Inspector Craig visits Transylvania
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.