PUZZLES
1) We are in an insane asylum where the only people we meet are doctors and patients. There’s a profound truth to this which we shall, for the moment, ignore. In the asylum, doctors and patients may be either sane (in which case they believe only what is true and know it as true, believing that what is false is false) or insane. The insane are entirely inaccurate in their beliefs (what they believe is true is false, and the propositions which they believe are false are true). Everyone says what he/she actually believes. In our first encounter, we hear a statement, which makes us believe that the speaker is a sane patient. We take immediate steps to have the person set free. Question: what is the simplest such statement?
2) We visit another asylum with identical inhabitants and interview four of them: Adel, Barbara, Cameron, and Denise. Adel believes that Barbara and Cameron are alike as to sanity. Barbara believes that Adel and Denise are alike as to sanity. We scratch our heads and ask Cameron whether he and Denise are both doctors, to which Cameron replies in the negative. Question: is there something wrong with this asylum?
The answers to this month’s questions appear below.
Answers
1) “I’m not a sane doctor.” There are other solutions, but unlikely more elementary than this. The explanation of the answer is that an insane doctor wouldn’t hold the stated belief if it were true, and a sane doctor wouldn’t hold the belief because it would be false. An insane patient couldn’t hold the belief, because the belief would be true. But if the speaker is a sane patient, the statement would be true and consistent.
2) If Adel and Barbara are both sane, then Barbara and Cameron are alike as to sanity, and so are Adel and Denise. This means that all four would be sane, in which case Cameron and Denise would be sane and thus alike. But suppose Adel and Barbara are both insane. This would mean that B and C are different from each other, and likewise A and D aren’t alike as to sanity. Hence C and D are both again sane and alike. Now suppose A is sane while B is insane. B and C would be alike, meaning that C is insane, but A and D would be different, meaning that D is likewise insane. Finally, if A is insane and B sane, then B and C are different from each other, which means that C is insane, while A and D are alike, which makes D also insane. Where have we got so far? If A and B are alike, then C and D are both sane. If A and B are different, then C and D are both insane. The logical possibilities mean that C and D are either both sane or insane. Whew! Now suppose they are both sane. This makes C’s statement true, the one in which C says that he and D are not both doctors. This makes either C or D a sane patient. If, on the other hand, C and D are both insane, then C’s statement is false, which means C and D are in fact both doctors, which makes them insane doctors. This asylum therefore contains at least one sane patient or two insane doctors.
