Sanity Riddle
Posted by Rob Herman at November 14th, 2006
A quick one to get back into the swing of things. Paraphrased from a riddle that is, I believe, from a book by Raymund Smullyan. The answer to this one is very simple but eluded me for a long time. It finally hit me in a flash of insight as I was walking around; intrepid reader John, who related it to me, ended up putting together a truth table.
In a certain mental hospital, there are doctors and patients, all mixed together. Some of each are insane and others are sane. Sane people always tell the truth, and insane people always lie.
There is an investigator in this hospital whose job it is to release sane patients. (He has another job, which is to admit insane doctors as patients. That isn’t relevant to this riddle, but it may come up in a sequel.) Sane doctors are not released (they are supposed to stay around and help the patients.)
You are a sane patient in this hospital, and you want to be released. The investigator does not know that you are sane, nor that you are a patient. Making only one statement, how can you convince the investigator that you are a sane patient?
Am I allowed to use a compound statement, i.e. something with the word and in it?
If compound statements are forbidden, then the solution may hinge on the nature of truth and falsehood adopted by the riddle. Can an insane patient say, “I am an insane doctor” because the statement, as a whole, is false? Or does the confession of insanity violate the rule that insane persons always lie?
Compound statements are definitely allowed. The truth or falsity of the entire statement is what is judged. For instance, an insane person can say “1 1 = 2, and the sky is pink,” because the whole statement is logically false. So, Ephraim Glass: yes, an insane patient could make that statement, and a sane doctor could not.
Reader Alatar mentioned offline that “I am not a sane doctor,” and not only is he correct but he has solved the riddle as well.
Hi, Rob. Here is an alternate solution for the Sanity Riddle: “If you were to ask me if I were a sane patient, I would say yes.”
Very much enjoyed dinner the other night. Happy New Year, and take care.
Ed