Continue with funny faces:  One boy, K, wants to find out which girl drew the funny face on his back. He knows that one and only one of his friends, E or M, is a detective who can help him with that. But K does not know what type of person his friend is: a knight who always tells truth, or a knave who always lies.

The three friends then have the following conversation:

• E: I am a knight and M is a knave 
• K asks E: Are you the detective? 
• E: Yes. 
• K asks M: Are you the detective? 
• M whispers to K's ear, either yes or no, but we cannot hear. 
• K: Now I know who is the detective 

Question: what did M say to K (yes or no?) and who is the detective?

Key observation:  Even though we don't hear directly what M said to K, we know that K can infer from M's reply to know for sure who is the detective.  So we can try out each of the two possible replies from M, yes or no, and see which one leads to a definite answer about who is the detective. We do that by filling out truth tables.

If M answers yes, then the truth table is:

actual case	E said		M said
	E is knight and M is knave	E is detective	M is detective
E is detective	T	T	F
M is detective	F	F	T

There are two possibilities: E is the detective or M is the detective, and they are listed in the first column of the table. The other columns show whether each thing that E and M said is true (T) or false (F). If E is the detective, then everything he said is true, so he is a knight, and M is a knave who told a lie. If M is the detective, then E is definitely a knave and M obviously is a knight and the detective. If M said yes to K, then both cases are possible, so K cannot know who is the detective.

On the other hand, if M answers no, then the truth table is

actual case	E said		M said
	E is knight and M is knave	E is detective	M is detective
E is detective	F	T	T
M is detective	F	F	F

If E is the detective, then both E and M are telling the truth when answering K's question: are you the detective. But E also told a lie when he said M is a knave. This situation cannot happen since E is either a knight who always tells truth or a knave who alway lies. So there is only possibility, which is the second case:  M is the detective, and everything E and M said is false. As a result, K now knows that both his friends are knaves and M is the detective.

Homework problem:  After a careful look,  K realized that the funny face must be drawn by only one girl. He suspected that girl is D. He told M about his suspicion and asked him to investigate, even though he knows M is a knave who always lies. M comes back and tells K that

"It is B instead of D who did it."

Question:

1. Did B draw the funny face (yes, no, maybe)?
2. Did D do it (yes, no, maybe)?