[Ah Beng]

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?

 
 


(If you still can't work out the solution after cracking your head for 3 days & 3 nights,  click here.)

[Bakegaku]

I figured it out to be Case 2 (as in, I solved in then checked it).  I didn't even think about case 1, though.

[Will]

Assume Lucy is an insane human:
This would be impossible because insane humans tell lies and Lucy would be telling the truth by saying they are both insane.

Assume Lucy is a sane human:
This would be impossible because Lucy says that that they are both insane, when Lucy is sane.

Assume Lucy is an insane vampire:
Lucy would be telling the truth, therefore Minna must be an insane human.

Assume Lucy is a sane vampire:
Lucy would be telling a lie, therefore Minna must be an insane human.

Lucy must therefore be a vimpire.