Knights, Knaves and Truth-Tellers

Knights and knaves puzzles put you on an island where every inhabitant is either a knight, who always tells the truth, or a knave, who always lies. You're given a few statements from a handful of people and asked to work out who is which — or sometimes just to work out one fact, without ever fully identifying everyone. The trick that trips people up is treating a statement as evidence about the world directly ("he says it's raining, so maybe it's raining"). Instead you have to reason about the statement conditional on the speaker's type: if a knight says X, then X is true; if a knave says X, then X is false. You then check which assignment of knight/knave labels makes every statement consistent with its speaker's type.

The classic self-referential trap is a person who says "I am a knave." A knight can't say this, because it would be a true statement admitting to being a knave — contradiction. A knave can't say it either, because it would be a false statement, meaning the speaker actually is a knave — but a knave's statements must be false, so this would make it true, contradiction again. The statement is simply impossible for anyone on the island to make, which is itself useful information in a puzzle where such a line appears attributed to a specific person: it tells you the puzzle intends a paradox device, or that you've mis-parsed who is speaking.

Compound statements are where most solving time goes: "A says: B is a knave, and I am a knight" requires testing A as a knight (both clauses must then be true) and as a knave (the whole conjunction must then be false, which happens if either clause is false) separately, then checking the rest of the puzzle for consistency under each case.