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.