Puzzle: Who murdered Fyodor Karamazov?

[Swammerdami]

Did someone mention logic puzzles? Here's one I composed two months ago.

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 tribe 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 fantasizing about the lovely Anastasia. But today he received an urgent summons: 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 had misgivings about attending another inquest. And these feelings increased 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 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?

[Swammerdami]

Three days and no response. :-( Please give me some feedback.

Was it too difficult? Uninteresting? Was the overly-long story-line a turn-off? Did some assume the puzzle was likely to be flawed anyway?

Should I provide a hint? The full solution? Or just go away quietly?

[hilarity n. suze]

Do Not provide the full solution. I am working on it. I haven't gotten too far but

I think Dmitri is not a Knight but could be a Knave. And about the mole, your left, or my left?

.

I think it is a fun puzzle. I like these things. I'm impressed. Xopowo!

[George Kaplin]

I like logic puzzles so I'll definitely give this a go. I've only just seen it and at first glance this looks like a real toughie so it might take me a couple of days.

[FlyingFood]

Hmmm. In Russia, puzzle solves you?

My answer:

All 3 are guilty. Dmitri is knucklehead, the others are Knaves. No moles exist. 2 & 7 are only true statements.

And some of the highlights of the logic:

If 6 is true, there are no knuckles so it the problem is simplified, and eventually you get a contradiction. So 6 false, Alexie is Knave or Knucklehead. Use 4 & 11, and number of guilty people, get some more contradictions. Establish that Ivan is not a Knight. Something something Dmitri is not a Knight. Use 2, 7, and Dmitri to set up a couple possibilities of number of Knuckleheads & number of guilty, go through all statements and figure out truth values

[Swammerdami]

Thanks for the interest from George, hilarity and Flying!

Originally Posted by FlyingFood

Hmmm. In Russia, puzzle solves you?

My answer:


And some of the highlights of the logic:

Well-solved, FlyingFood. But now I'm intrigued by something else. You joined the Board to post this solution? Are you a long-time lurker? ... Or are you following me around?

Welcome to the Geeb! I don't think you're allowed to send or receive PMs until you make a few more posts. There's a thread just for that, but I don't remember where it is. Perhaps you can post in the following thread where someone has been waiting almost 9 years to learn what brand of sherry is best for onion soup. https://www.giraffeboards.com/showthread.php?t=24938

[JackieLikesVariety]

Three days and no response. :-( Please give me some feedback.

I only just saw your post as I ignore the game room typically. but it is an interesting puzzle. I'm not even halfway into my first cup of coffee so no way I could hope to solve it yet, but maybe later?

anyway, good effort!

[Mahaloth]

My answer is:

Smerdyakov

My logic is that I read the book.

[The Mighty Quinn]

That was fun! My solving process was as follows:

1) D can't be a Knight, because he said that there were no Knights, a statement a Knight could never make.

2) If D were a Knave, the other two must both be Knights, because that's the only configuration for which the statements "there are no Knights" and "there is exactly one Knucklehead" could both be false. But I said there was only one Knight, so that can't be the case. D, therefore, must be a Knucklehead.

3) Let's start by examining the simplest cases. The other two can't both be Knights, because of I's statement above. Could they both be Knaves? Yes! So D's true statements were "there is one Knucklehead" and "there are no Knights".

Therefore, we know they must all be guilty, because I and A were both lying when they said they weren't guilty. And D can't have done it himself, because then his statements "exactly one of us is guilty" and "there is one Knucklehead" would both be true.

Also, we know that the victim did not have a mole on his right ankle, because if he did, then Knave A's "two of these statements are true" statement would be true. He also can't have a mole on his left ankle, since D's statement to that effect was paired with the known truth "there are no Knights".

4) Go through and carefully check the other 7 possible permutations to make sure Swammi didn't make an error.

So far I've gotten through Step 3, and am unlikely to go further.

Thanks for the puzzle!

[C2H5OH]

Originally Posted by The Mighty Quinn

That was fun! My solving process was as follows:

1) D can't be a Knight, because he said that there were no Knights, a statement a Knight could never make.

a knight can't make that statement, but a knuclehead could if it was his true statement and the other is false. or even if it was his false statement and the other were true.

[The Mighty Quinn]

OK, I checked all the other possible solutions, and, although I was excited a couple times, none of them work. Very elegant!

[C2H5OH]

Originally Posted by C2H5OH

a knight can't make that statement, but a knuclehead could if it was his true statement and the other is false. or even if it was his false statement and the other were true.

Oops. I see you covered that in 2). I haven't found any flaws in your thinking after all, but I haven't run through all the possibilities yet.

[Swammerdami]

Here is my own solution write-up:

Dmitri isn't a Knight [(7)]. If Dmitri were a Knave then there would be a Knight [(7) false] but no Knucklehead [(2) false; no room left for 2 Knucks], so (6) would be true and Alexei a Knight. But then (4) can neither be true [Ivan would be a 2nd Knight] nor false [Ivan would be a lying Knight]. So Dmitri is a Knucklehead. Alexei isn't a Knight [(6) is false]. If Alexei were a Knucklehead, then (2) false, so (1) true [Dmitri Knuck], so (3) false and (4) false [no Knights] so Ivan would be a Knave, so (10) false, (7) true, (1) true [above], so (12) true, so (11) false [Alexei Knuck by hypothesis]; but in fact (11) would be true -- Ivan would be the unique Knave. So Alexei is a Knave by contradiction. Ivan is a Knave [there must be a 2nd Knave [(11) is spoken by liar]]. (5) and (9) are now confessions of guilt; (3) is also false so there is a 3rd murderer. Three murderers; zero moles.