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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.

nahm8 fkn8
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2 Answers2

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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.

Mark
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  • 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 02 '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 02 '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 02 '18 at 16:31
  • Actually $B$ is always lying. Suppose $B$ is telling the truth. Then $A$ really said only $1$ of them is a truthteller. If $A$ is telling the truth then only $1$ must be telling the truth, but that's a contradiction because you supposed both $A$ and $B$ are telling the truth. And if $A$ is lying then you know more than $1$ of them is telling the truth. But it's impossible, because you supposed $A$ is lying and because $B$ is telling the truth you know $C$ is lying too. (because $C$ says $B$ is lying). So it's a contradiction once again. – Mark Oct 02 '18 at 16:48
  • And yes, $B$ doesn't tell you how many truthtellers there are. He just says what $A$ said. So he is lying about $A$. – Mark Oct 02 '18 at 16:59
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"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.

david rabinowitz
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  • 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 02 '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 02 '18 at 20:20