# How to write exactly one of us is telling the truth? (Liars and truthtellers)

I have a question regarding a version of the truthtellers and liars puzzle which I haven't seen anywhere before.

I'm stranded on an island and I know that this island has cannibals, but I don't know how to distingush between who are cannibals and who aren't. I run into three people on the island, A,B,C, and I ask them: "How many of you are truthtellers?" And A responds "flop flip" in his own native language. I then ask what it means, and B then says "It means only one" whereafter C says: "Don't trust B she is lying. Come with me I'm not a cannibal." How do I know what to do?

I want to set this up in a truth table, but I don't know how to formalize "only one of us is a truthteller" into logic.

If the question is should you come with $$C$$ or not then the answer is yes, you should. Just split into cases.

Assume $$A$$ is telling the truth. Then you know $$B$$ is lying, because if he was telling the truth then you would get that $$A$$ really told you only $$1$$ of them is telling the truth (which you assumed is true) which is a contradiction to that both $$A$$ and $$B$$ are telling the truth. So $$B$$ is lying and $$A$$ actually told you more than $$1$$ of them is telling the truth (he couldn't say none of them is telling the truth because he is telling the truth himself). So in that case $$C$$ is telling the truth and he is not a cannibal.

Now assume $$A$$ is lying and $$B$$ is telling the truth. We'll show this case is not even possible. In that case $$A$$ really told you only $$1$$ is telling the truth, but this is a lie. As you assumed that $$B$$ is telling the truth then you get that more than $$1$$ of them is telling the truth. So $$C$$ must be telling the truth. But then his words that $$B$$ is lying are the truth which is a contradiction to that $$B$$ is telling the truth. So this case isn't possible.

Finally assume that both $$A$$ and $$B$$ are lying. If you assume $$C$$ is lying then his words that "$$B$$ is lying" are a lie and $$B$$ is actually telling the truth. Once again a contradiction. So again $$C$$ is telling the truth and is not a cannibal.

So anyway, we can conclude that $$B$$ is always lying and $$C$$ is always telling the truth.

• I agree with what you've written, but do you know how to translate this into atomic formulas? Also is there a way to figure out what A is? – nahm8 fkn8 Sep 30 '18 at 12:11
• Sorry, but I'm not familiar with these formulas. Maybe the second answer you got might help. – Mark Sep 30 '18 at 12:14
• If C is telling the truth and A is lying won't B then be telling the truth since there'll only be one truthteller? Doesn't this mean that A has to be telling the truth? – nahm8 fkn8 Oct 2 '18 at 13:55
• If there is only one truthteller it doesn't mean $B$ is telling the truth. You don't really know what $A$ said. Maybe he lied all of them are telling the truth, so then $B$ also lied when he said that $A$ said "just $1$ is a truthteller". – Mark Oct 2 '18 at 14:02
• Ah okay. So the condition for B to be lying is if B lies about A even though B might indirectly be telling the truth? – nahm8 fkn8 Oct 2 '18 at 16:31

"only one of us is a truthteller" = $$(A\land\lnot B\land\lnot C)\lor(\lnot A \land B \land\lnot C)\lor(\lnot A \land\lnot B \land C)$$

Either only A is a truth-teller, or only B, or only C

iff B is a truthteller, than iff A is a truthteller, than "only one of us is a truthteller" = $$B \leftrightarrow (A \leftrightarrow((A\land\lnot B\land\lnot C)\lor(\lnot A \land B \land\lnot C)\lor(\lnot A \land\lnot B \land C)))$$

iff C is a truthteller, B is a liars = $$C \leftrightarrow \lnot B$$

So the truth table is:

$$\begin{array}{c|c|c|c} C & B & A & (B \leftrightarrow (A \leftrightarrow((A\land\lnot B\land\lnot C)\lor(\lnot A \land B \land\lnot C)\lor(\lnot A \land\lnot B \land C)))) \land (C \leftrightarrow \lnot B) \\ \hline 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 0 \end{array}$$

So as you can see, C is a truth-teller, while A can be a thruth-teller or liar.

• It seems correct. I've drawn the truth table and the formula returns true only when exactly one of them is telling the truth. Do you know how I deduct which truth assignment is valid? – nahm8 fkn8 Sep 30 '18 at 12:07
• I'll add a full truth table in a couple minutes – david rabinowitz Sep 30 '18 at 12:34
• I think I just realized something although I'm not sure if it's correct. Doesn't A have to be telling the truth? Because if A is lying then C is the only truthteller, but then B's statement is true? – nahm8 fkn8 Oct 2 '18 at 12:25
• B's statement is true if "flop flip" means only one, it doesn't matter whether "flop flip" is true. So if A was lying, since we proved that B was lying, it may be true that there is only one truth-teller, since A said something else ("I like cookies"), and B said that A said something, which he didn't. – david rabinowitz Oct 2 '18 at 20:20