A statement only a sane patient can make puzzle

Raymond Smullyan introduced sane and insane people in his book The Lady or the Tiger? in 1982. One might think that insane truth-tellers are equivalent to sane liars. Indeed, when talking about facts they both lie. Similarly, insane liars and sane truth-tellers tell the truth. It gets trickier when they talk about beliefs. An insane person believes that two plus two is not four. But being delusional they believe that they believe that two plus two IS four. In other words, an insane truth-teller will say, “I believe that two plus two is four,” which is a lie as it should be. Meanwhile a sane liar will say “I believe that two plus two isn’t four.” That means, when we look at statements about beliefs, the underlying statement is true for truth-tellers and false for liars independently of their sanity.

For convenience, let’s number each round of answers from the members of the asylum, starting from 0 (the initial statement about lovers). Let’s skip round 0 and look at rounds 1-3 (the questions relating to who is a patient).

The only person who thinks he is a patient is a partial in the sane stage.

• “I am a patient.” This can be sane lying, delusional lying, insane partial lying, and sane partial telling the truth.
• “I am not a patient.” This can be sane telling the truth, delusional telling the truth, sane partial lying, and insane partial telling the truth.

Patients in insane or sane state believe that they believe that they are patients. Doctors believe that they believe that they are not patients.

• “I believe I am a patient.” This can be sane lying, delusional telling the truth, sane partial telling the truth, and insane partial telling the truth.
• “I believe I am not a patient.” This can be sane telling the truth, delusional lying, sane partial lying, and insane partial lying.

Let’ denote the sanity of people just before round 1 by S (sane), D (delusional), Pi (partial insane), and Ps (Partial sane). We’ll denote truthfulness by T (truth-teller), L (liar), At (truthful alternator), and Al (lying alternator).

By analyzing rounds 1-3 about patients we get the following possibilities, where Y, means yes, and N means No.

We do not know who is who yet. But we know something. Namely, we can deduce information from some of the statements. Let us look at factual statements (not beliefs). Both DL and PiAl always tell the truth about simple facts. That means that Beth, Fiona, and Grace are Zack’s lovers. Now that we know some lovers we can start unraveling who is who from Round 4 (where they talk about their love life), working backwards. We start with Ann, who talks about Beth (the latter is a lover).

• Ann: Beth was Zack’s lover. Ann is PsAt in insane lying state, rather than SL. Ann is a lover.
• Ian: I believe Ann wasn’t Zack’s lover. Ian is PiL in sane state, rather DAl in true state. Ian is not a lover.
• Holly: Ian was Zack’s lover. Holly is SAt in lying state, rather than PsL in insane state. Holly is not a lover.
• Grace: I believe Holly wasn’t Zack’s lover. Grace is PiAl in sane true state rather than DL. Grace is a lover.
• Fiona: I believe that Grace wasn’t Zack’s lover. Fiona is DL rather than PiAl in sane true state. Fiona is a lover.
• Eve: Fiona was Zack’s lover. Eve is SAl in true state, rather than PsT in insane state. Eve is a lover.
• David: I believe Eve wasn’t Zack’s lover. David is PiL in sane state, rather than DAl in true state. David isn’t a lover.
• Cedric: David was Zack’s lover. Cedric is SAt in lying state, rather than PsL in insane state. Cedric isn’t a lover.
• Beth: I believe Cedric was Zack’s lover. Beth is DL, rather than PiAl in sane true state. Beth is a lover.

In the last round the asylum tenants only talked about their beliefs. That means we only need to know if they are telling the truth to decipher the true statements:

• The patients’ doors are locked from 11p till 7am and only doctors have keys to the patients’ rooms.
• There are no killers among the doctors.
• There was a person carrying the statue to the victim’s room at 2am.
• The person who unlocked the door is a doctor and a lover.
• The men are innocent.
• There are no killers among the delusional patients.
• All non-lovers are innocent.
• Only Beth, Cedric, David, and Ian are strong enough to lift the statue.
• The person who brought the statue to Zack’s room is a lover.

The person who unlocked the door, is a doctor and a lover. This could only be Eve. The person who carried the statue to the room must be one of Beth, Cedric, David, and Ian. We also know that the men are innocent. This leaves Beth to be the only person who could have carried the statue. We also know that the killer might only be a partial: Ann, David, Grace, and Ian. Given that the men are innocent, we are left with Ann and Grace. Given that none of them alone could have lifted the statue, both of them are killers.

The frequent mentions of the word THREE hint at the ternary extraction. Plus, the characteristics in the table do not match the order they were used in the puzzle. That means this order is important. We put people in alphabetical order and assign them values based on Radstein’s list. Then we convert the ternary numbers to integers and to letters.

This yields the answer, ALTERNATE .