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?
Monday, April 10, 2006
Smullyan revisited
As a further tribute to the genius of Raymond Smullyan, I present a somewhat more intimidating puzzle from his oeuvre. The hardest part is wading through the setup, but it's worth making the investment:
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8 comments:
A decidedly plebeian approach: Assuming that Lucy is honest and Minna is a liar, we have to check four possibilities to see if they could result in the responses given. If none do, then Minna must be the honest one.
1. Lucy sane, Minna sane
Honest and sane Lucy would not say "We are both insane" because she herself isn't.
2. Lucy sane, Minna insane
Honest and sane Lucy would not say "We are both insane" because she herself isn't.
3. Lucy insane, Minna sane
Honest but insane Lucy would not say "We are both insane" because she believes she is sane.
4. Lucy insane, Minna insane
Honest but insane Lucy would not say "We are both insane" because she believes they are both sane.
Lucy's statement is not compatible with any possibility when she is honest, so she must be a liar.
1) Check
2) Check
3) Bzzt! If Lucy is human (honest) and insane, then she will believe all false things and tell the truth about what she believes. Since "We are both insane" is false if Minna is sane - regardless of Lucy's sanity - you need more information here.
Back to the drawing board, Chumley.
So Lucy is honest and yet capable of saying both "We are both insane" and "I am sane"? That's -- uh, crazy!
Kindly restrict yourself to the problem as stated, Mr. Jillette.
See? I told you so.
You're both a little light on the support.
Suppose Lucy is human. Then she must be insane, for the reasons Steve gave above.
Also, Minna must be a vampire, because there is one vampire sister and one human sister. If Minna is insane, then both are insane, and Lucy's statement was true - which can't happen insane humans only say false statements.
Therefore Minna must be a sane vampire.
But sane vampires only say false things, and Minna's statement, "Of course not!", is true (since it is false that they are both insane). This is a contradiction.
Hence Lucy must be a vampire (insane), and Minna human (insane).
A human will always claim to be sane. If she really is sane, she'll declare the truth of it, and if she's actually insane, she'll claim the opposite of that. Either way, a human will never self-incriminate her sanity.
Since Lucy self-incriminates, she must be the vampire.
Also, in the book _Dracula_, Lucy Westenra becomes a vampire, while Mina Harker never quite does.
Eric, I agree with your conclusion, but there is one case where a human will self incrimidate. If it is an insane/human sane/vampire, the human will reply that both of them are insane. That's why we need the second question since a sane vampire will reply that yes they are both insane.
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