## 32.

Suppose A and B make the following statements:

A: We are all liars.

B: Exactly one of us is a liar.

Can we determine whether B is a knight or a liar?

Can we determine who C is?
32. We cannot determine who B is, but we can prove that C is a knight.

For the same reasons as in the previous problem, A must be a liar. Therefore, at least one of the other islanders, B or C, must be a knight. Let's find out who B is. He can either be a knight or a liar. Suppose he is a knight. Then his statement that only one of the islanders A and B is a liar is true. The only liar must be A, so C can only be a knight. Thus, if B is a knight, then C is also a knight. On the other hand, if B cannot be a knight, since all three islanders, as we already know, cannot all be knights. Therefore, C must be a knight in any case.