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While wandering, you encounter a pond, in which two demons are bathing. One is labelled 'honest', meaning they always tell the truth, and the other is labelled 'liar', which means they always lie. As you turn around and leave, you hear a demon's voice say "We aren't labelled correctly". Which of the demons said this?

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It must be

the demon labelled "liar".

There are two options:

  • If they are labelled correctly, then the one who spoke is a liar and labelled as a liar.

  • If they aren't labelled correctly, then the one who spoke is honest but labelled as a liar.

Note that the puzzle is not entirely well-posed, since it assumes that the only options are that either both or neither are labelled correctly, i.e. it precludes the possibility of both being honest or both being liars. There is a slight ambiguity in the English phrase "We aren't labelled correctly" - does it mean that they're both labelled incorrectly, or just that they're not both labelled correctly? - and the puzzle exists in this ambiguity.

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    $\begingroup$ What if they are actually both honest? $\endgroup$
    – Steve
    May 22 at 16:20
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    $\begingroup$ @Steve There's an implicit assumption in the puzzle, which should really have been made explicit but wasn't. I've added a paragraph to explain this. I did think about it while solving the puzzle, but not making that assumption would mean it isn't solvable. $\endgroup$ May 22 at 16:38
  • $\begingroup$ The argument in point 2 doesn't work, under the assumption that "we aren't labelled correctly" means that at least one of them is labelled incorrectly. If they aren't labelled correctly, then one of them is labelled incorrectly. Although the one who spoke is honest, one cannot say that this one is the one who is labelled incorrectly. In fact, it's totally fine that both are honest, and any of them can say that sentence. $\endgroup$
    – WhatsUp
    May 22 at 17:07
  • $\begingroup$ @WhatsUp See the final paragraph of the answer. The puzzle is slightly ill-posed in that it assumes there are only two options, both labelled correctly or neither labelled correctly. (Otherwise, it's unsolvable.) $\endgroup$ May 22 at 17:11
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    $\begingroup$ @WhatsUp I'm answering the puzzle as intended, and have edited to point out the ill-posedness for completeness. Not all puzzlers are as pedantic as they perhaps should be :-) $\endgroup$ May 22 at 17:19
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The demon labelled "Liar" because he's actually honest who's labelled as liar. So he must be the one who said it and he's right, in one case.

In another case, it becomes a vicious circle. If the honest one said and we believed , so he's trying to say in real he's actually a liar . It means he can't be believed and they're labelled correctly.

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This question was already answered. Yet, I thought to give my answer without seeing the solutions.

Both of them could have said that.

Because

Honest under the label liar is telling the truth

Or,

Liar under the correct label is trying to deceive you.

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