Detective Radstein was scouting vacation spots and ended up on a peculiar island. Inhabitants of this island are very particular about the truth values of their statements. There are three types of people: truth-tellers, liars and alternators. Truth-tellers only say things they believe are true; liars only say things they believe are false; and alternators switch at every statement between the two. The alternators switch according to their own internal counter. That means, if a stranger starts a conversation with the alternator, it is not known, a priori, whether the alternator’s first statement is true or false.

A murder was committed last night at the island’s asylum, and Detective Radstein was invited to investigate.

Independently of being obsessive about their statements there are three other types of people at the asylum: sane, delusional and partial. Sane people believe true facts are true. The medical system is very efficient on the island: all the doctors are sane, and none of the patients is completely sane.

The patients are either delusional or partial. A delusional patient only believes in facts that are not true. Moreover, they are very particular about their delusions. For example, if two plus two is four, the delusional patient believes that two plus two is not four. That is, they believe the exact negation of the true statement.

The partial patients combine traits of delusional patients and sane people. They switch the truthfulness of their beliefs after each statement they make. So a partial patient in the sane state can believe that two plus two is four, but after they make a statement, for example, that today is Wednesday, they switch to believing that two plus two is not equal to four. Similar to alternators, they switch their sanity according to an internal counter. That means, a stranger doesn’t know, a priori, whether the first statement is made in the sane or insane state.

When Radstein arrived at the asylum, he started by examining the victim’s room. The victim was a patient named Zack. He lived alone in his room. The victim was lying in his bed, his skull crushed. Next to the victim was a bloody bronze statue of a ballerina. It was obviously the murder weapon. Radstein put on his gloves and picked up the statue. It was very heavy: he could barely lift it. It would require significant strength to use this statue as a murder weapon. Radstein examined the victim. The man obviously died in his sleep. Radstein estimated the time of death as somewhere between 2am and 5am. All the windows were locked; the only way anyone could have entered the room was through the door, but the door lock had not been meddled with. Radstein also found a statue stand in the common room.

The asylum is small. There are only nine people there: Ann, Beth, Cedric, David, Eve, Fiona, Grace, Holly, and Ian. This is not counting the dead guy. It is worth noting that the members of the asylum do not talk about things they do not know about.

The detective went to a room where all the asylum folks gathered. They immediately started talking.

• Ann: Zack was my lover.
• Beth: Zack was my lover.
• Cedric: Zack was my lover.
• David: Zack was my lover.
• Eve: Zack was my lover.
• Fiona: Zack was my lover.
• Grace: Zack was my lover.
• Holly: Zack was my lover.
• Ian: Zack was my lover.

“That was amusing,” thought Radstein. He decided to control the conversation: “Yes or no, are you a patient?”

• Ann: Yes.
• Beth: Yes.
• Cedric: No.
• David: Yes.
• Eve: Yes.
• Fiona: Yes.
• Grace: Yes.
• Holly: No.
• Ian: Yes.

“With all these strange alternators and partials, why don’t I ask the same question again,” thought Radstein. Here are the answers.

• Ann: Yes.
• Beth: Yes.
• Cedric: Yes.
• David: No.
• Eve: No.
• Fiona: Yes.
• Grace: Yes.
• Holly: Yes.
• Ian: No.

“Hmm, will this help me find alternators and partials?” thought Radstein, “What happens if an alternator is also a partial? There is also the curious issue of their delusions. Is there a way to distinguish between insane truth-tellers and sane liars? What do insane people believe that they believe? Let me try something else.” He asked: “Yes or no, do you believe that you are a patient?”

• Ann: Yes.
• Beth: No.
• Cedric: No.
• David: No.
• Eve: Yes.
• Fiona: No.
• Grace: No.
• Holly: No.
• Ian: No.

While Radstein tried to collect his thoughts, the suspects got excited and started to discuss their love life again.

• Ann: Beth was Zack’s lover.
• Beth: I believe Cedric was Zack’s lover.
• Cedric: David was Zack’s lover.
• David: I believe Eve wasn’t Zack’s lover.
• Eve: Fiona was Zack’s lover.
• Fiona: I believe that Grace wasn’t Zack’s lover.
• Grace: I believe Holly wasn’t Zack’s lover.
• Holly: Ian was Zack’s lover.
• Ian: I believe Ann wasn’t Zack’s lover.

Then they moved on to the murder.

• Ann: I believe the patients’ doors are locked from 11p till 7am and only doctors have keys to the patients’ rooms.
• Beth: I believe there is a killer among the doctors.
• Cedric: I believe I saw a person carrying the statue to the victim’s room at 2am.
• David: I believe that the person who unlocked the door is not a doctor or not a lover.
• Eve: I believe there exists a man who is not innocent.
• Fiona: I believe there is a killer among the delusional patients.
• Grace: I believe there is a non-lover who is not innocent.
• Holly: I believe that only Beth, Cedric, David, and Ian are strong enough to lift the statue.
• Ian: I believe that the person who brought the statue to Zack’s room is not a lover.

Radstein came back to his hotel room to write a report. At the end of the report he wants to have a list for each person. In addition to solving the crime Radstein wants to figure out who is who. It is not clear if the person who unlocked the room and/or the person who brought the statue into the room also committed the murder. It is curious that for guilt too he needs three categories just in case.

The first bullet on the list is related to sanity. It will say whether the person is

• partial,
• delusional, or
• sane.

The second bullet is about truthfulness. It will say whether the person is

• an alternator,
• a liar, or
• a truth-teller.

The third bullet is related to guilt:

• an accomplice,
• guilty of murder, or
• innocent.

Please, help Detective Radstein.