is 5." What are they?
  3. A says: "At least one of us is a knave." What are A and B?
  4. A says: "B is a knight." B says: "A and I are different kinds." What are A and B?
  5. A says: "B and I are the same kind." What can you be sure of?
  6. Explain why nobody on the island can say: "I am a knave and 2 plus 2 is 4."
  7. A says: "If I am a knight, then B is a knight." What are A and B? (Remember lesson 2: an if-then sentence is false only when the if-part is true and the then-part is false.)
  8. Three islanders. A says: "Exactly one of us is a knight." B says: "Exactly two of us are knights." List every possible answer. Is anyone's kind certain?
Answers
  1. The sentence is true, so they are a knight.
  2. The sentence is false, so they are a knave.
  3. Case A is a knave: the sentence is false, so neither is a knave, which makes A a knight. Impossible. So A is a knight, the sentence is true, at least one is a knave, and it must be B.
  4. Case A is a knight: B is a knight, so B's sentence is true, so they are different kinds. But both are knights. Impossible. Case A is a knave: "B is a knight" is false, so B is a knave. B's sentence "different kinds" is false, and indeed both are knaves. It works. Both are knaves.
  5. Case A is a knight: the sentence is true, so B is the same kind: a knight. Case A is a knave: the sentence is false, so B is a different kind: a knight. Either way B is a knight. We cannot tell what A is.
  6. If a knight said it, the sentence would be true, so "I am a knave" would be true. Impossible. If a knave said it, it would be false. But "I am a knave" is true and "2 plus 2 is 4" is true, so the and sentence is true. A knave saying something true: impossible. Both cases fail.
  7. Case A is a knave: the sentence must be false, so its if-part ("I am a knight") must be true. But A is a knave. Impossible. So A is a knight. The sentence is then true and its if-part is true, so its then-part is true: B is a knight. Both are knights.
  8. Try each count of knights. A and B cannot both be knights (they would say different counts, and only one count is right). Case A knight, B knave: exactly one knight, which is A, so C is a knave. B says "two", false. Works: A knight, B knave, C knave. Case A knave, B knight: exactly two knights, so C is a knight. A says "one", false. Works: A knave, B knight, C knight. Case both knaves: A's "one" is false and B's "two" is false, so the number of knights is not 1 and not 2. Only C is left, so the number is 0 or 1. It cannot be 1, so it is 0: all three knaves. That works too. So there are three answers, and nobody's kind is the same in all three: nothing is certain.

Common mistake

Stopping after one case. If you try "A is a knight", find that it works, and stop, you may miss that "A is a knave" works too (as in the case table above). Always try both cases before you say what is certain.

Getting the opposite wrong. When a knave says "we are both knaves", the truth is not "we are both knights". It is "at least one of us is a knight". Use the opposite rules from lesson 1.

Where it leads