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This is a puzzle from this site: https://medium.com/i-math/vampires-insanity-transylvanian-sisters-88ce3516fd
I thought it was very nice, so I wanted to share it with you. WARNING: the above link contains the answer!

Inspector Craig of Scotland Yard was called to Transylvania to solve some cases of vampirism. Arriving there, he found the country inhabited both by vampires and humans. Vampires always lie and humans always tell the truth.

However, half the inhabitants, both human and vampire, are insane and totally deluded in their beliefs: all true propositions they believe false, and all false propositions they believe true. The other half of the inhabitants are completely sane: all true statements they know to be true, and all false statements they know to be false. Thus sane humans and insane vampires make only true statements; insane humans and sane vampires make only false statements.

Inspector Craig met two sisters, Lucy and Minna. He knew that one was a vampire and one was a human, but knew nothing about the sanity of either.

Here is the investigation:
Craig (to Lucy): Tell me about yourselves.
Lucy: We are both insane.
Craig (to Minna): Is that true?
Minna: Of course not!

From this, Craig was able to prove which of the sisters was the vampire. Which one was it and how did he do it?

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  • $\begingroup$ Humans always tell the truth? Hmm. Apparently, while I've met some vampires, I've yet to encounter a human. $\endgroup$
    – Paul Sinclair
    Commented Aug 11, 2020 at 16:59
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    $\begingroup$ As long as we're giving attribution, the original source is Raymond Smullyan's The Lady or the Tiger? $\endgroup$
    – Misha Lavrov
    Commented Aug 11, 2020 at 17:13

3 Answers 3

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Same conclusion, slightly different logic.

Since they disagree with each other, we know:

One is telling the truth and one is telling falsehood.

We also know one is human and one is a vampire. Combine that with the above, and we get:

Both are sane or both are insane. If exactly one were sane, and exactly one is a vampire, then both would tell the truth or both would tell falsehood.

So, if

Both are insane, then Lucy is telling the truth and must be the vampire.

However, if

Both are sane, then Lucy is lying, and must be the vampire.

QED

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Seems easy:

If Lucy tells truth - and they are both insane - she must be Insane Vapire and Minna is Insane human - that way is Minna lies and it seems to fit rules.

or:

If Lucy lies about both being insane - she might be sane vampire, or insane human.
But because one of them "must be" vampire - she can't be insane human
- in this case Minna would be either insane vampire - that contradicts "Lucy lies"
- or Minna would be sane vampire - that way Minna must lie in her statement "of course we are not both insane" - but she is sane, so not this case
So Lucy could be sane vampire - lies about "We are both insane" - so Minna tells truth and must be sane human.

Either way:

Lucy is vampire and Minna is human

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I think:

Lucy is a Vampire

reasoning:

Let's assume she is sane. Then she thinks she is sane and tell a lie (they can't both be insane). So she is a vampire. On the contrary if she is insane, she still believes to be sane. And if her sister is insane too, she tells a lie. She is a vampire! BUT if her sister is sane, she thinks that they are both insane (and both sane)... Damn insane people. She should be human... Her sister is sane and knows she tells the truth; we now have 2 humans. It's impossible. Bring me some garlic!

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    $\begingroup$ For last part - If Lucy would be insane human - lies about "we are both insane" and Minna could be sane human telling truth "of course not". Same way Lucy could be sane vampire and Minna insane vampire. So you need to know that there is just one vampire and one human. $\endgroup$
    – Jan Ivan
    Commented Aug 11, 2020 at 8:17
  • $\begingroup$ @JanIvan for the first exemple - If Lucy would be insane human, then she can't say "we are both insane": she would be liying as she thinks to be sane. For the second exemple Mina is insane, so she thinks herselfto be sane, as such the statement "Of course not!" would be true for her which she cannot say if she is a vampire. – $\endgroup$
    – Sechiro
    Commented Aug 11, 2020 at 9:01
  • $\begingroup$ If Lucy would be insane human and Minna not - Lucy would say "we are both insane" - since it is not truth and she is insane human. $\endgroup$
    – Jan Ivan
    Commented Aug 11, 2020 at 9:52
  • $\begingroup$ @JanIvan So she thinks she is sane, and that they are both insane... Damn insane people, i made a mistake. Eddited. $\endgroup$
    – Sechiro
    Commented Aug 11, 2020 at 9:58

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