Puzzle: Who murdered Fyodor Karamazov?
I've composed a wide variety of logic puzzles off-and-on for several decades. Mostly for fun, but
Dell Logic Puzzles paid me $100-$125 for very difficult puzzles. Here's one I composed two years ago for this message board! ... Getting only one solution and minimal applause I'll repost it.
In a certain district of Russia, many of the men are members of either the Cult of Knights or the Cult of Knaves. The Knights utter only true statements; the Knaves only false. Even when guilty of a crime they are, strangely, bound by this code.
A few years ago, Chief Inspector Plato was called upon to solve some crimes in that province, and usually was able to test guilt by asking
"If Someone of your cult was witness to this crime and I asked him if you were guilty, would he say Yes?"
He had trained the local police about this technique, and was seldom called to help anymore. He could spend much of his time contemplating how to attract interest from the lovely Anastasia. But today he was called in urgently. One of the notables had died in suspicious circumstances.
The local citizenry had adopted a counter-measure to inquisitive police: they would ignore the questions asked and just make arbitrary statements that happened to be True (if they were Knights) or False (if they were Knaves). This created a lot of extra work for the detectives but they had workarounds. For starters they often water-boarded the suspects for a while, then asked whether they wanted the torture to stop. This separated the Knights from the Knaves very easily! But they couldn't do that with the Chief Inspector in town!
Anyway, C.I. Plato was surprised that he was called on to attend an inquest. And he had mixed feelings when he learned that it was the death of Fyodor Karamazov that was being investigated.
You see, Fyodor was neither a Knight nor a Knave, but a
Knucklehead: He always spoke two sentences at a time, with exactly one of the sentences (either the 1st or the 2nd) true and the other sentence false. Although it was disrespectful to say so aloud, Plato was somewhat relieved that Fyodor was the deceased: the times he had questioned this old man got very confusing. But now it was Fyodor's murder that was under investigation.
Or was he even murdered? The local police had determined that Fyodor's three sons were all present at the death and all knew exactly what had happened. Either one, or two, or all three had participated in the murder of their father. Or maybe it was an accident, and all three were innocent. There was nothing whatsoever to go on; the butler had blundered and cleaned up all the material evidence.
Plato prepared to question the three brothers. "Which are knaves and which are knights?", he asked. The local police became apologetic. "Nobody knows, they kept to themselves all these years. And — sorry about this Chief Inspector — it's very possible that one or more of them are Knuckleheads like their father. At least we're sure that each brother is in one of the three cults. And of course the brothers know which cult each of their brothers are in."
What The F**k, Plato thought to himself in Russian, as he prepared his questions.
"Did you kill your father?", Plato suddenly yelled at Dmitri, hoping to take him by surprise.
Dmitri answered
- (1) Exactly one of us three brothers is guilty of the old man's murder.
- (2) Exactly one of us three brothers is a Knucklehead.
Plato couldn't make much of this. How many Knuckleheads were there? What if they are all Knuckleheads, he groaned. Great, thought Plato, just great. He tried again, shouting the same question at Ivan.
Ivan answered
- (3) Exactly two of us three brothers are guilty of the old man's murder.
- (4) Exactly one of us three brothers is a Knight.
Without being prompted,
Alexei chimed in.
- (5) I am not guilty of my father's death.
- (6) None of us three are Knuckleheads.
It was
Dmitri's turn again. He said
- (7) None of us three are Knights.
- (8) Papa had a mole on his left ankle.
"Go check the body for moles," said Plato decisively. "Fyodor has already been cremated" was the sheepish reply. On inquiry it turned out that nobody except for the sons had ever seen the old man with his boots off, even when he was taking comfort at the local brothel.
"Are you guilty of your father's death,
Ivan? A simple yes or no, please."
- (9) No, I am not guilty.
- (10) Papa had a mole on his right ankle.
By now the Chief Inspector was whimpering. "Can you help me,
Alexei?" That brother answered
- (11) Exactly one of us three brothers is a Knave.
- (12) There are exactly two true statements among (1), (7) and (10).
Now we might be getting somewhere, thought Plato. They're beginning to slip. But just then the Karamazov attorney showed up, telling his clients to shut up. And they'd all be leaving the next morning on a train to Paris if C.I. Plato couldn't crack the case by dawn.
Can you help? Which, if any, of the brothers killed their father? Which cults are they in, anyway?
