A Few Easy Ones from Raymond Smullyan
Many of you are familiar with the canonical ‘liar and truth teller’ puzzle. Raymond Smullyan
is a logician who has taken the basic idea and expanded it to produce an immense variety of challenging puzzles. He has even used variations on the theme to illustrate Incompleteness Theory and other logic esoterica.
Here are a few easy ones to give you a taste of his sumptuous banquet:
Dr. Tarr is a psychologist with the Department of Health. Her job is to inspect asylums to determine whether they are in compliance with the law. Asylums have Doctors and Patients. In a compliant asylum, all the doctors are sane and all the patients are insane. Clearly, an asylum with an insane doctor or a sane patient is Not A Good Thing.
Sane persons are correct in all of their beliefs. Insane persons are incorrect in all of their beliefs. Both sane and insane persons are scrupulously honest: they always state what they believe to be the case. Unfortunately, the asylums are very modern and do not use identifying devices such as uniforms, ID tags, or other devices to show which persons are doctors and which are patients. Nor is it possible to know whether a person is sane or insane by any means other than questioning them.
One day, after inspecting a number of asylums, Dr. Tarr was having a drink and cigar with her good friend Professor Feather. The professor found her work interesting and asked her to recount some of her findings.
“Well,” said Dr. Tarr, “at the first asylum I visited, I met an inhabitant who made a single statement. I immediately took steps to have them released.”
“Wait,” interjected the professor, “so you’re saying this person was not an insane patient?”
“Of course,” replied Dr. Tarr.
Professor Feather thought for a moment, then asked “How is that possible? This sounds like the old Liar and Truth Teller puzzle. This person either told the truth or they lied. But there are four possibilities for any person in an asylum: Sane Doctor, Insane Patient, Insane Doctor, or Sane Patient.
“Even if you knew whether they were lying or telling the truth, that would only narrow the matter down to two possibilities. For example, if they told a truth such as ‘two plus two equals four’, you would know that they were Sane. But how would you know that they were a Patient, not a Doctor?”
Dr. Tarr replied with a chuckle “I agree that I could not have deduced what to do based on an inhabitant saying ‘two plus two equals four’. But in this case, the patient was quite intelligent and thought of a single statement which could establish the fact that only a Sane Patient could make that statement.
“I’m sure if you think about it, you could construct such a statement. Name a statement which could only be uttered by a Sane Patient.”
Dr. Tarr and Professor Feather shared a chuckle over that one. Then the professor took a more serious tone. “But have you ever had to remove a Doctor from an asylum?”
“Yes,” said Dr. Tarr sadly, “it has happened. Doctors do go insane once in a while. Recently I had just such a case. As it happened, I was visiting an asylum for the very first time and the first inhabitant I met made a single statement. I immediately had the inhabitant transferred to a special institution for former Doctors.”
“Don’t say it!” exclaimed the professor, “I want to work it out for myself…”
“Another time,” continued Dr. Tarr, “I was visiting an asylum which had been placed on probation for irregularities such as Insane Doctors and Sane Patients. I asked an inhabitant ‘Are you a patient’, and she said ‘yes’.”
“What did you do next?” asked Professor Feather. “Did you need to do any more investigating?”
“I’m glad you worked that out. Another asylum was on probation and I decided to ask the very same question of the first inhabitant I met. This time, when I asked ‘Are you a patient,’ he replied ‘I believe so…’.” Do you think I knew enough to close the asylum?
Professor Feather thought about this one for a very long time.
Spoiler I: V nz abg n fnar qbpgbe.
Spoiler II: V nz na vafnar cngvrag.