# Meta Knights and Knaves Puzzle with Hats

I landed on the island of Knights and Knaves where Knights always tell the truth and Knaves always lie. I set out to find their two leaders, Raymond and Martin and I really wanted to talk with Martin.

I found them both at the same time, one in a Red hat and one in a Blue hat but I was not sure who was who and I was not sure which was Knight or Knave or both or neither.

I asked both of them: "Is Raymond a Knight?" Only the one with the blue hat answered, and I can't remember what he said. But I do remember that at that point I was able to tell who was Raymond and who was Martin.

What color was Raymond's Hat?

• I do not think that this question has an answer, because as long as the one with the blue hat answers with yes or no you cannot get any usefull information out of his answer. Oct 7 '15 at 7:57
• You can get information from the story based on the fact that I was able to tell who was Raymond and who was Martin, hence the meta part of it. Oct 7 '15 at 8:01
• This problem only makes sense if you assume the phrase "their two leaders" means the entire island has two leaders, rather than the more obvious interpretation that "their two leaders" are the leaders of the two factions, who must therefore be a knight and a knave. The "or both or neither" tells you the latter interpretation is wrong, but the concept of a single group having two leaders is at odds with the meaning of the word "leader". Oct 8 '15 at 15:00

If the Blue hat guy answered YES (Raymond is a knight), then there were many compatible scenarios:

• Raymond indeed was a knight, and Blue=knight=Raymond
• Raymond indeed was a knight, and Blue=knight=Martin
• Raymond was a knave, and Blue=knave=Raymond
• Raymond was a knave, and Blue=knave=Martin

If the Blue hat guy answered NO (Raymond is a knave), then there are essentially two compatible scenarios:

• If Blue=knight, then Raymond=knave and hence Raymond=red
• If Blue=knave, then Raymond=knight and hence Raymond=red

Answer: Since you were able to tell who was Raymond and who was Martin, the Blue hat guy must have answered NO, and Raymond was the Red hat guy. (But you could not deduce knighthood and knavehood of Raymond and Martin.)

Analytical approach:

There are 8 possible scenarios based on three variables.

1. Red=Raymond=Knight; Blue=Martin=Knight. Blue must answer "yes" to the question "Is Raymond a knight?" They both tell the truth; both are knights.

2. Red=Raymond=Knight; Blue=Martin=Knave. Blue answers "no" - Raymond is a knight but Martin (the speaker) is a liar.

3. Red=Raymond=Knave; Blue=Martin=Knight. Blue answers "no" - Martin (the knight) is telling the truth about Raymond.

4. Red=Raymond=Knave; Blue=Martin=Knave. Blue answers "yes". They are both liars, so one liar confirms that the other is a knight, i.e. lies.

5. Blue=Raymond=Knight; Red=Martin=Knight. As per scenario 1, they are both truth-speakers so the speaker says "yes".

6. Blue=Raymond=Knight; Red=Martin=Knave. Raymond says "yes", he himself is a knight.

7. Blue=Raymond=Knave; Red=Martin=Knight. Raymond says "yes", he is lying about himself.

8. Blue=Raymond=Knave; Red=Martin=Knave. Raymond says "yes" as per 7.

There are only two scenarios where the blue-hatted one says "no". In both of those scenarios, Raymond's hat is Red. Since I was able to tell who was who, I must have got the answer "no" from the blue-hatted one, which indicated to me that Raymond was the red-hatted one. If the answer had been "yes", I wouldn't have been able to make that decision.

He can only say yes if he's Raymond himself, because it'd lead to a contradiction otherwise. Similarly, he can only say no if he's not, so we can deduce who's who based on the answer.

If Blue answers yes, then there is no way finding out who wears which hat (for example, both can be knights and can wear either hat without causing a contradiction). But I did know who wore which hat, so Blue must have answered no. If Blue answers no, then he cannot be Raymond, because then Raymond would effectively be stating that he is a knave, and neither a knight nor a knave can do that. So Raymond must wear the Red hat, but we do not know knight/knavehood of Raymond and Martin.

Or, since the dude in the red hat did not answer, perhaps it was because

he did not know and therefore did not know how to answer truthfully or untruthfully depending on his normal inclination. Therefore, he could not be Raymond, as Raymond would have surely known which he himself was. That means Raymond wore the blue hat.

The fact that you couldn’t remember what the guy in blue said doesn’t matter, since you got your answer regardless. However, based on this

the only answer blue could have given was “yes”, as “no” leads to a contradiction if blue is in fact Raymond.

Therefore,

blue said “yes” and we don’t know if he is a knight or a knave, but you really don’t care since you wanted to talk to Martin anyway, who is the gentleman in the red hat, assuming he actually is ABLE to speak, otherwise his not speaking didn’t tell you anything.