Hello @mods , could you please resolve to N/A, given both AhronMaline's solution under one valid interpretation of the riddle, and their proof that no solution exists under another interpretation which hasn't been rejected as unreasonable.

New and third proposed solution, taken from @AhronMaline but without self-referential questions:

Questions are addressed to Alice.

Question 1: is it the case that either your favorite color is red, or 2+2=5, 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 2+2=5, 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 2+2=5, but not both?

Question 3: is it the case that either Bob's favorite color is red, or 2+2=5, 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 this doesn't work; "XOR 2+2=5" has no logical effect on a statement. Truth and Lie will give opposite answers on these questions, just as if you had omitted the second clause.

"XOR you will answer falsely" works because that clause is actually true for Lie but false for Truth.

@MindcraftMax I'm pretty sure you need self-refertial questions to solve it (if your interpretation allows it to be solved)

@MindcraftMax
>"Question 1: is it the case that either your favorite color is red, or 2+2=5, but not both?"

Is this one question for which there is a truth answer for any person and then if liar answers the question they say the opposite? This is what I assume to be the case.

However, is there a different interpretation where the liar lies to each part of the question then applies the Xor logic or even a third interpretation when they lie to each part then calculate the XOR and lie about the answer to that XOR calculation?

@ChristopherRandles I think it is clear that you parse the entire question as one statement, with an overall NOT for No answers, and this is the statement that must be true or false per the rules. That's what "telling the truth" and "lying" mean. There is no reason to invert clauses separately.

@MindcraftMax What's going on? When will this be resolved? I am becoming very impatient.

@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?

@MindcraftMax ping

@AhronMaline @robert @Elspeth @NBAP @Kodan Elspeth's strict interpretation doesn't just mean questions to the flipper are forbidden, it also implies you can't ask any questions at all, since you don't know if you're facing the flipper when asking.

I personally believe it renders this interpretation unreasonable.

And obviously, if you think it is reasonable, we therefore have a proof that it has no solution since we can't even think of a single question, which means it'll resolve NA (because NO for this interpretation, and YES for other interpretations in which Ahron Maline's solution applies).

For robert's initial objection that interpretation 1 could apply to Ahron Maline's solution, or more broadly that self-referential questions are not valid, see my previous reply.

If we end up with an agreement which interpretations are reasonable and if their solution applies to them, or find a better (i.e. non-self-referential) solution, I'll resolve YES; but for now we are in "NA or YES" territory depending on whether an interpretation has no solution (and its proof) or we always have a working solution for any of the interpretations.

We lack the reactions of other earlier commenters and market holders, and I want to allow all the time we need to settle the answer.

@MindcraftMax If I understand correctly, @robert 's proposed interpretation understand the rule as invalidating questions where the true answer depends on the coin flip. So ordinary questions would be okay.

But it does seem to mean you can't get any information out of Random, since the flip chooses between true and false answers that must be independent of the coin flip, which rules out self-reference so the flip must be between yes and no. We lose the specificity of Random "telling the truth or lying".

Assuming this, @robert proved in an early comment that three questions are not enough. So if his interpretation is judged "reasonable" then I guess we should NA.

(I'm arguing against myself here - I mainly want this to settle one way or the other, rather than keeping my mana tied up)

@MindcraftMax I would just sell shares rather than wait, but then if this NAs the mana gets clawed back and makes a big mess. So I hope you can resolve soon.

@MindcraftMax @AhronMaline accurately summarized my interpretation. The comment linked below defends it. I have one further argument if that is not convincing that it is reasonable. (Sorry for getting to this late, I was camping without internet)

https://manifold.markets/MindcraftMax/does-the-riddle-provided-in-the-des#kfsurtfa9p

@MindcraftMax I'm sorry to be a nudge, but please can this be resolved, either Yes or NA? What's the delay about?

@MindcraftMax Hi, could you please come to a resolution? No one seems to have anything new to state and I have loans to pay back.

@Velaris I finally had the time to look at Ahron's solution more deeply and managed to get rid of the self-referential bit, which was causing the disagreement that prevented me from resolving YES.

It’s quite simple in retrospect: just replace “you will answer this question falsely” with any non-self-referential contradiction, like “2+2=5.”
For instance, Question 1 becomes: “Is it the case that either your favourite colour is red, or 2+2=5, but not both?”

The way this solution works is still the same, but now the questions fit all reasonable interpretations of the validity rules – except, of course, if you consider those rules forbid asking any question at all, which I think nobody has argued after I raised the issue.

I’ll still wait until Saturday, since I posted this revised solution a bit late, to see if anyone finds another potential problem. Otherwise, I’ll resolve YES.

@MindcraftMax it seems you are ignoring my interpretation. Do you find it unreasonable because it relies on the interpreted spirit of what you wrote?

@MindcraftMax here is another interpretation:

one tells either a truth or a lie at random,

Usually, either ... or means exactly one of. In Ahron's solution, the random person is telling both a truth and a lie, which is not allowed.

@robert no, he flips his coin snd then answers either a truth or a lie, just as the other two would answer.

@AhronMaline but the answers if they were telling the truth or lying are the same. Therefore, when giving any the answer, they are telling both a truth and a lie

@MindcraftMax Thank you for the clear deadline.

By the way, it isn't the care :

Question 2: is it the care that either your favorite color is green, or 2+2=5, but not both?

@robert It's not "both a truth and a lie at once". It's a statement that may be consistently assigned as either a truth or a lie, like the statement "This statement is true". The assignment will then be whatever the speaker needs it to be to fulfill their Role. As I said initially, this concept of truth is not typically accepted in mathematical logic, but it is a common assumption in puzzles like this one.

@Velaris Haha I wish... the deadline was only based on @MindcraftMax 's new solution. They were hoping to avoid making the call on the interpretational question. But as I pointed out, the new solution is incorrect. (I considered keeping silent about that but figured that @robert would notice the error anyway, so speaking up moves things forward faster). So the unpleasant call will need to be made, and we haven't seen a deadline for when that will happen.

@AhronMaline What does it mean to be a truth or a lie? If a truth has to have the intention to tell the truth, and a lie the intention to tell a lie, then you are right. If telling the truth versus a lie is about outcome, then it is both a truth or a lie at once. It comes down to interpretation...

@robert what does "about outcome" mean? If we don't accept "assigned" truth values, then this sort of self referential statement is simply meaningless. If we do, then once the assignment is made, the statement becomes true or false but not both.

Again, it's the same as with "This statement is true".

@AhronMaline about *the outcomes

"This statement is true" is both true and false.

one tells either a truth or a lie at random, decided by a fair coin flip

We don't know how it is decided. The random person could flip the coin to determine their answer, then only give that answer if it exactly one of true of false (and repeat otherwise or something). That is up to interpretation.

@robert What? Now I'm just confused about what you're trying to say. But the phrasing seems very clear: the coin flip selects between "truth" and "lie", and the flipper then answers accordingly.

I still don't know what you mean by "both true and false", but anyway it doesn't matter. There is nothing in the description that forbids the flipper from saying such things. If True and False are allowed to say "this statement is true", then Random is allowed as well.

@AhronMaline

If True and False are allowed to say "this statement is true", then Random is allowed as well.

Why is that true? The description forbids Random from saying a statement that could both be a truth and a lie. True and False are not subject to such a restriction.

But the phrasing seems very clear: the coin flip selects between "truth" and "lie", and the flipper then answers accordingly.

If we accept that Random is forbidden from saying answering a question in a way that could be both (or neither) true and false, then the coin flip selecting between "yes" and "no" randomly and between "truth" and "lie" are indistinguishable. We do not know Random's process. They could be doing this.

@robert How are you proposing to parse the words of the description? It says he either tells a truth or a lie, decided by a coinflip. So the coinflip is choosing between these two behaviors. Which in turn presumably use the same concept of truth used by the other two players. In some cases the two lead to the same final answer; why would there be any problem with that?

@AhronMaline

But the question does disallow telling both a lie and the truth. I am giving an example of an internal mechanism where answering a self-referential question leads to both a truth and a lie, or neither (which are both disallowed). This example is mostly irrelevant, so it is probably fine ignoring it.

As I said before, if you define truth and lie to be about intent, then the line we are discussing does not disallow self-referential questions. However, if we define truth versus lying in the language of logic (I.e. saying yes to "is P true" is telling the truth iff P is true and say yes to "is P true" is lying iff P is false) then answering such a question is both telling the truth and lying. This is what I meant before by describing truth telling by outcome.

The flipper’s answer is considered truly random

This cannot be the case when asking a question that will be answered the same way if telling the truth or lying.

@robert I am also using the "language of logic". The reason that both the truthteller and the liar are able to say "this syatement is true" is because once the truthteller assigns it to be true, then the statement - that it's true - does in fact become logically true. The liar assigns it to be false, in which case the claim that it's true is in fact false. By the same token, neither of them are ever allowed to say "this statement is false", because it has no self-consistent assignment as a true or false statement.

I really don't see any hint in the text that would forbid the flipper from doing the same thing as the others for this question.

The line about "truly random" means that we shouldn't treat the coinflip as a pseudorandom but deterministic process, with some hidden preixisting outcome. There is no reason to read it as ruling out cases where the randomness doesn't matter.

In short, I think this new interpretation is completely unreasonable.

@AhronMaline If you accept the reading of a truth xor a lie as reasonable and take the "saying yes to "is P true" is telling the truth iff P is true" definition of truth versus lie (which is a legitimate definition of truthfulness), then it follows that you cannot ask such questions to Random.

Also, at this point, we have all been staring at this question for long enough that we cannot properly evaluate what is reasonable.

@robert So first of all no, I don't know where you got that xor from. The text says or, not xor. But in fact that's not the point - the coinflip does choose between two options which are mutually exclusive. To tell the truth or to lie. It's just that for this particular question, those two choices lead to the same final yes-or-no answer, for different reasons. There is no such thing as "both true and false".

And I am quite confident that I would have judged this as unreasonable on the first day I saw the riddle.

Note that I have not said this about your first proposed interpretation, that the true answer can't depend on the coinflip. There I am not sure whether it should be called reasonable. @MindcraftMax still needs to judge on that question. But this new one is just baseless IMO.

@AhronMaline I agree it is stretching it, but the only reason that it is stretching it is that the spirit of the question, not the meanings of the words in the question themselves, make it clear that xor was not intended.

I proposed it because if my other interpretation is deemed unreasonable because it relies on the spirit of the question, then this one, for consistency, should be deemed reasonable.

@robert why do you say that your first interpretation "relies on the spirit of the question"? It certainly was not what @MindcraftMax had in mind, as the discussion made clear. And the goal was to find tricks to get information, so why would the "spirit" be to rule out such tricks?

@AhronMaline the problem is that the discussion has made clear that MindcraftMax intended for there to be loopholes, but that was not true for the question's wording. Without the discussion, one cannot know that the intent was for there to be loopholes, and the question reads like the spirit was for there to be no loopholes.

The reader of the question cannot tell what the intention was. It should not matter what is intended, only what is written.

@robert @AhronMaline you are completely right, this solution actually doesn't work, "xor contradiction" isn't doing anything.
In my head I had calculated the truth table with "xor your answer to contradiction is no"; except obviously if we actually change the wording of the question to that, we end up with @NBAP's solution and its pitfalls all over again, as @ChristopherRandles's interrogation would apply here.

About "this statement is true", or phrased as a question "is your answer true?", I don't think there's a problem. The real issue would be with "is your answer Yes?", because the truth-teller could answer both "Yes" or "No", meaning the question is not valid (and on top of that the liar can't answer neither "Yes" nor "No").

At least I now have a clearer picture of what went wrong: I see now that the issue lies in the fact that AhronMaline and NBAP found propositions whose truth value were different if you asked the flipper when they're telling the truth and when they're lying. If you can find such a proposition, and it doesn't rely on conditioning on simulated questions and isn't self-referential, then you'll probably have found a satisfactory solution under all reasonable interpretations of the riddle.

On the contrary, if we find at least one reasonable interpretation where we can prove that no such proposition exists, then we'll be a step closer to a full proof (we would then still need to see how many bits of information are required vs how many can be retrieved; I tried to think of a brute-force examination of the truth table of all possible classical propositions, but obviously there are 36 possible combinations of truth-teller/flipper/liar and R/G/B, so 2^36 possible cases just for the first question. Unless we can exploit some strong symmetries, a proof based on that idea seems far-fetched…)

That means we're still far from a clear resolution unfortunately, as I don't think there's agreement on AhronMaline's solution being good enough, from I have read in the comments.
And we're still nowhere close to an N/A resolution, as we need a proof that no solution exists in another reasonable interpretation.

@MindcraftMax It still just boils down to whether you consider @robert 's first interpretation to be reasonable. We cannot decide that by further discussion; it's a judgement call that needs to be made, and you are the judge. Please decide one way or the other about this.

In this thread, Robert has also suggested another, new interpretation that would rule out my solution So I guess you need to rule on that one as well. But I think that one's clearly unreasonable, and Robert has also admitted it's a "stretch", so hopefully that second judgement call will be easy.

If you do judge Robert's first interpretation to be reasonable, then we should be able to resolve N/A. That interpretation understands the "no conditioning on the coinflip" rule as explicitly ruling out all questions in the category you mentioned - propositions whose truth value depends on the coinflip. That means that for all valid questions, the proposition must have an external, existing truth value. In which case, the coinflip between truth and falsehood decides between Yes and No answers, and so the flipper's answer provides no reliable information.

If we take that to be the rule, then Robert proved long ago that there is no solution. I'll copy it here (BTW, how do you do block quotes?):

"Suppose that you have a strategy that solves this riddle. There must be some last question on which your guess depends i.e. if the answer to this question is a yes, your determination of the favorite colors is different than if the answer is no. Therefore, this question cannot be asked to your random friend.

Therefore, after those at most three questions, out of 36 possible assignments of truth telling/favorite colors, there are at most 4 it can be. 4/36 = 1/9 < 1/8, so this is impossible and a successful strategy does not exist."

I'll explain a bit more: at the end of the process, you need to know the three colors, and also that the last person you spoke to is not the flipper. If you don't know that, you couldn't have learnt anything from their answer. So you must have ruled out two out of the six Role assignments, leaving only four, along with the fully determined colors. 3 bit of information cannot possibly be enough for that.

@robert @MindcraftMax Hmm, maybe the reasoning that "you need to know the last answerer is not Random" isn't airtight. So here's another proof:

Call the first answerer A. You begin with 36 possibilities altogether. On 12 of them, A is Random, so whatever you ask, Yes and No answers are both consistent with those 12 possibilities - they cannot be ruled out. So at best, you can ask a question such that each answer rules out 12 of the other 24 possibilities.

Now of those 24, 12 have B as Random and 12 have C. If you want, you can set it up so each answer rules out precisely the 12 possibilities for either B or C to be Random. For example, ask A "if I were to ask you 'is B Random', is it possible that you would answer Yes?" But we know this doesn't solve the puzzle. It leaves you with someone who definitely isn't Random, but no information at all about the colors. Then you have all six color possibilities and only two more question, which is not enough.

On all other choices for your first question, at least one of the two answers will leave you with all three friends still possibly Random, each with at least 1 of the original 36 possibilities. In fact, you should choose a question such that this happens on both answers: if your best-case answer rules out all the 12 possibilities for, say, B to be Random, and your worst-case answer doesn't rule out all 12 for C to be Random, that would mean that in the worst case you rule out at most 11 possibilities, along with not deciding the Roles. It's better to have each answer rule out 12, mixed between the possibilities of B and C being Random.

So now we are at the second question, with 24 remaining possibilities. No matter who you ask, they may be Random, so there is at least one possibility that you won't be able to rule out. At best, you might find a question that rules out 12 possibilities in the best case and 11 in the worst case.

Thus in the worst case, you are left with at least 13 possibilities when reaching your last question! Then the best you can do is to rule out 7 in the best case and 6 in the worst case. So in the worst case you are left with 7 possibilities, meaning you do not know all the colors.

@AhronMaline Okay, that's great! It took me a long time to process, but it looks to me like there's no loophole in this reasoning.

Still, it's only the second part of the proof. For it to be complete, there should be a reasonable interpretation under which no question can possibly involve a proposition that could distinguish between the flipper telling the truth and them lying, in particular it should exclude self-referentiality.

I'll accept more hand-wavy reasoning for this part, as it probably boils down to how we interpret "reasonable" anyway ^^ But now we're talking, I think we're actually much closer to a resolution.

I might do a poll if the arguments end up not being convincing enough, between something like

"1) AhronMaline's answer OK in every reasonable interpretation of the riddle (→ Yes)",

"2) Reasonable interpretation exists such that no proposition has a distinct truth value for flipper telling truth vs lie, all else being equal (→ N/A)",

"3) can't settle because needs further proof for claim in 2) (→ no resolution yet)".

@MindcraftMax There is nothing to prove on that step, because the interpretation in question understands "conditioning on the coinflip" to mean exactly that! My solution would be invalid precisely because the proposition in question has a truth value that depends on the coinflip, hence is "conditioned on it".

If you find that hard to see, perhaps that is a sign that this shouldn't be considered a reasonable interpretation!

@MindcraftMax Please see this comment as justification for that interpretation.
https://manifold.markets/MindcraftMax/does-the-riddle-provided-in-the-des#kfsurtfa9p

The fact that you cannot see it is because you know what you meant when you wrote question, not because a reasonable person could not interpret it as such.

@AhronMaline @robert I'll wait till Saturday to see if there's any objection from anybody, and resolve N/A otherwise (or sooner if someone gives a more convincing interpretation yet).

@MindcraftMax I take it this means you have judged the above interpretation to be reasonable, even though it effectively reduces the flipper to simply saying Yes or No at random. Very well, so be it. At least the wait will be over and my invested mana freed up.

@MindcraftMax if you @ mods (remove the space), they can resolve this NA for you.

@robert the mods would never, some of them hate n/a'ing.

@Velaris I'll try anyway, don't see what else I can do.