## 34.

Once again, we have three islanders A, B, and C, each of whom is either a knight or a liar. Let's agree to call two islanders of the same type if they are both knights or both liars. Suppose A and B make the following statements:

A: B is a liar.

B: A and C are of the same type.

Who is C: a knight or a liar?
34. Suppose A is a knight. Then his statement that B is a liar must be true, which means B must be a liar. But then B's statement that A and C are of the same type is false, so A and C are not of the same type. Therefore, C is a liar (since A is a knight). Thus, if A is a knight, then $C$ is a liar.

On the other hand, suppose A is a liar. Then his statement that B is a liar is false, which means B is a knight. Therefore, B's statement that A and C are of the same type is true. From this, we conclude that $C$ is a knight (since $A$ is a liar).

Thus, we have proven that regardless of whether A is a knight or a liar, C must be a liar. Therefore, C is a liar.