Recipe

Solve a knights-and-knaves puzzle

Use when: every speaker either always tells the truth or always lies, and you must decide who is which from what they say

Goal: the one assignment of knight or knave to each person that makes every statement consistent with its speaker

Taught in Knights and knaves

1. 1

   Let each person's initial stand for "this person is a knight." Its negation means "this person is a knave."

   Example

   Ana and Ben: A = Ana is a knight, B = Ben is a knight. "Ben is a knave" is \neg B.

   Why: everyone is one of the two kinds, never both and never neither, so one letter per person captures everything.

2. 2

   Write what each speaker says as a formula over those letters.

   Example

   Ana says "At least one of us is a knave": \neg A \vee \neg B.

   Why: the claims have to be in symbols before you can check them against the speakers' kinds.

3. 3

   For a statement S by Ana, write A \leftrightarrow S.

   Why: a knight's claims are true and a knave's are false, which is the same as "Ana is a knight if and only if S."

4. 4

   Pick a speaker, often the one who talks about the most people. Try "knight" and then "knave," and follow each case.

   • the case forces a clash — cross it off.
   • the case settles everyone with no clash — keep it as a candidate answer.

   ⚡ Shortcut

   With two or three people, a truth table of the speaker rules does the same job mechanically: keep the rows where every rule is true.

   Why: every person is a knight or a knave, so checking both cases covers every possibility.

5. 5

   For each speaker, confirm that a knight's claim comes out true and a knave's comes out false.

   Why: a case can look finished while one statement you never used is broken.