# Catch the liar(s)

Alice: "Bob is lying, so is Charlie!"

Bob: "Charlie is lying, so is Alice!"

Charlie: "Alice is lying, so is Bob!"

Who is/are lying?

• -1. you changed the question after some answers were posted. I even believe you changed the question after you accepted an answer. With this new wording it is also possible that everyone is telling the truth – Ivo Beckers Feb 12 '16 at 11:23
• Have to agree with Ivo Beckers on this one: You made an edit that completely changed the contents of the question. This is considered a big no-no. Downvote on my part aswell. – Tim Couwelier Feb 12 '16 at 11:27
• @IvoBeckers No wonder the answers looked weird. – Lawrence Feb 12 '16 at 11:36
• Since you've accepted an answer that addressed the old question, how about reverting to the original question and posting the new version as a new question? – Lawrence Feb 12 '16 at 12:43
• I've rolled back this question to its initial version, and reopened it - in the future, please avoid making significant changes to a question after it's been successfully answered. Thank you! – user20 Feb 12 '16 at 18:41

There are only eight cases so it's easy to check by hand which cases are true:

case 1: They are all lying
This is impossible because that would make their statements true

case 2: They are all telling the truth
this is also impossible because it contradicts all statements

case 3,4,5: two people are telling the truth
This is not possible because person 1 says person 2 lies and vice versa

case 6,7,8: two people are lying
This is possible! The two person that lie, lie about the fact that the truth teller is telling the truth, while the truth teller correctly points to the other two as liars.

So there is one truth teller and two liars, and the truth teller could be anyone

• case 2 is possible... – Ben Feb 12 '16 at 11:16
• @Ben yeah it is. But the OP actually changed the question after I posted this answer I see! Before the change case 2 was not possible – Ivo Beckers Feb 12 '16 at 11:20

Exactly one of them is telling the truth but we can't tell which. For example, if Alice is truthful, then Bob is a liar because he is wrong about her and right about Charlie. Charlie is right about Bob but wrong about Alice. This works no matter who you start with.

The other cases:

No two can be telling the truth because each is calls the other two liars. They cannot all be liars because then they would all be telling the truth.

• >! If i'm correct, I think also everybody can be telling the truth... If they are all telling the truth, then nobody is lying, so all 3 implications have false assumptions, so they're true. – Piotr Pytlik Feb 12 '16 at 10:40
• I'd put my money on Bob lying. If you search on Google for "XXX is Lying", then Alice returns 1.4M results, Bob 32.4M and Charlie 14.2M which suggests that people called Bob lie more than twice as often as people called Charlie, and Alice usually tells the truth (compared to the other two)! – Gordon K Feb 12 '16 at 11:02
• Just wanted to say: ignore this answer. This is an (accepted) answer to the original question. The question has since been edited to an entirely different one and this is no longer relevant. – Hugh Meyers Feb 12 '16 at 12:49

The statement of the puzzle has been changed several times. My answer below fits with the following version of the puzzle.

Alice: "If Bob is lying, Charlie is lying!"
Bob: "If Charlie is lying, Alice is lying!"
Charlie: "If Alice is lying, Bob is lying!"

Who is/are lying?

We use the words "predecessor" and "successor" with respect to the cycle Alice -> Bob -> Charlie -> Alice.

• If person $X$ is lying, then the successor of $X$ is saying the truth (as his/her statement ends with "then $X$ is lying").
• If two (or more) persons were lying, then one of them would be the successor of the other one; contradiction to the above statement.
• If exactly one of them is lying (say Alice) and the other two (say Bob and Charlie) are saying the truth, we get the contradiction from Charlie: "If Alice is lying, Bob is lying!" (Charlie says the truth; Alice lies; hence Bob must lie).

The only remaining possibility is:

All three are saying the truth (which indeed is fine).