Solution

PUZZLE CORNER

CLUB of DOWNERS GROVE

Kiwanis Monthly Challenge

A new puzzle will appear each month.

When you have solved the puzzle, you may submit your solution.
The solution will be given the following month, along with credit to those who have solved the puzzle.

Puzzle #6 for September “Inspector Craig Visits Transylvania”:

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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 is 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?

Text Solution #6:

Lucy's statement is either true or false. If it is true, then both sisters are really insane; hence Lucy is insane, and the only insane Transylvanian who can make a true statement is an insane vampire. So if Lucy's statement is true, then Lucy is a vampire.

Suppose Lucy's statement is false. Then at least one of the sisters is sane. If Lucy is sane, then since she has made a false statement, she must be a vampire (because sane humans make only true statements). Suppose Lucy is insane. Then it must be Minna who is sane. Also Minna, by contradicting Lucy's false statement, has made a true statement. Therefore, Minna is sane and has made a true statement so Minna is human and, again, Lucy must be the vampire.

This proves that regardless of whether Lucy's statement is true or false, Lucy is the vampire.