In a remote island there are only two types of people that live on it, Knights and Liars. Knights always tell the truth. Liars always speak falsely.
Five suspects are interrogated in connection with a murder. Their statements are as follows:
Al: "Cal and Dan are lying."
Ben: "Al and Elmer are lying."
Cal: "Ben and Dan are lying."
Dan: "Cal and Elmer are lying."
Elmer: "Al and Ben are lying."
Which suspect do you know for certain to be a liar?
Assume that A is telling the truth.
Then B and E are both lying (based on their own statements), and C and D are both lying (based on A's statement).
But then C and D's statements are true, which is a double contradiction.
Our initial assumption is wrong. We know that A is lying.
We could stop here, and just trust that only one person is a proven liar, but let's press on.
Assume that B is telling the truth.
Then C and E are lying (based on their statements), and A and E are lying (based on B's statement).
Then D is telling the truth (based on his statement), so A is lying, which is consistent.
So, If B is telling the truth, D does also, and all others lie.
Assume that C is telling the truth.
Then A and D are lying (based on their statements), and B and D are lying (based on C's statement).
Then E is telling the truth (based on his statement), so A is lying, which is consistent.
So, If C is telling the truth, E does also, and all others lie.
Assume that D is telling the truth.
Then A and C are lying (based on their statements), and C and E are lying (based on D's statement).
Then B is telling the truth (based on his statement).
So, If D is telling the truth, B does also, and all others lie.
Assume that E is telling the truth.
Then B and D are lying (based on their statements), and A and B are lying (based on E's statement).
Then C is telling the truth (based on his statement).
So, If E is telling the truth, C does also, and all others lie.
In summary, there are only two possibilities:
C and E tell the truth and all others lie
or
B and D tell the truth and all others lie.
A is the only one who cannot be telling the truth.
Full Orthodox Solution (spoiler)
