@MindcraftMax what do you say? It's been a week. Do you think @robert's "strict interpretation" of the rules is reasonable (probably leading to an N/A resolution), or can we resolve YES? Or wait longer?

I'm almost certain that this riddle is an easier version of the one in Sheafification of G's video on the topic: https://www.youtube.com/watch?v=uUPr07ThSH0 . So, considering you can solve it in 3 questions even if you replace true/false with either foo/bar or bar/foo it should be pretty easy to say yes.

Solving one of the logic puzzles of all time!

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@AndrewPeterson960c it's different. In that one the labels have to do with truth telling, and here they do not

So, @MindcraftMax how is this going to work? Will you give a ruling on how to interpret the "no attemting to fix the coin outcome" rule? Or maybe a deadline for ppl who want to argue NO to make their case?

Otherwise this could just sit here indefinitely, waiting for new suggestions that never come.

@AhronMaline I'm considering your solution to be correct, and I don't think the questions you stated are infringing the validity criteria but I'll let at least one week to see what the others think and make sure we come to an agreement before I resolve.

@MindcraftMax according to the rules you laid out in the description, you cannot resolve YES unless someone has a solution for interpretation 1.

@MindcraftMax if you do not understand my proof that no solution exists, please get an llm to help you understand it

@MindcraftMax It seems to me that this solution violates the following criterion.

Questions must not rely on knowing or conditioning on the outcome of any coin flip.

Furthermore, any solution that contains language along the lines of "Is your favorite color red if and only if you're obliged to tell the truth on this round" would similarly violate that condition.

@robert NBAP's solution relied on interpretation 2c (or 3, if you found it reasonable), that's why I couldn't resolve based on his answer.

However, AhronMaline's solution is more general in that it is independent of the interpretations ChristopherRandles laid out, since it avoids nested or hypothetical constructions (like "If I asked you" as in NBAP's, or "If you were to tell the truth"…).

For a more thorough explanation anticipating two possible objections:

Here the questions are of the form: "Is A xor B true?", equivalently "Is one and only one of the following assertions true: A; B?", where A is a statement on their favorite colour, and B is a self-referential statement.

Evidently, you could say the flipper might need multiple coin flips, one for each internal sub-proposition in their reasoning – even though there's no hypothetical involved, and only a single (true or false) proposition is actually given as a final answer. But aside from stretching the interpretation quite far, this would lead to undefined behaviour, as multiple normally equivalent paths to the answer could now yield different outputs (e.g. "A xor B" vs. "(A or B) and not (A and B)").

Therefore, given A and B atomic statements, I find it only reasonable that the flipper uses a single coin flip to answer "A xor B", and not more.

BUT on the other hand, one could think self-referential statements are problematic. If you consider that a question like "Are you going to answer this question falsely?" requires the respondent to project themselves flipping the coin to fix the answer, and then flip the coin another time to decide whether to answer truthfully or not; that reintroduces the possibility of ChristopherRandles's different interpretations.

Except that, with interpretation 1, we arrive at a contradiction: if the coin flip that determines the actual answer to the question differs from the result of the simulated coin flip, their answer would be the paradoxical "Yes", rendering the question unvalid (violating rule #1).
But the problem lies elsewhere: when considering the flipper needs to simulate answering the question separately, "this question" in the simulated flip is no longer the actual question being replied to aloud: they're no longer answering the self-referential question they were asked.

So to conclude, I don't believe the different interpretations are at play anymore with AhronMaline's solution.

That said, I still believe one could be unsatisfied with self-referential questions. Even though it isn't explicitly forbidden by the rules, it may be argued that self-referentiality gives rise to other issues that disqualifies its use.

Let's see what others have to say.

@Elspeth I agree that "Is your favorite color red if and only if you're obliged to tell the truth on this round" violates the validity rule #2.

But AhronMaline's questions don't rely on conditioning, they use the proposition "you're going to answer the question falsely" as part of the question. See my reply to @NBAP @Kodan @AhronMaline from 3 days ago if you think this relies on "knowing the outcome of any coin flip".

@MindcraftMax

https://manifold.markets/MindcraftMax/does-the-riddle-provided-in-the-des#b2brvc1jfr5

@Elspeth and I are not the only people who see that there is a "strict" interpretation of the rule "knowing the outcome of any coin flip" that forbids asking such questions. Are you arguing that this is not a reasonable interpretation?

@robert According to your strict interpretation, is there any possible way to get information from Random? Or at least, the first-glance appearance that such a way could exist? If not, then the distinction that Random tells the truth or lies, rather than saying yes or no randomly, is irrelevent. Which argues against this interpretation, perhaps to the point of making it unreasonable

@AhronMaline I read the spirit of the riddle as you cannot get any information from the random person. I agree that the exact wording is a bit awkward for that, but I interpreted that as a result of it being hard to be precise in English.

I interpret the bullet point

– Questions must not rely on knowing or conditioning on the outcome of any coin flip. The flipper’s answer is considered truly random: neither you nor the other friends can infer anything from the flipper’s internal process, nor may a question attempt to "fix" the outcome of the coin flip by hypothetical reasoning (e.g., questions like “If next time the flipper speaks, they were to tell the truth, then...”)

To mean "One can probably think of loopholes to this. Don't do those," but in imprecise language because precision is hard and it is hard to think of all the loopholes one might want to prevent. Hence, the strict interpretation is warranted, even if other parts of the question are slightly awkward if the strict interpretation were intended.

Even if this was not @MindcraftMax's intention, the reader cannot distinguish between this and purposefully adding loopholes, making it a reasonable interpretation.

Riddles aren’t real, there’s no such thing as a riddle without a solution because some guy (usually the one who posits the riddle) ultimately decides whether any given answer to the riddle is the correct version of correct

@LukeShadwell That's why I made sure to also present clear resolution criteria in my own riddle and based on agreement, because I know it's the kind of things that always happen.

I'm even the kind (even though I'm clearly not in the majority for thinking that) to consider for example that the famous Sleeping Beauty problem has no true answer between the halfer and the thirder positions, precisely because it is not a well-posed problem to me (I consider there's an issue between the way we define a probabilty, the Beauty's singular identity/consciousness, and maybe even other things that are fishy).

But sometimes it's the interesting point of those riddles, if a riddle had a straight-forward singular answer then it would probably not be as interesting and clearly not an open/unresolved problem (and so not worthy of being a Manifold post).

Ok, I have an answer under one hopefully-reasonable interpretation: I will assume that if it is possible to consistently assign two different truth-values to a particular answer, the person giving the answer gets to make that assignment. For example, if you ask "wiil you answer this question truthfully?" then the truthteller will say yes and consider that the truth, and the liar will also say yes, considering that a lie. Under this assumption, the flipper must also answer yes to this question, on either outcome of the cointoss.

This interpretation enables a "truth serum": for any proposition P, you can ask "is it the case that (P XOR you will answer this question falsely)?"

The truthteller, or the flipper if the coin says to speak truth, must answer truly, and make truth-assignments consistent with that. So they will judge the second clause to be false, and then, to make the full statement be true, they must answer with tje correct truth value of P. But the liar, or the flipper when lying, needs to make the full statement false while also judging the second clause as true, so they too must give the true answer to P.

Thus you can find out the truth for any P with one question. Now you have 3 questions to decide between 6 possibilities; more than enough. And it doesn't even matter who you ask.

@AhronMaline I believe you found a way to circumvent the way @NBAP was relying on a particular interpretation of the problem, and your answer seems to me like it's indeed solving the riddle in any way one interprets it.

Still, I'd like a fully written out solution including the precise questions before I pin the comment, and I'll let other people tell what they think, like whether or not the questions respect the validity rules and if the solution respects any possible interpretation of the riddle.

I'll resolve the market based on agreement between us on validity of the solution.

@MindcraftMax well, of course there is the interpretation that self referential questions like that are logically meaningless and hence invalid. The interpretation I was using is fairly common in puzzles, but not in mathematical logic (or really in common sense).

Writing the exact questions is easy, but I don't have time right now. I'll do it tonight.

@MindcraftMax OK here goes: as mentioned, in this approach it does not matter who you ask the questions to. For specificity assume all questions are adressed to Alice.

Question 1: is it the case that either your favorite color is red, or you will answer this question falsely, but not both?

If yes: Alice' favorite color is red. Now we only need one more question.

Question 2: is it the case that either Bob's favorite color is green, or you will answer this question falsely, but not both?

If yes then the colors are RGB, if no they are RBG.

If the answer to Question 1 is no, then Alice's color is not red. Then the other two question are

Question 2: is it the care that either your favorite color is green, or you will answer this question falsely, but not both?

Question 3: is it the case that either Bob's favorite color is red, or you will answer this question falsely, but not both?

If yes-yes the colors are GRB, if yes-no they are GBR, if no-yes BRG, if no-no BGR.

@MindcraftMax we can also make it less self-referential by replacing "answer falsely" with "answer in a role that at that point requires you to lie". That should be well-defined even in mathematical logic, I think.

OTOH, that change also brings it closer to being an "attempt to 'fix' the outcome of the coin flip by hypothetical reasoning" which you forbade.

@AhronMaline It's a bit unclear to me what "this question" refers to in this context. When you write:

> is it the case that either your favorite color is red, or you will answer this question falsely, but not both?

Do you mean this?

"Is the following proposition true: 'Your favorite color is red XOR you would answer falsely if asked "Will you answer this question truthfully?"'"

In any case, it's a very clever insight, but I'm not sure that it gets around the obstacle I ran into, namely that it depends on how/when Random flips their coin(s).

@NBAP "this question" = the full question I am currently asking you. There is no hypothetical involved; it's about how someone will actually answer.

@AhronMaline Ah, I see. So it's essentially the same as saying, "Is it the case that red is your favorite color XOR you will answer this question as False would (i.e. falsely)"?

I understand, though now my concern would be that this runs afoul of the rule against conditioning on the result of the coin flip. It's not immediately obvious, but it seems to me that this is logically equivalent to the following:

"Is it the case that red is your favourite color XOR (you are False OR [you are Random AND the result of your coin toss was tails])?"

It's not clear to me if this is the kind of thing @MindcraftMax had in mind when they prohibited conditioning on the result of the coin toss, but perhaps they can clarify.

@NBAP since you don't know what coin flip result corresponds to what behaviour, I think it's not equivalent to that wording; also if you think about it, gaining information from the random answerer could always be broken down in this way, and so you wouldn't be " allowed " to ask questions to them, which isn't in the spirit of the riddle imo

If Random chooses to act like truth or to act like lie, it basically makes them vulnerable to the truth serum, which allows for @AhronMaline 's answer

@NBAP yeah it's not very clear how that rule is meant to work. If you interpret it very strictly it would mean you can't ask anything where the true answer depends on the coin toss. But that would also rule out the "if I would ask you" trick (in the versions where it works), since there also, we are effectively conditioning on the hypothetical answer which depends on the coin toss. In fact I'm pretty sure it would similarly rule out any way at all of getting information from Random. But the description does seem to imply that there's some way of doing that; otherwise why specify that he chooses truth versus lying and not yes vs no?

So I think we need some much narrower interpretation of what's called "fixing the coin toss". And I think asking about the truth of the answer itself is different enough to not get hit by that.

@NBAP @Kodan @AhronMaline I prohibited conditioning on the outcome of any coin flip, like in my example "if next time the flipper speaks, they were to tell the truth,…".

I believe this doesn't prevent from asking a question involving their current answer, because once the flipper knows the result of the coin toss, their answer no longer depends on the result of any (subsequent) coin toss, and you didn't "fix" the result of the past coin toss.

And if in fact, you considered it was actually conditioned on the result of a coin toss – the one that just happened – then like you said, no matter the question even if it isn't self-referential, we would always need to consider it is breaking the conditioning rule, since the answer depends on the result of the coin toss anyway. Which obviously isn't reasonable (you would never be able to ask any question).

So to me, your solution is perfectly fine to me, even though indeed the questions are self-referential (but I didn't forbid that). If people have a more satisfying answer, feel free to give another go (not sure it is possible though).

I'm still waiting for a bit to see what other people think of this solution before resolving, I want to be sure an agreement has been reached on the validness of the questions and the fact it is indeed a correct solution (although for the latter I think we can already agree, since it's easy to check the solution, way easier than checking the attempts at a proof that there is none).