A logician named John went to the Land of Knights and Liars. He met two inhabitants, A and B. He asked A two questions.
John:Has B ever said that you are a liar?
A answered, either "Yes" or "No."
John:Is B a liar?
A answered, either "Yes" or "No."
The next day, another logician named Jack went to the Land of Knights and Liars. He met the same two inhabitants, A and B. He also asked A two questions.
Jack:Has B ever said that you are both liars?
A answered, either "Yes" or "No."
Jack:Is B a liar?
A answered, either "Yes" or "No."
One of the two logicians, John and Jack, could figure out what types A and B were, but the other logician could not figure out their types. What are A and B?
Here is a table of the obvious possible answers. (Hope I made no mistakes!)
There seem to be only 4 cases
A B Q1 Q2 Q3 Q4
- - -- -- -- --
K K N N N N
K L Y Y Y Y
L K Y Y Y Y
L L Y N Y N
It seems that neither logician can determine the types if he gets two answers of Yes.
So, one of them did not get two Yes answers, and the other did!
The only way I can make this work is if B has at some point in the past said that A is a liar, but he has never said that they are both liars. (Probably he was never asked). It can't be the other way around, because saying that they are both liars can be logically viewed as including an assertion that A is a liar.
Then the truth table becomes:
A B Q1 Q2 Q3 Q4- - -- -- -- --K K N N N NK L Y Y N YL K Y Y Y YL L Y N Y N
So, Logician 1 receives two Yes answers and cannot figure out their types.
So Logician 2 is then the one who figured out their types. From his last question, which was answered Y, he knows that A and B are a Knight and a Liar, in some order. If He receives a Yes answer to question 3, then he cannot work out whether B is a Knight or whether he is a Liar who just never said that Both A and B are liars. So he must have received a No answer to question 3. If B were a Knight, then A is a Liar, and he would have received a Yes answer to Question 3 So, logician 2 knows that B is a Liar, making A the Knight.
Final Solution:
A is a Knight, B is a Liar, B has said that A is a Liar, but B never said that they are both Liars.
With a little interpretation (spoiler)
