109. 

For each of two people A and B, it is known that they are either a knight or a liar. Suppose A makes the following statement: "If I am a knight, then B is a knight."

Can it be determined who A and B are: who is the knight and who is the liar?

110

A is asked: "Are you a knight?" He replies: "If I am a knight, then I will eat my own hat."

Prove that A will have to eat his hat.
109. These four problems are based on the same idea, which is as follows. Let $P$ be any statement, and let A be any inhabitant of the island of knights and liars. Then if A makes the statement: "If I am a knight, then $P$," he must be a knight, and the statement $P$ must be true! This is hard to believe, and we will prove our surprising assertion in two ways.

1) Suppose that A is a knight. Then the statement "If A is a knight, then $P$" must be true (since knights always tell the truth). Therefore, A is a knight, and it is true that if A is a knight, then $P$. From these two facts, we conclude that $P$ must be true. Thus, taking as premises the assumption that A is a knight, we obtain as a conclusion the statement $P$. Thus (taking into account fact 4 about implication), we have proved that if A is a knight, then $P$. But this is precisely what A asserted! Therefore, A must be a knight. And since we have proved that if A is a knight, then $P$, we conclude that $P$ must be true.
2) Another way to convince ourselves of the truth of our assertion is as follows. Recall that from a false statement, any statement follows. Therefore, if A is not a knight, then the statement "If A is a knight, then $P_{\text {}}$" automatically becomes true and, therefore, could not have been made by a liar. Thus, if someone, about whom it is known that he can be either a knight or a liar, makes such a statement, then he can only be a knight and the statement $P$ must be true.

Let us apply this principle to our problems. Let us start with problem 109. If we take as $P$ the statement "B is a knight," it is clear that A must be a knight, and his statement must be true. Therefore, B is a knight, and we get the answer: A and B are both knights.

In problem 110, let us take as $P$ the statement "A will have to eat his hat." We see that A must be a knight and that he will have to eat his hat. (Thus, it is proved that although knights possess undeniable virtues and righteousness, they can nevertheless be somewhat foolish.)

Answer to problem 111: A is a knight.

The correct conclusion that can be drawn in problem 112: the author is again hoaxing the readers! The conditions of the problem are contradictory: the statement "If I am a knight, then two plus two is five" cannot belong to either a knight or a liar.