Knights & Knaves

On this island, everyone is one of two kinds — and you can only hear what they say.

Knights always tell the truth. Knaves always lie.

Tip: click a speech bubble to hear the islander speak — A and B have different voices.

Puzzle 1 — warm-up

One islander speaks. Knight or knave?

"Two plus two equals five."

Your vote: is A a knight or a knave?

"Two plus two equals five" is false — and only knaves lie. So A is a Knave.

Puzzle 2 — two islanders

A speaks about the pair of them. B stays quiet.

"At least one of us is a knave."

…says nothing…

Your vote: which is the knight, which is the knave?

If A were a knave, "at least one of us is a knave" would be true — but knaves can't tell the truth. So A is a Knight, the statement holds, and that makes B the Knave.

Puzzle 3 — the twist

Now they talk about each other. Reason carefully.

"B is a knight."

"A and I are of different types."

Your vote: who's who?

Suppose A is a Knight. Then "B is a knight" is true, so B is a Knight too — but then B's "we're of different types" would be true, making them different. Two knights can't be different types: contradiction. So A can't be a Knight — A is a Knave. Then "B is a knight" is a lie, so B is a Knave. Check B: a knave saying "we're of different types" is lying, so they're the same type — both knaves. ✓ Everything fits: both are Knaves.

Puzzle 4 — three islanders

Harder: three of them, all talking. Follow the chain — assume, and watch it break.

"B is a knave."

"C is a knave."

"A and B are of the same type."

Your vote: label all three.

This one is yours to crack — no answer today. Reason it through after class, post your argument on Ed Discussion or bring it to office hours, and we'll settle it next time.

(instructor — show solution)

Try A is a Knight. Then "B is a knave" is true, so B is a knave — so B's "C is a knave" is a lie, making C a Knight — so C's "A and B are the same type" must be true. But A is a knight and B a knave: different types. Contradiction.
So A is a Knave. Then "B is a knave" is a lie, so B is a Knight; B truthfully says "C is a knave," so C is a Knave; and C lies that "A and B are the same type" — indeed they differ. ✓ Everything fits: A knave, B knight, C knave.