A Knights and Knaves Starter Set

On this island, knights always tell the truth and knaves always lie. You meet two people, A and B. Work out what each one is.

The method

There are only four possibilities for two people. Suppose each one in turn, then look for a contradiction. Whatever survives is the answer. No flash of genius needed.

Puzzle 1

A says: We are both knaves.

If A were a knight, the statement would be true and A would be a knave, which is a contradiction. So A is a knave and the statement is false. That means they are not both knaves, so B is a knight.

Puzzle 2

A says: At least one of us is a knave.

If A were a knave, the statement would be true, and knaves cannot say true things. So A is a knight, and the statement is true. Since A is not the knave, B is the knave.

Puzzle 3

A says: B is a knave. B says: A and I are the same kind.

Suppose A is a knave. Then B is a knight, so B’s statement is true and they are the same kind, which contradicts what we supposed. So A is a knight and B is a knave, and B’s statement is indeed false.

Why bother

Every one of these runs on a move called proof by contradiction: assume something, follow it carefully, and see if it breaks. It works just as well on a budget, a plan or an excuse.