Talk:The Hardest Logic Puzzle Ever

Explanation of the question(s) used in Boolos Solution
The idea is to use the effect of "iff" to design a question that will direct the True and False to have same answer about anything he include in the question. (As shown in the illustrated diagram)

Logicians have introduced the useful abbreviation "iff," short for "if, and only if." The way iff works in logic is this: when you insert iff between two statements that are either both true or both false, you get a statement that is true; but if you insert it between one true and one false statement, you get a false statement.

The key point is that he designed a question using iff so that the answer is directly tell the truth about anything he wanna know (the “X” in the question) from the True and False guys.

So, he used this designed question to ask True and False about the location of the random guy, and so he was able to easily proceed with his solution in the other two questions (Q2 & Q3). Brilliant!

The idea is to use the effect of "iff" to design a question that will direct the True and False to have same answer about anything he include in the question. (As shown in the illustrated diagram) Actually, you can reach to the same thing, by a simpler question: (Q1: Ask A ‘’if you ask the none-random guy (from the other two) ‘is B=R’, then would you answer be ja?”), which is based on a similar idea, which is: (The golden rule applied here: T+F=F; that’s if you ask T (or F): if you ask the other one, F (or T) about a truth (which they both know), then the truth is the opposite of the answer you got.)  — Preceding unsigned comment added by Essam.abdellatif (talk • contribs) 13:45, 17 April 2013 (UTC)

What's the point of any of this? It appears to be Original Research and has nothing to do with improving the article. -- Jibal (talk) 11:47, 5 May 2017 (UTC)

Another solution which is more rational and simple
This is a more rationale and simple solution of the hardest logic puzzle (invented by Raymond Smullyan and solved by G. Boloos, 1996). It is just has a different road-map (milestones) and a different set of questions that the original solution has. Its road-map of this solution is: (1) first 2 moves has one goal (goal1) which is to identify the “random” guy in 2 questions, and then, (2) goal2 is to identify who is the True/False by the 3rd question.

The first goal (goal1) has 2 milestones (one per each question: (1) identify one of the none-random guy by the 1st question, (2) identify the other none-random guy, and hence identify the random by the 2nd question



And it's based on the following 3 key concepts:

key#1: The golden rule applied here: T+F=F; that’s if you ask T (or F): if you ask the other one, F (or T) about a truth (which they both know), then the truth is always the opposite of the answer you got from them.

'''Key#2: If Q(something), ja? then answer(ja)=yes, and answer (da) =no.'''

But, please notice that :the answer you get here is just yes or no; it does not mean they are True or False about the thing is asked about. The rule of this is working as follow: Any question about "something" (that you don't know, and wanna know actually) and you include the part " would you answer by ja?"(whatever ja=yes, or ja==no) in the question, then when you get the answer "ja", it means two things (not just one):


 * ja, this thing is true, which is answered by a True person (which means this thing you are asking about is really the truth), or,
 * ja, this thing is true, which is answered by a Lair (which means this thing you are asking about is really not the truth)
 * Indented line
 * Indented line

Finally the 3rd key concept is that if I questions you about something that I already know it's truth and include "would you answer ja" (regardless ja=yes, or ja==no) ) in the question, then who answers "ja" he is True, and who answers "da" is False. So, Key#3: If Q(truth, that you already know), ja? then answer(ja)=True, and answer (da) =False.

The goal1 (first milestone) is achieved in 2 steps (using Q1 and Q2 and applying key1 and key2), while goal2 is achieved in the 3rd step (using Q3, and key3)

And, the questions set of the solution is as follow:

Q1: Ask A ‘’if you ask the none-random guy (from the other two) ‘is B=R’, then would you answer be ja?” '' So, if we got the answer ja, it means: And in both cases, we are definitely sure that B is not R.
 * either A is truly the one of the none-random guys, and hence the truth is the opposite of their answer (i.e., C=R)
 * or, A is the random guy and both B and C is not R.

Similarly, if we go the answer da, it means: And in both cases, we are definitely sure that C is not R.
 * either A is truly the one of the none-random guys, and hence the truth is the opposite of their answer (i.e., B=R)
 * or, A is the random guy and both B and C is not R.

So, we are now, in both cases, identified one of the three who is a none-random (either B or C, in each of the ja and da answers)

Now, we are going to direct the second question, to that none=random guy we knew from Q1 (i.e., either B or C), as shown in the illustrated diagram.

Q2: ask the none-random guy identified in Q1 “if you ask the none-random guy (from the other two) ‘is A=R’, then would you answer be ja?”

And again, the answer we get is just the opposite of the truth we are looking for (and this time, in Q2, we already excluded the random guy, so we are having just one scenario per each answer (ja/da) in the 4 paths (cases) in Q1&Q2), and therefore we can easily identify the other none-random guy and hence, or of course the random guy.

Q3: I hold a blue pen and ask any one of them (Q3): if I asked you is this pen blue, would you say "ja"?
 * If the answer is "ja" then he is the True guy and the other one is the False. Or,
 * if the answer is "da" then he is the False guy and other one is the true.

(And that what's required!)

The attached diagram illustrates the solution with it's logic-tree of the possibilities and conditions.

I said this solution is more rational and simple because the original solution is a but complicated (and even I'm not sure if it's correct) and also some other solutions, such as the one which is based on the "embedded question lemma" I found it incorrect.

What's the point of any of this? It appears to be Original Research and has nothing to do with improving the article. -- Jibal (talk) 11:47, 5 May 2017 (UTC)

Is there a problem in the solution which is based on the "embedded question lemma"?
I guess given solution which is based on the embedded question lemma, has a serious mistake here!

They conclude from the answer of Q1 that: "Either way, C is not Random." and that's not true. Why not true? Simply and clearly because (and forget about the "ja/da" and also "including the word "ja" or "yes" in the question) when you ask B 'if A is random?', then we have 8 possibilities (4 per yes and 4 per no) and two of them are completely missed in the given solution, and they are as follow:
 * If answer is yes: B is T and A is R, B is F and C is R, or B is R (with 2 possibilities A and C are T and F whatever who is F/T)
 * Similarly, If answer is no: B is T and C is R, B is F and A is R, or B is R (with 2 possibilities A and C are T and F whatever who is F/T)
 * Similarly, If answer is no: B is T and C is R, B is F and A is R, or B is R (with 2 possibilities A and C are T and F whatever who is F/T)

So, we can never ever can conclude that C is not R by this question! (whether you use ja/da or modify the question to include "would you say yes?" or would you say ja?")



Also, please notice the following about including the word yes/no in the question: If I hold a blue pen, and ask you if this is a blue pen, would you say "yes"? You truly answer me "yes", and "no" if you want to lie. The same thing if I use "no" in the question; the true answer is "no", while the false answer is "yes".

But, and please take care of this, if hold a red pen, and ask you if this is a blue pen, would you say "yes"? The quite opposite thing will happen: true is "no" and false is "yes", and if I use "no" in that question, the answer would be "yes" (true) and "no" (false" >>>> the rule is reversed here! (when I ask about untrue fact)

So, you gotta notice that the difference between these two rules:

Rule#1: Any question about "something" (that you already know) and you include the part " would you answer by ja?"(whatever ja=yes, or ja==no) in the question, then when you get the answer "ja", it means only one thing. It means that this thing is true, and therefore this "person" is saying the truth; the truth that you already know about it (not yet looking for)

Rule#2: Any question about "something" (that you don't know, and wanna know actually) and you include the part " would you answer by ja?"(whatever ja=yes, or ja==no) in the question, then when you get the answer "ja", it means two things (not just one):
 * ja, this thing is true, which is answered by a True person (which means this thing you are asking about is really the truth), or,
 * ja, this thing is true, which is answered by a Lair (which means this thing you are asking about is really not the truth)

I guess the this solution wrongly use these two rules the thing leads to wrong conclusion (starting from Q1)

When you ask A: if B=R, would you say ja? So, the answer ja means 2 things not one (of course we already considered that A might be R, and hence his answer ja and da is meaningless, as correctly clarified in the soltuion):
 * (1) ja means the True guy (A) is answering that: true, B=R. (i.e., A=T, B=R, C=F)
 * (2) And, ja also means that the False guy (A) is answering that: true, B=R (and the truth is that C=R, but he is lying) (i.e., A=F, B=T, C=R)

Similarly, the answer da to that question, means also two things (not just one):
 * (1) da means the True guy (A) is answering that: false, B is not R. (i.e., A=T, B=F, C=R)
 * (2) And, da also means that the False guy (A) is answering that:false, B is not R (and the truth is that C=R,, but he is lying) (i.e., A=F, B=R, C=T)

And of course we still have the other 2 possibilities (R, T, F) and (R, F, T) if A is R himself!

So, you can never conclude that C (or even A or B) is not R, by this question (Q1 in the solution), whether the answer was ja or da!

Isn't that so? Am I missing something here?

I don't know how late I am to answering this, but for what it's worth, I believe this comment and accompanying chart have missed something important about the embedded question – it serves two purposes simultaneously. One is that it lets you treat "same answer" as equivalent to "yes", and "different answer" as equivalent to "no"; the original writer here got that. But the otherpoint is that it extracts the truth out of any consistent speaker. True will tell the truth about telling the truth, of course, but (here's the key) False will lie anout lying, and thus give the truth. This means that any question along the lines of "If I asked you Q, would you say ja?" will result in a truthful answer to Q such that "ja" can be treated as "yes" (unless you are talking to Random, in which case it's ambiguous). ± Lenoxus (" *** ") 19:28, 26 December 2013 (UTC)

What's the point of any of this? It appears to be Original Research and has nothing to do with improving the article. -- Jibal (talk) 11:48, 5 May 2017 (UTC)

I would just like to add that I came to the same conclusion as Lenoxus. We must be missing something because it seems unlikely that an incorrect answer would be accepted as correct for so long. 2605:E000:141D:C2FD:F137:CBA7:CA78:10A8 (talk) 00:13, 16 February 2020 (UTC)


 * You are missing one of the levels of indirection. If you go up to False and say "does 2+2=4?" then he will say "no", because he is lying.  If you ask him "What would you say in the hypothetical scenario where I asked you 'does 2+2=4?'?", then he will say "yes", because he is lying about the lie that he would give in your hypothetical scenario. Antistone (talk) 22:17, 11 May 2021 (UTC)


 * Well put. I'll add a suggestion for anyone working through this who is struggling on this point. I drove myself batty when solving this puzzle by phrasing it, "If I asked you ..." -- I would look at notes I had written earlier and wonder, "by 'his answer,' did I mean to the actual question or to the embedded hypothetical?" I found that giving the hypothetical to someone else, e.g., "If FDR was here and he asked you ..." made it much easier to keep things straight.

flow chart
that flow chart is unhelpful and is explaining the trivialization no the real solution. i suggest it be deleted or one be made for the actual solution section, if people find them helpful. —Preceding unsigned comment added by 71.196.245.150 (talk) 04:06, 6 January 2010 (UTC)

Apparent Single-Question Solution to the Clarified (Boolos) Puzzle
PhysicistQuery 09:08, 23 July 2007 (UTC): It appears that a single (logically trivial) question of the following nature solves the "completed" version of the puzzle:

"This question is for the god called False:

Is exactly one of the following two statements true?

Statement1: Either A is called True or you are A and B is called True

Statement2: Ya means Yes

Clearly, this is not the intended solution, but I cannot find why it should not be valid. Can someone either find the catch, or suggest why it has not previously been proposed? (My guess would be the wording of the puzzle and the powers of suggestion)

lwr314: We briefly considered such `questions' and decided that they were not legal. For one thing, it breaks 'each question must be put to exactly one god' since you are putting that `question' out there and each God thinks 'am I False?' and answers if he says 'yes' to himself. The other Gods are silent. So you are really asking three questions there since you are asking it to each God

Perhaps a bit simpler would be:

"This question is for the God called False:

E("Is the God to your left (in cyclic order) True?")"

Whichever God speaks is False and his answer determines the other two.

But it really is three questions. It isn't a very good three question solution either since it uses silence -- it is easy to solve it in two questions using silence.

PhysicistQuery 09:12, 24 July 2007 (UTC) I'll accept that the question by name can be interpreted as four questions, although I'm not clear how that is different from a question by position given that you possibly cannot identify the front of a god (and they may have undefined special powers). It's not worth re-arguing at this point, as it is uninterseting except insofar as it highlights how the question's wording might be tightened

lwr314: If you face each God and ask it, that is three questions (why would you not be able to repeat the same question to more than one God?  That is the oddest constraint ever.  You've got to stop with these weird rules.  Like i said before, we can add as many of these arbitrary rules as we like -- may favorite is ``No use of the letter `e' in your questions'', the puzzle doesn't say that we can, so to be safe...).

PhysicistQuery 21:52, 24 July 2007 (UTC) You write "why would you not be able to repeat the same question to more than one God? That is the oddest constraint ever"

I agree that it is an odd constraint, but it is not mine. The question reads "each question must be put to exactly one god". "Put" does not necessarily describe a single event, so your own objection as I quote it illustrates very nicely why the interpretation I used holds up. It is of the class that you must allow for, but cannot rely on.

lwr314: What? A question is a question, not a question form or a collection of questions that have all the same words in the same order. When i ask `Is the sky blue?' at 3pm and then i ask `Is the sky blue?' at 5pm, i have asked two distinct questions -- distinct since they have different properties (one of which is the time of asking). So, `putting a question to a God' is necessarily a single event. Just use normal English instead of some crazy new language and there will be no problems of interpretation.

Three does not necessarily mean the integer 3, actually all the puzzle meant was that the number of questions needed to be 3 in Z_5. So 8 or 11 or anything 3 modulo 5 is a fine number of questions. It didn't say it was talking about Z, must account for it being Z_5, or Z_2 even -- then i only get 1 question.

That is not a reasonable interpretation, just as your interpretation is not reasonable.

PhysicistQuery 14:15, 25 July 2007 (UTC) You are being condescending, and at the very least verging on rudeness. Please stop and reconsider.

This interpretation of the word "question" is not uniquely mine. Not only did you yourself use the word "question" in specifically the sense you are now decrying, but also Webster gives "an interrogative sentence or clause" as a principal definition for "question"; you can find this for yourself on the web. If you wish to argue this with the compilers of Merriam Webster, please take it up with them.

lwr314 (continued) i just don't understand how you read all these weird notions into a perfectly normal English sentence. Clearly, the clarification "each question must be put to exactly one god" is meant to disallow questions that are directed at more than one God simultaneously.

If you want to continue espousing these weird interpretations, i recommend making a new Wikipedia entry to house them (perhaps titled `fyz's interpretation of the text of the Hardest Logic Puzzle Ever translated into the native fyz tongue').

Arbitrary non-standard interpretations of words are not interesting.

PhysicistQuery 14:15, 25 July 2007 (UTC) So far as I know, solutions using the "don't know" option were not considered by Boolos. Nor did he repeat a question in any of the direct senses f. Not having godly powers, I cannot say, but it is possible that he relied on one of the areas that might disallow this, i.e.: uncertainty about interpreting the gods' answers; potential interpretations of "question"; ditto of "yes/no question"; or, "god" meaning some undefined but extended power. In any event, you cannot have it both ways - either this is about Boolos' paper, or it is about the Puzzle. If the former, that is what you should call it, and perhaps include valid critiques of the wording (including the additional solutions it allows); you could even justify excluding solutions to a different puzzle that allows the use of uncertainty. If it is about the puzzle, the encyclopaedic article should of course stick to a combination of published material and contribution by invited acknowledged experts (unfortunately, I don't think that Wikipedia has a mechanism for the latter); but I believe the talk section remains a suitable place for discussion of alternatives and ambiguities.

f When creating questions, I would personally try to create then so that even the indirect equivalence (any combination of logical and semantic steps) was avoided. In the end, however, I would abandon the more remote steps if they precluded finding an answer - the remote equivalence one would come first, but I'd be in some doubt about the ordering for the others. Fortunately, the case does not arise, as there is no need even for this type of duplication.

It seems to me that you are strongly wedded to unique interpretations - by unique I mean single, rather than unusual. If this is a puzzle, the solution is generally considered to be wrong if it does not work with all meanings that can be ascribed using dictionaries (combined with Fowler).

lwr314: Yeah, and `question' used to mean `dog-like creature with raised ears' in Zawenzian English. If you want to go interpret the puzzle in some English other than the English which Boolos' spoke and understood, that is fine. However, you are then discussing a *different* puzzle.

PhysicistQuery 14:15, 25 July 2007 (UTC) Perhaps you should check your facts before indulging in sarcasm? See above.

lwr314 (continued) 'The Hardest Logic Puzzle Ever' refers to a puzzle by Boolos!

That includes his interpretation. A pile of text on paper is not a puzzle, it is a pile of text on paper. An interpreted pile of text on paper can be a puzzle. Boolos' interpreted his pile of text on paper and that is called `The Hardest Logic Puzzle Ever'. If we deviate far from his intended interpretation then we are discussing a different puzzle. You are deviating massively and it is tiresome and uninteresting.

PhysicistQuery 14:15, 25 July 2007 (UTC) You are being condescending, and verging on rudeness. Please stop and reconsider. The actual point is answered elsewhere. (N.B. Are you saying that your paper where you repair Random doesn't deviate from Boolos' article?)

Fyz (24July continued) BTW, the idea of worrying that the constraints are odd or capricious seems very strange to me in the overall context of this puzzle.

lwr314: What? Why can you not understand that we can add any number of other arbitrary constraints? Should we not worry that they are odd? You just refuse to understand this point. You cannot use the letter `e' in any question! In old English a `puzzle' was something that needed to be done under water, so you need to solve this puzzle under water to be `safe'!

PhysicistQuery 14:15, 25 July 2007 (UTC) There is a difference between the constraints being added or being dictionary/Fowler interpretations of the wording of the question. The constraints you recognise are already odd - so oddness cannot in itself be a reason for rejecting an interpretation.

lwr314 i don't particularly like it as a three question solution since those are not yes/no questions (in the usual English sense -- ``Are you going to say `no' to this question?'' is a yes/no question in the usual English sense). You can go and change the meaning of `yes/no question', that is fine, but then you are speaking a *different language*, not English.

PhysicistQuery 21:52, 24 July 2007 (UTC) I agree with that - my oversight. It's only a valid question if you regard it as only being addressed to the god who is named.

Fyz: However, even if you allow this, the "two-solution" question remains unreliable, both because the gods might know what Random will answer, and because nothing in the question says that True cannot say "I don't know" - which could sound very like Yes or No to people unversed in the language. (And there is the issue of whether this is a yes-no question - which also affects the "single" question if you regard it as addressed to other gods than intended)

lwr314: What? How could it possibly be relevant that ``I don't know'' could be misheard as `yes' or `no'?

PhysicistQuery 21:52, 24 July 2007 (UTC) I thought of some possible objections, but that one quite took my breath away. The question doesn't say "... to devise some questions that would determine the identities if you could understand the answers ..." but "... to determine the identities ...". this is precisely the type of interpretation that you have to allow for, but cannot rely on.

PhysicistQuery 20:02, 23 July 2007 (UTC) Hmm - I think that is pushing things more than a little, especially if you support the view that repeating the same question could be permitted. When giving tutorials, I have frequently addressed a question by name to an individual in the group, and I have no doubt that everyone present would have considered that I put the question specifically to one individual. It's a bit like saying that, because the other gods might hear when I address a question to A that I have addressed it to them also. My view would be that if True, False, and Random are the gods' names, less consideration is likely to be required as to whether they are True, Random, or False than whether they are A, B, or C (or equivalently, whether you are 'facing' them or one of the other gods). My view is that, if you wish to avoid that possibility, just as for the delayed response and the nature of Random, you would be better to include it in the basic wording of the puzzle.

lwr314: There is a big difference between the situation you describe and the situation at hand; namely, you do not know which God you are addressing, you know his name, but you don't know which God that name picks out. The question `must be put to exactly one God', not `put to the name of exactly one God'. *put to the God*, meaning you look at him and you ask the question. You h

The other objection is the reason i was writing `question' in quotes. That is not a question. It is like a conditional question (as the recipient of the question is unknown to the asker when it is asked -- his name is known, but not *him*). All we have to do to rule out odd things is to speak and understand English.

Fyz: Actually, I did think of a potential problem - there is nothing that allows you to assume that all the gods are within hearing range for a single question; however, that objection should not work for a god a, and also requires the sort of puzzlers' "must make no assumptions that can be refuted by a tenable interpretation of the question" that appear to be so alien to other contributors to this talk page.

lwr314: It's not about `making no assumptions that can be refuted by a tenable interpretation of the question'. It is about speaking English and understanding English.

PhysicistQuery 09:12, 24 July 2007 (UTC) Put it how you will. Understanding English in this context includes both understanding what the words must inevitably imply, and what they may imply. You can make use of the first, but your solution must be tolerant of the second, although it cannot make use of it.

aMy understanding that their being puzzle-gods would invalidate this objection, as being a 'god' includes that they would know (including hearing) everything relevant, otherwise they would be oracles (special knowledge of the future only), or priests (behaviour constrained by duty)

The above, like my taking the view that nothing may be assumed that a tenable interpretation of the puzzle would disallow, rests on my experience studying in the same group as a dedicated group of puzzlers very many years ago. (As you have probably gathered, I am not normally a dedicated puzzler, but was tempted by the title of this puzzle.)

While in discussion, you were pretty heavy on the use of splitting as an extension. Personally, I would regard this as less trivial than duplicating the double negation - and at least it renders some purpose to the Ya-Da extension. (In fact, once I had thought about it I would probably include it anyway, as it doesn't delay the latest point at which you know the identities, and it gives a 1/3 probability that you discover the meanings of Da and Ya, as I tend to be greedy where titbits are concerned.)

BTW, wouldn't left-right in cyclic order require that you know the left and right of a god - or include the definition that left and right are as seen by yourself. (Next in cycle ABC might be simpler, but still fallible as a perverse god would simply return his answer for Random on every occasion). In any event, most of the complexity is in living with Ya/Da - four words versus parentheses is not a bad trade for a verbal inter-species question.

DoctorCaligari 29 December 2007

It seems to me that any question to a god regarding either the identity of another god or how another god would respond to another question assumes more than the given information. We aren't told, nor is it fair to assume, it seems to me, whether the each of the gods are aware of each others' identities and answering patterns. —Preceding unsigned comment added by DoctorCaligari (talk • contribs) 09:21, 29 December 2007 (UTC)

Anonymous assertion and discussion regarding "More rigorous interpretation of the puzzle?"
Anon: Everything below should be removed. This is not the puzzle. It is simply Fyz's unique re-interpretation of the puzzle. No academic articles published on the puzzle have supported Fyz's interpretation. The Hardest Logic Puzzle Ever is the puzzle published by George Boolos (which can be found Here). Any "controversy" over Conditions 1 and 3 below are settled in that article. Condition 2 (which would be better worded as "The gods will grant an answer to any yes-no question that they can answer") is nowhere to be found in the article. The possibility of unanswerable yes-no questions by all knowing gods was probably just not considered. But this possibility provides for very interesting solutions that have philosophical implications related to the Liar's paradox. Furthermore, as lwr314 has proven, Fyz's added constraints in conjunction with the standard interpretation of the puzzle makes it unsolvable and thus uninteresting. I leave it to others to fight with Fyz. I have made the case against her already before but she seems stubbornly committed to a misinterpretation.

Fyz 16Jul: It appears that you have taken it on yourself to remove the entire contribution. Please explain on what authority and expertise you do this. Note that I am by no means alone in believing that the puzzle solved by Boolos is not the same as that posed by Smullyan - see for example http://www.uweb.ucsb.edu/~rabern/SSHardPuzzle.pdf, which is reference 4 of the main article. What I have done is to elucidate why Smullyan's puzzle as written might be worth more attention than it has heretofore received. I now understand that everything in the encyclopaedic section needs to be referenced to other work, so you would be correct to remove my extension to the puzzle as written.

Anon the monkey: Look, man, there is no such thing as "Smullyan's puzzle as written". You have made this up. There are Smullyan puzzles as written but none of these are The Hardest Logic Puzzle Ever. There is Boolos' puzzle, i.e. The Hardest Logic Puzzle Ever, which is inspired by (or Boolos' re-working of or Boolos' take on or Boolos' memory of) certain Smullyan puzzles. I don't know how else to convey this. It is getting silly.

The Hardest Logic Puzzle Ever is not equal to some puzzle x such that x was authored by Smullyan.

The Hardest Logic Puzzle Ever is equal to some puzzle x such that x was authored by Boolos and was based on some puzzle y such that y was authored by Smullyan.

PhysicistQuery 21:37, 18 July 2007 (UTC) I know I won't convince you - you are wedded to the puzzle as you first met it. And maybe that is my issue too - I first met it without the clarifications - and came up with the confused gods and Boolos' solution on my way to the answer that made use of what you describe as broken random. The "difficulty" of the puzzle as I met and interpreted (see note next para) it was in the presence of red herrings. Given that I had come up with both solutions presented on the article page on the way, I think you must agree that the unclarified puzzle was more "difficult" for me than the clarified and fixed one would have been.

Note: this interpretation used the standard method for puzzles - that the solution must work for any tenable interpretation of the wording of the puzzle, and nothing that is not in the wording is used if it can possibly be avoided. Under that method Boolos' "clarification" really is a significant change - and I can't be the only person to see it like that, or he wouldn't have needed to answer that question in the first instance.

A more rigorous interpretation of the puzzle?
(by Fyz) A safe solution to any puzzle would make no assumptions that are not fully supported in the text of the puzzle. Although the puzzle as formerly interpreted remains both demanding and interesting, the derivation of solutions that would allow the gods' identities to be ascertained under other assumptions as well as under those made is even more demanding - and in my view more valid in terms of the puzzle as written, rather than as interpreted by Professor Boolos. This section is an attempt to outline the range of conditions under which a valid solution must still work. (The detail below is being gradually updated)

Condition1: The gods may answer any question at any time that is soon enough for you to reach the desired conclusion, and will answer in such a way that you know they are answering one of your questions. (These last two constraints are not explicitly supported, but are implicit in the puzzle being soluble). Note that this is not saying that this is what will happen - only that a satisfactory answer must still work if it does.

Justification: There is nothing in the question to indicate that the gods will answer each question before the next one is posed; indeed, they may not answer the questions in the same order that they were asked. That means that if you ask any one god two questions, there is no way to be certain that you will no which of his answers applies to which question. Note also that this possibility was specifically excluded in Professor Boolos' clarification, but its inclusion makes for a more robust solution - and a more demanding puzzle.

Condition2: The question as posed must be answerable truthfully by an answer that is either yes or no.

Justification: Again, we are taking into account a worst-case interpretation - we have no reason to suppose that the gods will not behave according to this very narrow interpretation of a yes/no question. In addition, for safety we should assume that including any question of this nature may allow the gods to answer none of the questions, as they can regard you as having breached the conditions of the puzzle. Note specifically that this possibility excludes the existing solutions outlined in "Exploding God-Heads". )

Condition3: "one god per question" can mean not simply that each of your three questions should be addressed to a single god, but that you cannot repeat the same question addressed to different gods. For safety, "the same" should be interpreted in the widest manner possible.

(Notes: If two questions yield the same information, they could be regarded as being the same question. Also, questions with identical wording that are addressed to different gods could be regarded as identical.  On that basis, we should cater for the possibility that any sequence of such logical and semantic steps between two questions would allow the gods to treat them as identical, and thus not be obliged to answer. As an example, that would make the following two questions identical: Question to god A: "Is B Random?"; and, Question to god C: "Is A either True or False").

As this "rigorous" interpretation of the question appears to be recent, I am not immediately posting the answer here. I will however say that it appears to require that the behaviour of Random needs to be as worded in the puzzle - Random's answers are either true or false, rather than randomly da or ya. For those more inclined to doubt that such a solution is possible, my attempt at a solution may be found at http://cr4.globalspec.com/thread/8840#newcomments post #59. (The wording of each "Statement 1" is logically OK, but could be improved to show it's relation to the original puzzle - see below for details)

Fyz

A more difficult version of the question
(which also has a solution)

Just add the following to the end of the question:

The gods will not answer any question until after all three have been posed, and will not answer any of the questions if the answers to any two imply the answer to the third.


 * I have a more difficult version, too: You don't know the words 'da' or 'ja' - You only know that there exist translations of 'Yes' and 'No'. So the first question can not contain "...Does 'da' mean 'yes'..." but rather "...is in the ghosts alphabet the translation for 'Yes' before the translation for 'No'...". The difficulty is that in the end the outputs "da. da. da." and "ja. ja. ja." are the same cause you don't know the other word. de:User:Träumer 17:32, 11 February 2008 (UTC)


 * That seems unsolvable given that you can only get 4 different sequences of answers: "111", "112", "121" and "122". But 4 is not enough to differentiate between the 6 possible god configurations. RedNifre (talk) 00:47, 22 March 2015 (UTC)


 * You can make that even harder by only allowing all three questions to be addressed to the same god. RedNifre (talk) 00:47, 22 March 2015 (UTC)

A generalized version of the original puzzle ?
Here's a generalized version of the famous Smullyan/Boolos Hardest Logic Puzzle [in its regular form]. I don't know of a solution yet, I even don't know if it is indeed a harder puzzle although it looks like (but I have a conjecture based on my work on the original one, using utterance operators). So here it is, for what it is worth:

Suppose we are confronted with n+1 GODs, name-numbered G0 ... Gn. G0 always tells the full truth, i.e. G0(p)=p all p. G1 deviates randomly from G0 on some statements and/or some occasions, i.e. G1(p)<>G0(p)= p for some p. G2 deviates still more from G0 but in such a manner that deviations by G1 are retained. Etc. Gn will deviate from G0 on all statements, so he is the real and only full liar. This can easily be formalized in terms of sets and implications.

So there will be a hierarchy of ever more untrustworthy Oracles in between one good God and one evil Devil (G-O-D). Of course, all G-O-D's know their own identity as well as the identity of the other G-O-D's (or at least their immediate precursors and/or successors in the hierarchy?) Is there a simple way to reveal each creature's identity by asking just n+1 regular questions (although there are clearly (n+1)! different possible orders)?

Note: If n=3, we have the original puzzle, which is indeed solvable in 3 regular questions by first eliminating the unreliable G-O-D and then asking two more questions. I guess we can do a similar thing for general n by focussing in on G0 and Gn by n-1 questions (eliminating on our way all unreliable G-O-D's), then asking again n-1 questions using the double negation technique, and finally deciding between G0 and Gn; so this results in 2(n-1)+1 questions (correct for n=3). Can we do better? Koyaanisqatsi II (talk) 12:24, 19 June 2008 (UTC)
 * Since there are (n+1)! possible outcomes, you need at least lg((n+1)!)=THETA(n log n), which is asymtotically more than 2n-1. To be concrete, for 8 creatures there are 8!=40320 possible cases, while you can only distinguish between 32768 different cases by posing 15 questions.Honnza (talk) 08:02, 22 November 2008 (UTC)

Discussion of more difficult version
Anon: [HOW IS THE LAST CONSTRAINT RELEVANT?] Fyz: It is merely an additional constraint that makes life a bit more difficult - just like each of the constraints in the question. Once you try working with the full, worst-case interpretations of all the constraints you should see how it works together to make a really tight puzzle.

N.B. that the first clause of this extension is not strictly necessary.

Anon: [POST YOUR SOLUTION. I'D LIKE TO BREAK IT.] Fyz: I will be pleased for you to try. I will place it in due course - but I'd like to give others the opportunity to claim first independent solution - placing my own would "spoil" this. N.B. Have you checked out my solution to the strong interpretation of the original puzzle yet?

Fyz

Return to more rigorous interpretation?
lwr314: It is not possible to solve the puzzle if 'The gods will not answer any "question until after all three have been posed'. It isn't even possible if the Gods answer yes/no instead of ja/da.

Here is the simple proof of this fact:

Assume (to reach a contradiction) that we can solve the puzzle thusly. Then we have 3 questions Q1, Q2, Q3 that we can put to gods A, B, C respectively and after we have asked them all get responses R1, R2, R3. Since one of the gods is Random, we gain no information from his response. Since each response distinguishes at most 2 possibilities (yes or no), and only 2 responses give information, we can distinguish at most 4 possibilities. But we have solved the puzzle and thus distinguished 6 possibilities. This is a contradiction.

Here I have assumed the modified puzzle where Random randomly answers yes/no instead of Boolos' broken random. It may be possible to solve it under such conditions using Boolos' broken Random.

Fyz in reply to lwr314: As you rightly say, yes/no and ya/da are irrelevant to the solution. But, like most of the objections to my signed contributions, you are not working with the question as posed; unlike the others, you appear to recognise this.

Working with the question as posed: clearly, the very possibility of a solution relies on the fact that Random answers either truly or falsely. You can therefore formulate the question so that Random's answers are meaningful. Simply modify one of the standard "double negative" statements (for example) as follows: "When you answer this question, you speak truly".

lwr314: See my response on the discussion page for a response to this.

Another thing to note is that since you are using Boolos' broken random, by the methods in the 'Random's behavior section', your 'more difficult puzzle' is equivalent to the following trivial puzzle. Fyz: What you write is not my "more difficult puzzle", just my more rigorous interpretation of the original. The "difficulty" in the original puzzle as written is more in interpreting the puzzle than finding a solution, so this rather bypasses that.

lwr314 & Fyz: Three gods A, B, and C are named, in some order, '1', '2', and '3'. The gods always speak truly. Your task is to determine the identities of A, B, and C by asking three yes-no questions (the gods may interpret a yes-no question to be one that can be answered truly with a yes or a no); each question must be put to exactly one god and no two questions can be semantically or logically equivalent. The gods might not answer until all questions have been posed. The gods understand English and will answer in English.

Fyz: for the "more difficult puzzle" add "independent" between "three" and "yes-no", with independent meaning that the answers to two questions must never imply the answer to the third.

lwr314: This puzzle can be easily solved as follows.

Ask A: "Is A named '1'?" Ask B: "Is A named '2'?" Ask C: "Is B's name less than C's name?"

Now wait for their answers. Ok got them.

From the first two responses we can determine A's identity. Now the third question determines the order of the other two.

Fyz notes on lwr314 solution (updated 16 July): that gives an elegant solution to the puzzle as originally posed (and ignoring 'clarifications'), as we can use (for example) position in an alphabetical English dictionary to establish an ordering. However, it doesn't answer my "more difficult version of the question", because the question to B is unnecessary if A is named '1', and Answer2 => answer1

Of course, if we take the equivalent reduction of Boolos' question as interpreted by him, we have the following, which might be regarded as being of similar difficulty: Three gods, A,B and C are named in some order, '1', '2', and '3'. Your task is to determine the identities of A, B, and C by asking three yes-no questions, each addressed to precisely one god, which the gods will answer in English immediately after you have asked the question. God '1' and god '2' always answer truly, but god 3 will answer yes or no entirely at random.

This puzzle is readily solved as follows: Q1 Ask A: "Is B named '3'? If the answer is yes, then ask the following questions of C; if the answer is no, ask them of B (in your preferred order): Q2: "Is A named '3'?" Q3: "Are you named '1'"

The first question determines a truthful god, the other two determine which god carries which name. There is also no real problem in modifying this to eliminate repeated semantics.

My points are:

a) that "The Hardest Logic Puzzle Ever" as posed has had to be modified in at least two aspects to allow this answer to be safe; and

b) that the solutions to each of the above problems (once defined) are of similar difficulty, because each requires a slight lateral step - the one based on the text needs you to provide an ordering for the gods; and the one following Boolos' interpretation requires you to recognise that you can identify a consistent god by answering a suitable first question. I wouldn't know how to say which solution is the more difficult.

c) as in many puzzles, a large part of the difficulty in solving the original puzzle comes in the interpretation of the question, of which Smullyan's section was typically sparse. In this case the task is to discover what requirements must be satisfied to provide a safe solution based purely on the text of the puzzle.  Only when you have done that does the the limited from of randomness as defined by Professor Smullyan "speaks truly or falsely" become necessary.  I would assert that it is only the difficulty of a safe interpretation of the puzzle that distinguishes the difficulty of the two puzzles.

Fyz

lwr314: Of course, this is precisely the solution on the main wikipedia page as taken from `A Simple Solution To The Hardest Logic Puzzle Ever'. The embedded question lemma* reduces the (modified) puzzle to what you have stated and the solution from there is the same.

Fyz: Indeed - and no more difficult than the other

lwr314: The point is that Boolos' (unmodified) reduces to an even more trivial puzzle where all the gods tell the truth and you just have to ask them who they are.

Fyz: No, that is the puzzle as reinterpreted by Boolos to say the gods answer immediately.

lwr314: Reinterpreted by Boolos! The article is about 'The Hardest Logic Puzzle Ever' as coined by Boolos. Reinterpreted by you is more accurate.

Fyz 16Jul: Frankly the article looks more like someone else's puzzle reinterpreted by Boolos, and then partially fixed (to ignore broken Random) with further re-interpretations (so that Boolos' solution becomes relevant) by others. (Partial only because the wording of the puzzle was not changed to match the problem actually being solved)

lwr314: Even with your added constraints, the Boolos' puzzle reduces to the triviality I stated above. The reason the (modified) puzzle is `hard' is that your questioning needs to be dependent on the answers given -- it isn't just static.

Fyz: The standard computer bisection methods for finding a word in alphabetical order are non-static in that sense. I can't see how that is hard. My position is that, once reduced and the meaning explained, neither puzzle is hard - but if you ignore Boolos' "clarification", the interpretation of the original puzzle becomes more demanding.

lwr314: Agreed, neither puzzle is hard (that's why i said `hard' in quotes). However, modified Random introduces an element not usually found in puzzles. That no-information-giving quality of Random is what i find interesting in the puzzle.

Fyz 16Jul: Agreed that obtaining information from an answer where you don't know of any bearing on the question is the interesting aspect of the 'modified Boolos' version of the puzzle. Equally, although I am not a regular puzzler (too many of them are void) I have not met a puzzle where it is necessary to find a way to split the set; if it already exists elsewhere, I'd be interested to know.

lwr314: In response to your points.

(b) You don't have to provide an explicit ordering on the Gods -- that was just a convenient simplification. Replace the third question with,

Ask C: "Is (B's name `1' and C's name `2') or (B's name `1' and C's name `3') or (B's name `2' and C's name `3')?"

Note that this question is semantically identical, but did not require you to think of ordering them. All you had to do was ask precisely the question you wanted answered. Solving this puzzle is totally trivial, I just asked two questions and then looked at what needed to be determined with the third to get the answer. Totally static.

Fyz: Isn't that one of the ways to define a cyclic ordering in a set of three?

lwr314: i did say that it was semantically equivalent to the ordering question, so yes. The point was that one could have come up with the question without considering ordering anything. If you think of what the first two questions give you, the third question is free as you know precisely what group you need to divide in two.

Fyz 16Jul: For what it is worth, here is a more economical question (the one I presented under the CR4 link):

Ask C: "Is (B's name '1' or is B's name '2' and 'A's name 1)

Fyz - specifically to lwr314 : Instead of fighting over the precursor (you won't persuade me as to difficulty, because I only came up with the "rigorous" interpretation when I found the exploding heads and then the two-stage solution too easily for what was supposed to be a "hard" puzzle, and the only "difficulty" for me was to become certain I had a literal interpretation that covered all bases):

lwr314: You found the two question solution too easy? i thought that was the most interesting and non-trivial part of the whole thing. You can solve Boolos' original puzzle with two questions too (and i mean his clarified one, since that is the puzzle he presented in the paper that this is all based on).

Fyz 16Jul: It was the first solution I came up with, almost as I read the question. I gave a hint on this as in the session where I read the question (I was away from my desk, so not logged in), and elaborated somewhat later (you can find all this in the CR4 thread if you have time to burn). Of course, without Boolos' timing constraint, it needed to be three questions. Then I realised that it could be possible for gods True and False to know what answer Random would give, and decided there must be another question buried there. Remember that I did this before looking at any independent work. Sorry to be repetitive, but in the end, the main challenge in the question (as presented and without clarifications) was in defining the range of constraints that had to be satisfied - so it's not strictly a logic puzzle at all.

I'm beginning to regret becoming involved in the first place - so I'll go and do some work for a while.

Fyz: Why not have some fun solving what I believe to be a harder puzzle (the literal interpretation of the original puzzle with worst-case interpretations, but including the extra constraint on the independence of the questions)?

lwr314: Because i can sit here and add arbitrary constraints all day long to make it `harder', but not more interesting.

(1) Nowhere in the statement of the puzzle did it say that i was allowed to use a word (say 'is') in more than one question. So, to be `safe', each English word can be used at most once over all questions.

(2) Nowhere in the statement of the puzzle did it say that i was allowed to have questions that mention other questions or responses to questions.

(3) Nowhere in the statement of the puzzle did it say that i was allowed to have questions containing colons.

(4) Nowhere in the statement of the puzzle did it say that i was allowed to use the letter `e' in my questions.

(5) Nowhere in the statement of the puzzle did it say that i was allowed to have a solution in which none of the questions involved an ordering on {1,2,3}.

You get the idea. All you have done is come up with a particular solution to a trivial puzzle and then insist that all solutions must have properties in common with your solution. Did i ban your solution yet? If not, i can keep going until i do. Arbitrary constraints are not interesting. They would be if they amounted to something more than a minor annoyance (and the puzzle could still be solved), but yours don't.

Fyz 16Jul: Of course you didn't, and you know it. But of course you must be aware that there could already be an interpretation of the question as it stands that that forbids explicit included questions (as most solutions are already worded to avoid this, I didn't go into that). The implication that the interpretation and extension started from the answer and worked backwards would appear more applicable to the "standard" solution based on Boolos' clarifications. I certainly got to the end points by asking "what if" questions - first about tenable interpretations about the meanings of the constraints on godly behaviour in the puzzle as set; it was the fact that "broken Random" resulted in a trivial puzzle if you did not do this that set me off. The jealous gods (no redundant questions) was set off by a not-quite-plausible interpretation of "one question" - whether it is a valid question if you already know the exact answer.

lwr314: Of course i didn't?  i don't think you understood. Are you sure that none of your questions used the letter `e'? The puzzle did not explicitly state that you could use e's, so to be `safe' we shouldn't use them.

Nowhere in the puzzle did it explicitly state that a solution generated by any mind other than mine was valid. So, to be `safe', we should only consider using solutions generated by my mind.

Question on original text
Question:

Could someone who knows the solution put it in a nice neat little box or logic tree something? As it stands now, there's many more than three questions listed in the Solution section and I can't pick out which ones you need to ask when. Thanks!

I think it should be made more clear whether the gods themselves know which god is which ... It isn't obvious.

Attempt at answer by Fyz

They may not 'know' in a sense we recognise - but we have to assume they will actually answer our questions, and if so the fact that True always speaks truly etc. guarantees the information we need. One full solution to the problem as interpreted by Boolos might be:

Q1 to A: Is the following statement true? B is named Random iff you are true iff ya means yes

If answer to this is ya, then ask following questions of C, if da, ask them of B

Q2: Is the following statement true? A is named Random iff you are true iff ya means yes

Q3: Does ya mean yes? (True will answer ya, False will answer da)

Exploding heads
Interestingly my friend used the same expression to refer to a different situation which is not clarified in this article: Assume that god A is True, and god B is Random (although of course we don't know this). Ask god A, "What would B say if...?" How does A answer? Does A impersonate B, providing a random answer? Or does his head asplode? This question must have been covered somewhere? Stevage 15:04, 23 May 2007 (UTC)

The "Exploding god-heads" section title is a spoiler
The "Exploding god-heads" section title is a spoiler. It can be seen from the contents at the top of the page and immediately gives an idea of how to solve the puzzle. It should be renamed to something like "A simple solution?".

Discussion of "Reservations"
Fyz: Before entering the discussion on the individual points, I think it will help to make general comment on this type of puzzle. These puzzles are generally intended to be solved as if your life depended on the solution being watertight. If you want to minimise the chances that you will fail, you need to assume that the gods are not on your side, and look for the most awkward interpretation, not the easiest.

If, therefore, you make assumptions that are not explicitly supported in the question, and that are not necessary to create a solution, you have done the equivalent of placing your life in danger unnecessarily. The only reason, therefore, for rejecting any interpretation of the puzzle that makes it harder to answer is that: either the interpretation is logically untenable; or, it makes the puzzle impossible to solve.

Anon: [IT IS CLEAR THAT ANSWERS SHOULD IMMEDIATELY FOLLOW THE QUESTIONS; BOOLOS' SECOND CLARIFICATION WOULD NOT MAKE SENSE OTHERWISE. **What the second question is, and to which god it is put, may depend on the answer to the first question. (And of course similarly for the third question.)**]

Fyz: Yes, that was Boolos' view. Remember that Boolos did not originate the very elegant and spare wording of the main puzzle. Mine is that if he needed to make unnecessary assumptions, he hasn't really solved the puzzle.

Anon2: Fyz says "Boolos did not originate the very elegant and spare wording of the "main puzzle". This is just false. Boolos is the first to publish this puzzle in the form we see here. It is a direct quote from Boolos. This is boolos' puzzle and his clarification sof his own puzzle. Boolos' makes all these similar assumptions in the paper that he solves the puzzle in. As far as what "The hardest logic puzzle ever" is and the intent of its author I think it is very clear that this article gets things right. Now, there is nothing wrong with considering variants of the Boolos puzzle and seeing how their solutions look, but those are different puzzles.

Fyz 2: Boolos clearly states that the wording of the bulk was due to Professor Smullyan, and that Professor McCarthy was responsible for the extension. That makes Boolos the populariser of the puzzle, not its creator. In any case, it is not unheard of for the original presenter of a problem or statement to be unaware of the full implications of their work (and that is not necessarily to detract from their contribution - vide Fermat)

Anon: [BOOLOS MAKES IT CLEAR THAT ONE GOD CAN BE ASKED MULTIPLE QUESTIONS, SEE CLARIFICATION ONE]

Fyz: Indeed he does. But in order for the order of the answers to be maintained you need to make an unnecessary assumption.

Anon2: What assumption?

Fyz 2: As implied in that statement - the assumption is that the gods will answer the questions in the order of presentation. Nothing in the puzzle says they will, and the puzzle allows a solution without making this assumption - I think that is a pretty good description of "unnecessary"

Anon: [WHERE DOES IT SAY THIS!] (that you cannot make use of exploding heads)

Fyz: I was a bit too cryptic - see amendment. My meaning was that the question says that the questions are yes-no questions. If you ask questions that do not have valid yes-no answers, any god with an ounce of self-preservation instinct would regard these as not conforming to the required constraints.

Anon2: What is a "valid yes-no answer". Questions of the form "Are you going to answer this question by saying 'yes'?" are perfectly good yes-no questions. It just so happens that the particular gods in this puzzle have some trouble with certain yes-no questions. What this really proves is that there can't be a god who answers all yes-no questions with the truth/lie (essentially a variant of the Liar paradox). Thus, the puzzle must be amended. Exploding heads seems like the course of least resistance. But there are other ways to amend the puzzle.

Fyz 2: A valid answer is one that is true, or at least conveys useful information. That doesn't mean that the gods have to give that answer - as stated in the question. The question "How many fingers do I have" is not a valid yes/no question - although you could answer "Yes" - but it would not be valid, as it does not convey any useful information. Similarly, if I asked you "Will this coin land heads when I toss it", neither "Yes" nor "No" would be a valid answers, because they are neither true, nor do they convey any information. The valid answer here is "I don't know" which is true, although not helpful in predicting the outcome.

Anon: [THEN THIS CLEARLY WAS NOT THE INTENT]

Fyz: Were you the author? If not, you should take it to be a puzzle that was intended to be reasonably challenging, and that every phrase was intended to convey specific constraints. As the question is answerable with all the constraints interpreted in these more cautious ways, it is reasonable to assume they were intended.

Anon2: Agreed. What I meant was that if you are interpreting the puzzle such that it is unsolvable (which I suspect you are) then that clearly was not the intent.

Fyz 2: This interpretation of the puzzle is solved, and I have presented a solution as stated at the link on the main article (though I shall improve the wording when eventually I import it to wikipedia - I have not yet done so merely to avoid accidentally spoiling anyone's efforts to solve it independently). If you can find a hole in my solution, I would be happy to hear about it.

Anon: [IN ALL OF SMULLYAN'S BOOKS HE MAKES THE SAME "ASSUMPTIONS" YOU FIND OBJECTIONABLE]

Fyz: The only 'assumption' I'm objecting to is answering before the following question needs to be asked (and related to that in the order the questions are posed). Professor Smullyan certainly made that assumption in some cases, as when you meet people when walking around. But that is justified by the context. Here, the assumption is unnecessary, therefore it would be uncharitable to assume that Professor Smullyan had failed to anticipate the more difficult version that the wording supports.

Anon2: Again see Boolos (the author of the puzzle) clarification 1.

Fyz 2: see above comments - my view is that Boolos popularised a very elegant puzzle, but his "clarifications" possibly oversimplified it puzzle

Anon: ["EXPLODING HEADS" IS OF COURSE JUST A COLORFUL WAY OF TALKING ABOUT THE CASE WHERE THE GODS ARE UNABLE TO ANSWER. THE POINT IS THAT THE PUZZLE DOES NOT RULE OUT THIS POSSIBILITY. OF COURSE THE PUZZLE COULD BE CHANGED TO ACCOUNT FOR THIS LOOPHOLE, BUT UNLESS YOU KNOW HOW TO WIELD THE POWER OF THIS LOOPHOLE IT DOESN'T HELP MUCH]

Fyz: In my view, as explained above, the puzzle rules this possibility out in any way that has been tried so far (questions that don't have valid answers that are either yes or no). My first attempt to solve the problem made use of confusion, but when I realised that this would reduce the number of required questions to two, I started reading the question more carefully.

Anon2: I don't think that you came upon the two question solution (it's impossible for modified random and very difficult for original puzzle). If you have solved it in two using exploding heads I would be very interested to hear it. I have solved it in such a manner but have not posted it yet.

Fyz 2: If you allow the narrow interpretation presented by Boolos, and the gods do not have special knowledge of each other or of the future, then it is possible for both cases. One version that solution is:

Ask A: Is the following statement true: If I ask B a question, he will answer truly iff you are True iff da means yes If A is confused ask the equivalent question of C with A as target, else ask the question of B with C as target.

Anon: [THIS DOESN'T MAKE SENSE]:

Fyz: I'm not certain which bit you mean: "random does not mean unknowable" Generate a random sequences of numbers, writing each one down. "the random decision may be taken as you pose the hypothetical question" the fact that the decision has been taken at a specific time does not make it less random - so long as it was taken on a random basis. Another way of looking at this is that Random may at some time in the future be asked this very question, and the gods are not subject to time in the same way we humans are.

Anon: {THE INTENT IS SURELY JUST TO MAKE CLEAR THAT YOU CANNOT ADDRESS MULTIPLE GODS AT ONCE]

Fyz: Would you risk your life on the gods interpreting the statement in this way if you could provide a solution that did not require this interpretation?

Anon: [WHY? "IS THE SENTENCE WRITTEN ON THAT CHALKBOARD TRUE?" SEEMS TO BE ASKING VERY DIFFERENT QUESTIONS WHILE IN THE PRESENCE OF DIFFERENT CHALKBOARDS!]

Fyz: Again, that is a possible interpretation. So you would be happy to risk your life on asking each god in turn (for example): "Are you True?"

Anon2: I am not too worried about this because it doesn't seem to help to ask the same question to multiple gods.

Anon: [UNCLEAR WHAT THIS IS SUGGESTING].

Fyz: I'm suggesting a sequence of identities that would make questions "the same", for example: To A: "Is B Random" {logically equivalent to} To B "Are you Random" {semantically equivalent to} To A "Are you Random" {logically equivalent to} to C "Is A random" {logically equivalent to} to C "Do you consistently answer either Truly or Falsely" (I allow inversions in logical equivalence, because they give the same information.

Anon2: Some of the questions use the same words, some of them have the same meaning. You want to rule out both. Ok. But using such things doesn't help anyway. What's all the fuss about here?

Fyz: I agree this is totally extreme. If there was no solution under these constraints, I would back off.

Anon2: Of course there is a solution under this "constraint" I think that all the solutions I have seen don't violate this constraint.

Fyz 2: It probably doesn't matter, but very few of the existing ones do, because it makes (for example) the following questions equivalent: Ask A: Is B Random? Ask C: Is A either True or False?

An extra note by Fyz Please don't capitalise - apart from being hard to read and seeming like rude shouting, it makes me think instinctively that you need the extra emphasis because your argument is weak, which doesn't make it easy to provide a straightforward answer.

Anon2: I LIKE TO WRITE TRUTHS IN CAPS!

Fyz 2: Most of your truths were objections to my truths... You should be aware that the effect on others is that you are shouting me down. In solving a puzzle, one should try puzzle it out without recourse to authority (e.g. Boolos), and on the basis that your solution must allow you to know which god is which under any tenable interpretation of the wording of the question, and without adding any unnecessary constraint on the gods' behaviour. (Help is allowed - but you should make your own decisions as to the validity). I am certain that you will find that most of your truths turn out to be highly conditional on unnecessary assumptions. Perhaps you would like to go back and remove the capitals from those contributions which you now regard as points for discussion?

Anon2: So I feel like this whole 'Reservations' and the section after doesn't really belong in the actual article, especially in the present tone. I don't know the proper tags, but something about original research or unencyclopedic tone. Anyways, is there any reason this should be here?

Fyz 2: Maybe the expression should be tidied up - I would be happy for someone to do this, provided the sense remains (I may do it myself when time permits). However, the article is about the (reputed?) "Hardest Logic Puzzle Ever", not about a deliberately restricted interpretation of that puzzle. My article references an area of discussion of this topic (the CR4 link) as well as a solution.

Anon2: I think it should be removed. It is more apt for a discussion board. Somewhat interesting interpretive issues but in the end misguided.

Fyz 2: If your life depended on a robust solution, and all you had was the puzzle as presented, would you be happy to go with the solutions in the original article? As my solution would also work under the original interpretations, I know I wouldn't.  If you agree my solution would work under those conditions, I'd be interested to know how you can regard my contribution rather than the originals as "misguided" - especially given the name of the puzzle.

Anon2: The article should include popular interprative variants, though. SO if one of these is a common (mis)interpretation it should be included. E.g. the modified puzzle is so common it must be included (and it was clearly Boolos' intent although the actual statement gets it wrong).

Fyz 2: I agree that Boolos' original interpretation and the various solutions should be included - although, as stated above, Boolos would be the populariser rather than the originator of the puzzle. But it still seems to me that the most robust solution to the puzzle as worded should be the prime solution, and that Boolos' "clarifications" should be relegated to intelligent misinterpretations.

lwr314: Perhaps you did not notice that i added a little to your section. This discussion is moot since your proposed change makes the puzzle unsolvable.

It is not possible to solve the puzzle if 'The gods will not answer any question until after all three have been posed'. It isn't even possible if the Gods answer yes/no instead of ja/da.

Here is the simple proof of this fact:

Assume (to reach a contradiction) that we can solve the puzzle thusly. Then we have 3 questions Q1, Q2, Q3 that we can put to gods A, B, C respectively and after we have asked them all get responses R1, R2, R3. Since one of the gods is Random, we gain no information from his response. Since each response distinguishes at most 2 possibilities (yes or no), and only 2 responses give information, we can distinguish at most 4 possibilities. But we have solved the puzzle and thus distinguished 6 possibilities. This is a contradiction.

Here I have assumed the modified puzzle where Random randomly answers yes/no instead of Boolos' broken random. It may be possible to solve it under such conditions using Boolos' broken Random.

Fyz response to lwr314: Why do you assume a modification to the puzzle? Obviously, with that modification the "rigorous" interpretations make it insoluble. The puzzles I am proposing are precisely as published at the beginning of the article, as 'written by Boolos. The only difference for the rigorous version is that I suggest the solution should be resistant to any tenable interpretation of the puzzle as written. My "more difficult" version simply adds one more constraint - "the gods will not answer any question if the answer to any two of the questions could make the third question redundant". (You may wish to edit your additions to the article, or possibly move them - I'd have no problem with your moving my reply at the same time).

lwr1: i assume the modified puzzle because not doing so trivializes the puzzle. In fact, the unmodified puzzle can be solved in two questions. The modified puzzle was the intention of Smullyan.

Fyz: How do you know this?

lwr1 (continued) Your constraint is unnatural and is not supported by the text of the puzzle (that is, the constraint that the gods will not answer any question until all have been posed).

Fyz: I am not imposing a constraint - they may answer immediately after you have asked the question. But there is nothing in the question to say that this is the case, so I can't see it is sensible to assume it if the question as posed does not require it. If possible, you should provide an answer that relies only on things that are actually specified in the question. What I am asking is - given the wording of the question, and if your life depended on it, which solution would you now choose? If you don't consider that covering all aspects is a relevant criterion in puzzle solving, please say so now, and we can terminate our discussion without further rancour.

Lwr1 (continued) You are talking about a *different* puzzle. This is fine and could be interesting, but to argue that your interpretation is *the correct* interpretation ('would you stake your life on it...', blah) and say that other interpretations are *wrong* is detrimental.

There are things in the statement of the puzzle that can be given multiple (all valid) interpretations. One is not 'more correct' than another. For example,

(1) You could define 'yes/no question' as 'a question to which the only possible answers a yes and no AND there is a correct answer'. This is the interpretation you would like to take. That is fine, although it precludes gaining any insight into how and why questions like the liar question break down (understanding why the Liar's Paradox breaks is an active area of research). Also, this model isn't fair to modified Random as he can answer questions with no correct answer just fine (Boolos' random is also interesting with such breaking questions as he only breaks some of the time).

You could define 'yes/no question' as 'a question to which the only possible answers are yes and no'. In doing so, you don't cut off interesting lines of research (such as using exploding heads to solve Boolos' original problem in two questions).

Both are valid. There is just nothing interesting to do with the first (beyond solving the puzzle -- which is trivial).

Fyz: "Both are valid". You are making precisely my point here - if you want a "safe" solution to a puzzle your answer must satisfy all valid interpretations

--

Lwr1 15Jul `safe' from what? monsters? Boolos' broken Random trivializes the puzzle even with your added constraints. So who is it now that is going to be eaten by the monsters? Someone who takes Random to be truly random or someone who trivializes Random based on a literal interpretation of a mistake of English on Boolos' part (it was clearly a mistake as he solved the puzzle assuming modified Random)? With your intense fear of the monsters i'd seriously consider whether or not trivializing Random will ruin their mood.

-- Fyz 16July: Safe in the mathematical sense - relying only on the axioms as presented. Also safe in the sense of providing an answer if your life depends on getting it right - like the multiple exit problem where the doors behind the gods lead to freedom, death, or permanent imprisonment.

lwr314: No axioms were presented! Just a bunch of English containing numerous words that have no rigorous definition.

Fyz 16Jul: and I thought I was pedantic - in the mathematical sense it would be the axioms; in the case of the puzzle it is the actual constraints presented in the puzzle. It is only the open interpretation that makes it interesting - and this sort of issue does have consequences in mathematics also

Fyz: In my view the problem as redefined by Boolos is equally 'trivial' - that is what I solved first (in two ways) before deciding that any "difficulty" should actually lie in a "safe" interpretation of the question. I'll accept your remarks that all my ideas are more trivial than Boolos' redefinitions if you continue to maintain that view when once you have solved my "more difficult" version that extends my literalist interpretation - but with the added requirement that the third answer is needed regardless of the order in which the answers are presented.

lwr314: i don't get the point of that restriction, like `you can only solve the puzzle with sub-optimal questions'? This is not interesting. You can add tons of arbitrary restrictions to make it harder and harder (to a point). Actually you could pare it down so that there was only one (up to logical equivalence) way to solve it. But why would i spend time doing this when there is so much more interesting mathematics to be done? This is just a finite (6) piece game.

Fyz 16Jul: I think you demonstrate below that are entitled to your view - you have shown the necessary step of duplication. However, your description is quite complex, and the actual solution is rather simple. (It is easy if you are already familiar with finite games. Equally, the unbroken Random is easy if you are familiar with conditional behaviour.  I would hazard that more people are familiar with the latter, which would make it easier for more people)

But surely the restrictions are arbitrary already, so what makes this different? Would you be happier if I referred to the gods being very touchy and no one wishing to offend the other by making their answer unnecessary? As I'm sure you well know, the additional step of splitting some of the cases actually has relevance in some mathematical proofs; that alone gives this version some relevance.

lwr314: 'splitting some of the cases actually has relevance in some mathematical proofs'. What does this mean? Logic has relevance in proofs? There is nothing deep or interesting about 'splitting cases'. It isn't like some sort of lemma you apply in a proof 'now by the theory of splitting some of the cases...'. It is something that you do from obviousness.

lwr 314: Boolos' clarifications make the puzzle trivial too! He broke Random. We fixed him.

Fyz 16Jul: Here we are pretty-much in agreement, Boolos broke the question. If we start from the question as posed, his clarification makes removes the ambiguity that stops it from being totally trivial. (fyz break)

lwr314: i don't know what you are talking about. Boolos' puzzle (that is including his clarifications) is totally trivial. You just use the embedded question lemma and get correct answers to all of your questions. Is that what you are referring to as 'not totally trivial'? Or is it something else?

Fyz: Apologies - a lot got lost in editting. It should have read along the following lines: Boolos made the question trivial when he required the immediate answer to the version with broken Random. The subsequent repair to Random recovered a meaningful question. On the subject of triviality - is seemed to me that having double negative requiements both for False and for ya/da does not add anything to the logical problem - though it could potentally test the puzzler's skills at wording. Do you beliee this to be less trivial than creating a compact logical test for my proposed extension? Am I missing something here?

(fyz continued) If we start from the question as you wish it to be, it is the question, not the clarification, that breaks Random. (fyz break)

lwr314: i don't understand this sentence. by 'the way you wish it to be', do you mean 'the way Boolos' stated the puzzle'? In other words, you mean 'the way the puzzle is'?

Fyz: I mean "the way the puzzle is incuding the clarification" - which, BTW, Rabern does not include in the "quotes" that are usually taken to define the extent of the puzzle. Maybe that is at the root of our different views as towhat consitutes the pzzle

(fyz continued) I hope you would agree that the only way to truly fix Random (and the puzzle) to match your requirements is to rewrite the puzzle so that Random is randomly yes-no and the gods all answer before you need to ask the following question (immediately would do).

lwr314: Are you saying this because you think you are telling me something, or are you just realizing this? We replaced Boolos' clarification B3 with B3* and kept the rest, that is how we 'truly fixed Random' (see our paper 'A Simple Solution To The Hardest Logic Puzzle Ever', if you want to know what i agree with).

Fyz: Had I realised previously that lwr314 meant you were Landon Rabern, a lot of misunderstanding might have been avoided. Yes, you fixed Random to create a non-trivial version of Boolos problem - I'd simply like to see Boolos 'Clarification' and the fixing recreated in an elegant single statement of the complete nontrivial version the puzzle. While you are about it, I suppose you could also pare down the language description (as I see no need to say what ya and da mean if they are answers given by True and False answer to yes-no questions).

lwr 314: It is not possible to solve the puzzle with your added restriction when the gods answer yes/no. It is necessary that they only answer ja/da (otherwise you don't have enough lee-way to use sub-optimal questions).

Fyz 16Jul: That is the method of splitting I had in mind, because it has the potential for generating additional information in one third of the cases. However, we could create a split in other ways if necessary, for example by asking whether "god A is older than the others or they are infinitely old", or whether "Random spoke truly or falsely the last time he answered a question that I was not party to".

lwr 314: The added constraint makes the puzzle equivalent to the following trivial game with 12 pieces.

1  2  3   4  5   6  7   8  9  10  11 12

Exhibit 3 partitions {A1, B1}, {A2, B2}, {A3, B3} of the pieces into 2 sets so that,

(1) Selecting one set from each of the three partitions and intersecting them gives a set containing elements from precisely one row (e.g. A1 \cap B2 \cap A3 must contain elements from precisely one row).

(2) Selecting one set from each of any two of the partitions and intersecting them gives a set containing elements from more than one row.

Fyz 16Jul: Don't you need to define eight partitions?

lwr314: No, you need 3 partitions, each into two sets. Here are a few partitions that work.

{1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12} {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,7,8}, {2,4,9,10,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,7,8}, {1,4,9,10,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,7,8}, {2,3,9,10,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,7,8}, {1,3,9,10,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,9,10}, {2,4,7,8,11,12}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,9,10}, {1,4,7,8,11,12}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,9,10}, {2,3,7,8,11,12}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,9,10}, {1,3,7,8,11,12}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,9,10}, {2,4,5,6,11,12}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,9,10}, {1,4,5,6,11,12}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,9,10}, {2,3,5,6,11,12}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,9,10}, {1,3,5,6,11,12}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,5,6,11,12}, {2,4,7,8,9,10}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}, {2,4,7,8,11,12}, {1,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,5,6,11,12}, {1,4,7,8,9,10}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}, {1,3,7,8,11,12}, {2,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}, {2,4,7,8,11,12}, {1,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,5,6,11,12}, {2,3,7,8,9,10}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}, {2,3,7,8,11,12}, {1,4,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}, {1,4,7,8,11,12}, {2,3,5,6,9,10}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,5,6,11,12}, {1,3,7,8,9,10}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,3,7,8,11,12}, {2,4,5,6,9,10}, {1,4,9,10,11,12}, {2,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,3,7,8,11,12}, {1,4,5,6,9,10}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}, {1,3,9,10,11,12}, {2,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {1,4,7,8,11,12}, {2,3,5,6,9,10}, {2,4,9,10,11,12}, {1,3,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10}, {2,3,9,10,11,12}, {1,4,5,6,7,8}  {1,2,3,4}, {5,6,7,8,9,10,11,12}, {2,4,7,8,11,12}, {1,3,5,6,9,10}, {1,4,9,10,11,12}, {2,3,5,6,7,8}

{1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12} {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {3,5,6,9,10},{1,2,4,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {1,2,4,7,8,9,10}, {3,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {3,5,6,11,12}, {1,2,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {1,2,4,7,8,11,12}, {3,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,7,8}, {4,5,9,10,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {4,5,6,9,10},{1,2,3,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {1,2,3,7,8,9,10}, {4,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {4,5,6,11,12}, {1,2,3,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {1,2,3,7,8,11,12}, {4,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,7,8}, {3,5,9,10,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {1,5,6,9,10},{2,3,4,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {2,3,4,7,8,9,10}, {1,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {1,5,6,11,12}, {2,3,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {2,3,4,7,8,11,12}, {1,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,7,8}, {2,5,9,10,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {2,5,6,9,10},{1,3,4,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {1,3,4,7,8,9,10}, {2,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {2,5,6,11,12}, {1,3,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {1,3,4,7,8,11,12}, {2,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,7,8}, {1,5,9,10,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,7,8}, {2,3,4,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,7,8}, {2,3,4,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,7,8}, {2,3,4,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,7,8}, {2,3,4,9,10,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,7,8}, {1,3,4,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,7,8}, {1,3,4,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,7,8}, {1,3,4,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,7,8}, {1,3,4,9,10,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,7,8}, {1,2,4,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,7,8}, {1,2,4,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,7,8}, {1,2,4,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,7,8}, {1,2,4,9,10,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,7,8}, {1,2,3,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,7,8}, {1,2,3,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,7,8}, {1,2,3,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,7,8}, {1,2,3,9,10,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {1,2,4,7,8,9,10}, {3,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {3,5,6,11,12}, {1,2,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {1,2,4,9,10,11,12}, {3,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,9,10}, {4,5,7,8,11,12}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {1,2,3,7,8,9,10}, {4,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {4,5,6,11,12}, {1,2,3,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {1,2,3,9,10,11,12}, {4,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,9,10}, {3,5,7,8,11,12}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {2,3,4,7,8,9,10}, {1,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {1,5,6,11,12}, {2,3,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {2,3,4,9,10,11,12}, {1,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,9,10}, {2,5,7,8,11,12}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {1,3,4,7,8,9,10}, {2,5,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {2,5,6,11,12}, {1,3,4,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {1,3,4,9,10,11,12}, {2,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,9,10}, {1,5,7,8,11,12}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,9,10}, {2,3,4,7,8,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,9,10}, {2,3,4,7,8,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,9,10}, {2,3,4,7,8,11,12}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,9,10}, {1,3,4,7,8,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,9,10}, {1,3,4,7,8,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,9,10}, {1,3,4,7,8,11,12}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,9,10}, {1,2,4,7,8,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,9,10}, {1,2,4,7,8,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,9,10}, {1,2,4,7,8,11,12}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,9,10}, {1,2,3,7,8,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,9,10}, {1,2,3,7,8,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,9,10}, {1,2,3,7,8,11,12}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,7,8,9,10}, {4,5,6,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,7,8,9,10}, {4,5,6,11,12}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,7,8,9,10}, {3,5,6,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,7,8,9,10}, {3,5,6,11,12}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,7,8,9,10}, {2,5,6,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,7,8,9,10}, {2,5,6,11,12}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,7,8,9,10}, {1,5,6,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,7,8,9,10}, {1,5,6,11,12}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}, {1,3,4,7,8,11,12}, {2,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}, {1,3,4,9,10,11,12}, {2,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,9,10}, {2,3,4,6,11,12}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}, {2,3,4,7,8,11,12}, {1,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}, {2,3,4,9,10,11,12}, {1,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,9,10}, {1,3,4,6,11,12}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}, {1,2,3,7,8,11,12}, {4,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}, {1,2,3,9,10,11,12}, {4,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,9,10}, {1,2,4,6,11,12}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}, {1,2,4,7,8,11,12}, {3,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}, {1,2,4,9,10,11,12}, {3,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,9,10}, {1,2,3,6,11,12}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}, {1,2,4,7,8,11,12}, {3,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}, {1,2,4,9,10,11,12}, {3,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,6,11,12}, {4,5,7,8,9,10}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}, {1,2,3,7,8,11,12}, {4,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}, {1,2,3,9,10,11,12}, {4,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,6,11,12}, {3,5,7,8,9,10}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}, {2,3,4,7,8,11,12}, {1,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}, {2,3,4,9,10,11,12}, {1,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,6,11,12}, {2,5,7,8,9,10}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}, {1,3,4,7,8,11,12}, {2,5,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}, {1,3,4,9,10,11,12}, {2,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,6,11,12}, {1,5,7,8,9,10}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,11,12}, {2,3,4,7,8,9,10}, {2,5,7,8,11,12}, {1,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,6,11,12}, {2,3,4,7,8,9,10}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,11,12}, {1,3,4,7,8,9,10}, {1,5,7,8,11,12}, {2,3,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,6,11,12}, {1,3,4,7,8,9,10}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,11,12}, {1,2,4,7,8,9,10}, {4,5,7,8,11,12}, {1,2,3,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,6,11,12}, {1,2,4,7,8,9,10}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,11,12}, {1,2,3,7,8,9,10}, {3,5,7,8,11,12}, {1,2,4,6,9,10}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,6,11,12}, {1,2,3,7,8,9,10}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,3,7,8,11,12}, {4,5,6,9,10}, {3,5,9,10,11,12}, {1,2,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,2,4,7,8,11,12}, {3,5,6,9,10}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,3,4,7,8,11,12}, {2,5,6,9,10}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,3,4,7,8,11,12}, {1,5,6,9,10}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}, {1,3,4,9,10,11,12}, {2,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {1,5,7,8,11,12}, {2,3,4,6,9,10}, {2,5,9,10,11,12}, {1,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}, {2,3,4,9,10,11,12}, {1,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {2,5,7,8,11,12}, {1,3,4,6,9,10}, {1,5,9,10,11,12}, {2,3,4,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}, {1,2,3,9,10,11,12}, {4,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {3,5,7,8,11,12}, {1,2,4,6,9,10}, {4,5,9,10,11,12}, {1,2,3,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}, {1,2,4,9,10,11,12}, {3,5,6,7,8}  {1,2,3,4,5}, {6,7,8,9,10,11,12}, {4,5,7,8,11,12}, {1,2,3,6,9,10}, {3,5,9,10,11,12}, {1,2,4,6,7,8}

If you wanted to make a really `hard' puzzle, just pick one of those and add the constraint that a solution is not a solution unless it is that one.

Fyz: I hope you auto-generated those! Apologies that I misunderstood "Partition". I should have known better as you were dealing with sets. [My excuse (and it's not a good one) is that I have recently been working in an area where partition meant "partition" is used to mean what is mathematically the elements of a set within the partition.] But the intended point of my constraint within the puzzle is that there are some very simple ways to split the set. For some (with different backgrounds from yours) the idea of using uncertainty is itself a significant extension to the problem. For others, finding a simple way to perform and describe that split would be more of a challenge. I suspect that, for set theorists, it's basically trivial.

Regarding creating a solution from a single one of those possibilities - no, I think that would be easier than finding an elegant puzzle-style solution to the open question. All you have to do is convert the constraints into the sets and then describe them. This could be a very messy set of questions if you select the wrong sets - any technical interest (you may disagree) lies in choosing a set of sets with a simple verbal solution

lwr314 continued: In regards to my method being complicated. It really isn't, it is just the general form of the puzzle. With my formalism above, no thinking is required to solve the puzzle, so it really is easy.

Fyz: I meant complicated in the sense of allowing massive numbers of options to appear. For example, I personally would find it simpler to use the name of A (in your solution to the unclarified/broken_Random puzzle) that gives just three possibilities rather than working with the full six.

Fyz: I just found your update that included the clarifications in a second set of quotes. I think that will help, though it would be even better if at least the principal clarification*** could somehow be included in the same set of quotes as the initial text. Might an interim solution might be to state that "the actual puzzle is not precisely as stated above, but must be adjusted according to the clarifications below"?


 * "What the second question is, and to which god it is put, may depend on the answer to the first question. (And of course similarly for the third question)"

lwr314: If you would like a solution with short questions, just look at the above and find a set of partitions that can be compressed (described simply) such as what i did before with the ordering.

lwr314 continued: To see to equivalence, just consider the following table:

ja = yes     ja = no    ABC           ABC 123          123    132           132     213           213    231           231    213           312    321           321

So, the top left 123 says "ja = yes AND A is named `1' and B is named `2' and C is named `3'". This is how i always think about the puzzles. Now you just need to divide it into a bunch of groups of two with yes/no questions (use the embedded question lemma to elicit meaningful answers from everyone). So it is equivalent to the above. Condition (2) above is your added restriction. It is easy enough to translate the partitions above back into questions, but why bother?

In the above partition format it would be easy to find more constraints that made such a set of partitions even harder to find and then you could translate the constraint back into English to make a still harder puzzle. But, it is all really the same static thing. You see how there isn't much here? We were really concerned with the Liar's Paradox and using it to solve the puzzle in two questions.

Fyz 16Jul: We obviously have a different view of what constitutes difficulty - nothing can resolve that. You appear to believe that the mere requirement for conditional action makes a puzzle difficult. As so much information in day-to-day life are used to define conditional action, this seems rather strange to me. So far as I'm concerned, if there is a difficulty to the unbroken Random puzzle, it is in realising that you can derive information from the first question. Certainly, Randall expressed that almost as a relevation when he presented his solution for the unbroken Random version (in the CR4 site referenced elsewhere).

lwr1 continued

(2) You could define 'question' as 'an equivalence class of semantically equivalent inquisitive utterances' as you want to. Or you could define 'question' as 'an inquisitive utterance'. The latter is the normal way to go. But the former could be something interesting to investigate too.

Fyz: Again, I don't particularly lean to either interpretation. I just take the view that a safe answer will work with all tenable interpretations - and the possibility of such interpretations would be a good reason for defining Random's behaviour in the way that Smullyan did - after all, it would be both shorter and more natural to write "whether Random replies da or ya is an entirely random matter" than the very specific "speaks truly etc" that was actually used.

Moved from article 17 July 2007
-- Fyz: Rather than "Clarify" the puzzle in so many ways that actually change its meaning, would it not be sensible to split the question into two completely different versions:

One would be exactly as posed, which has a relatively simple solution (see discussion) once the worst-case interpretations of the questions has been ascertained; and

One where the question itself is modified to fit Boolos' answer and the various clarifications?

Anon Reply: have you read the puzzle?

It clearly states that "...whether Random speaks truly or falsely is a completely random matter". The modification is to change this to 'whether Random replies affirmatively or negatively is a completely random matter'. There is no conflict between Boolos' statement of the puzzle and his clarifications. There is only a conflict between the way Boolos intended to have Random work (evidence: look at his solution) and the way he must actually work (see this section).

Fyz Re-reply: There are two significant modifications to the puzzle - one by Boolos (specifying immediate answers) is a restriction that trivialises the original puzzle as written, and the other is the one required to repair Random to make Boolos' solutions appropriate (though his solution does actually work for the trivialised puzzle). The two puzzles are the original one (as stated and without Boolos clarifications), and the one with Boolos' "clarification" integrated and Random mended as described. I believe that both belong here, because the first is as originally stated and is non-trivial, and the second is supported in ref 4 as well as in the above text. (However, I don't believe that this discussion should stay in the encyclopedia).

Anon Reply continued: Also keep in mind that the clarifications are part of the damn puzzle!

Fyz Re-reply continued: Any clarification that changes the meaning (in common with blasphemy) is completely unnecessary, and had time permitted would in itself have caused any referees to require the original 'paper' to be amended for consistency. We are not doing either the puzzle or Boolos' memory any favours by slavishly reproducing the original, which could be regarded as a "working document".

lwr314: Boolos' original paper was consistent. His solution to the puzzle that he posed (with the clarifications, since those are part of the puzzle that he posed) was correct. Apparently, he just did not realize that useful information could be acquired from Random as he defined him (surely, in his mind he had the assumption that Random gives no information as most commentators on the puzzle have assumed). Here are the facts:

(1) The article concerns 'The Hardest Logic Puzzle Ever' which is the name Boolos' gave to a puzzle that he published a solution for (the puzzle containing his clarifications). The article is about *that* puzzle.

(2) Modified Random fits in with the article since Boolos' solution applies to that no-information-giving Random as well.

(3) Exploding heads fits in with the article since they give a method of solving Boolos' puzzle in two questions.

(4) Your modifications do not fit in with the article since Boolos' solution does not solve your puzzle. You are discussing a *different* puzzle. As i have shown on the discussion page, this different puzzle is not interesting. One of your versions is a completely trivial modification which is uninteresting and for the other you have just added an arbitrary constraint to make the puzzle slightly harder. As i showed you over there, i could make it even harder by adding more arbitrary constraints. With the constraints we are discussing a different puzzle -- it does not belong in this article.

(5) i'll say this again since you don't seem to understand. `The Hardest Logic Puzzle Ever' refers to a puzzle by Boolos' which includes his clarifications.

PhysicistQuery 21:29, 17 July 2007 (UTC) (to lwr314): I note from your article and references that Boolos first published the puzzle in 1992, but the clarifications appear only in 1996. I cannot see that one would have solved the puzzle in the "approved" manner during this period - although it is likely that some (as did I when I only had access to that part of the puzzle - see the hku website) would have created and felt forced to reject those solutions before proceeding to the "uninteresting" version. Would that make two different puzzles with this name, or would you regard intent as more important than text in this case?

BTW, I made some entries as 86.145.9.205 when I was away from my desk - and forgot my login.

lwr314: Is it true that the clarifications are not in the 1992 version of his paper? In a footnote in the 1996 paper he says `A version of this article, translated by Massimo Piattelli-Palmarini, appeared in La Repubblicaon 16 April 1992 under the title “L’indovinello più difficile del mondo.”'. i assume that by `version of this article' he means the puzzle and solution (since he says `article' and not `puzzle'). If this is the case, then he surely included the clarifications as well before solving the puzzle. i have not seen the Italian version to know for sure.

PhysicistQuery 10:42, 18 July 2007 (UTC) You could easily be right that this is a misinterpretation of the references - I have tried (and so far failed) to find a copy of the original "La Repubblica" article.

However, Boolos wrote "Before ... let me give answers to certain questions". To me that suggests that the puzzle must have been initially presented without clarifications - I have no way of knowing whether that was in 1992 in La Repubblica or earlier.

Anon the monkey (to Fyz): IT wouldn't matter at all IF the Italian version did not include the clarifications (although there is no reason to think it didn't), since there is no conflict or inconsistency between the concise statement of the puzzle and the clarifications. In the concise statement Random is broken and in the clarifications Random is broken...oh ,so you must be on this 'the gods don't answer right away' thing again. That just seems like a weird thing to be worried about. When you ask someone a question they respond. So if you are trying to finding an earlier version of Boolos' article that doesn't include the clarifications, and therefore doesn't "rule out" that the gods don't answer right away and thus vindicates your weird interpretations...well then you are really...well just face it you had the consice statement, you came up with a wierd take on the puzzle that had a solution and now are trying to do something. good job solving it but its not the puzzle. your a smart guy. get over it!

PhysicistQuery 16:29, 23 July 2007 (UTC) Well AtM - I will try not to respond in kind, because clearly we have different experiences of puzzles and puzzlers. I have not personally been a puzzler for many years - but this one struck me as having potential. Many years ago I was part of a mathematics group that included a number of dedicated puzzlers; here it was that I was persuaded that the solution to a puzzle was simply not valid if it did not cover all tenable interpretations of the puzzle as presented. That is all I have done, starting from the versions that appear as numbers 1, & 2 on my web search for "hardest logic puzzle". OK, so Boolos adds some constraints, and the Raberns repair one weakness in Boolos' version (see "Apparent single-question solution..." for another possible one - I'll be interested to discover the flaw). The other thing was that these puzzlers without exception would have been looking for a "smart" solution (such as the 'exploding heads' or Boolos' solution) - so Smullyan's puzzle means they would find both before being forced to the simple set theoretic version. Naturally, you will take a different view if you don't accept these approaches to puzzling. (Personally, I regard the addition of da/ya as a simple secondary double negative - and a trivial addition unless that feature has added value)

Weird picture
Why is there a tapestry of Odin, Thor, and Freyr, incorrectly labeled "True, False, and Random", in this article? I can't imagine what purpose this illustration hopes to accomplish. Xezlec (talk) 02:58, 18 March 2008 (UTC)
 * OK then, removing it. Xezlec (talk) 01:49, 15 May 2008 (UTC)

new solution?
Is this a valid solution?

Ask god A "If I asked you 'does 2+2=4' twice in a row, would you answer both questions with a lie?"

If he is True or False, he will respond with whichever word means no and you will know that the other word means yes.

Then ask him "does 2+2=4?"

From the answer you can determine whether he is True or False

Then ask "If I asked god B 'does 2+2=4' would his answer be a lie?"

If god B is Random, god A will not be able to answer because he cannot read Random's mind and does not know whether Random will provide true or false information in response to that question.

If god B is True or False, god A will answer the question.

If you know the identities of two gods, you know the identity of the third

If god A is Random, he will not be able to answer the first question because he does not know what the "coin in his brain" will land on one or two questions from now. He will not have enough information to either lie or tell the truth.

Then, you ask god B, who must be either True or False, the same question and based on his answer you will know what "da" or "ja" means.

Finally, ask god B "does 2+2=4?" You will know from his answer whether he is True or False

Unfortunately, this approach is less rigorous than the original solution by Boolos because it does not include a mechanism to guarantee an answer. There is no way to know if the god is simply taking a long time to answer or not answering. While it is certainly true that the god must answer after a finite amount of time, so that if you wait long enough you can solve the puzzle, this solution does not provide a way to ever know when the god will speak.

If you know the identities of two gods, you know the identity of the third —Preceding unsigned comment added by 166.61.238.40 (talk) 00:45, 1 July 2009 (UTC)

"Go to the god who was identified as not being Random by the previous question (either A or C), and ask him: "If I asked you 'Are you True?', would you say 'ja'?". Since he is not Random, an answer of 'ja' indicates that he is True and an answer of 'da' indicates that he is False."

This would not work. If you go to the god identified as not being Random by the previous question and say, "If I asked you 'Are you True?', would you say 'ja'?" this would cause problems. We can assume that after asking the very first question you would know what 'ja' and 'da' means. So, let's assume that 'ja' is yes and 'da' is no. The problem is that False would say 'ja' because he would lie and say he is true and True would also say 'ja' because he would tell the truth and say that he is True.

It would be simpler to ask "are you the random god?" the truth god would answer no and the falsity god would answer yes —Preceding unsigned comment added by 71.192.108.214 (talk) 23:59, 27 October 2010 (UTC)

Error in "The Solution?"
In "The Solution" section following thing is written towards the end -

Ask god B, "If I asked you 'Is A Random?', would you say 'ja'?". If B answers 'ja', then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is indeed Random. Either way, C is not Random.

Suppose A always tells the truth, B always lies and C is random. Further suppose, 'ja' means 'Yes'. In that case if B is asked the question "If I asked you 'Is A Random?', would you say 'ja' ('Yes')?" B being a liar (s)he would say 'ja' ('Yes'). But still C is random. This contradicts the conclusion of the last line in the paragraph above. In fact even if 'ja' means 'No' the conclusion is still wrong. Pranab.Chakraborty (talk) 17:45, 9 November 2010 (UTC)

There is a mistake in the reasoning here. Lets suppose A is True, B is False and C is Random -- and assume `ja' means yes. Notive that if you were to ask B `Is A Random?', B would respond `yes', since A isn't Random and B is a liar. So when you ask B, "If I asked you 'Is A Random?', would you say 'yes'(ja)?", B will not respond with `yes', since that would be the truth, instead he responds with `no'. The reasoning fails to appreciate the following:

``For any yes/no question Q, asking either True or False the question

* If I asked you Q, would you say 'ja'?

results in the answer 'ja' if the truthful answer to Q is yes, and the answer 'da' if the truthful answer to Q is no (Rabern and Rabern (2008) call this result the embedded question lemma)." Max.Deiter —Preceding unsigned comment added by 67.176.95.82 (talk) 15:53, 1 December 2010 (UTC)

I agree. Thanks for the explanation Max. Since there is no "Error in the Solution" this talk can be completely deleted also. — Preceding unsigned comment added by Pranab.Chakraborty (talk • contribs) 18:59, 12 December 2010 (UTC)

Xor's hammer (blog)
has a good post about this puzzle, including a version with a hierarchy of gods controlling the head explosions of the lower level gods. Best quote:
 * Similarly, if we suppose that SuperGod is in charge of deciding whether or not God’s head blows up, SuperGod’s head is in danger of blowing up due to a clever self-referential question, and so forth. So we may actually extract an unbounded amount of information from a single yes-or-no question by choosing the question carefully and then observing how much of the universe is destroyed by our asking it.

link: http://xorshammer.com/2008/08/18/hardest-logic-puzzle-ever/

67.122.209.190 (talk) 07:00, 17 January 2011 (UTC)

Not the Hardest Logic Puzzle in the World
This puzzle is simple do to the fact of if a person said "True, is the god to your left false?" or "False, is the god to your left Random?" or even "True, is the god to the far left Random?" after finding out which god is wich you need only ask any two random questions you would like to ask. If anything you can even figure out what "Da" and "Ja" mean.

If anyone is wondering my name is Cody Joseph Heisel. I have a facebook account. —Preceding unsigned comment added by 75.60.88.115 (talk) 06:21, 31 January 2011 (UTC)


 * You can't direct your first question specifically at True, False or Random, because before asking any questions you don't know which of them is which. Equinox ◑ 11:19, 11 June 2017 (UTC)

The solution uses four questions, not three.
The introduction says that you are allowed to ask three yes-no questions. You can put more than one questions to one god, but "hence, some god is not asked any question at all". So the questions must still be three in total.

The solution which is posted requires four questions, not three:

1. Does da mean yes iff... (etc) to God A 2. If I asked you "is A Random?" would you say yes? to god B 3. If I asked you "are you True..." to god C. 4. If I asked you "is B Random..." again, to god C.

So what's up with that? — Preceding unsigned comment added by 84.38.12.50 (talk) 22:56, 5 July 2011 (UTC)


 * Acutally, it is only three questions. The first one of your questions is not part of the solution. Only 2, 3 and 4 are needed. Using the lemma described in the article, you do not need to figure out what ja and da means to solve the puzzle

History date change (Doctor Who)
The history section says that the puzzle Knights and Knaves was popularised in a film of 1986. I want to point out that it featured in the Doctor Who story Pyramids of Mars broadcast in November 1975. Perhaps the section should be edited to state this. Custardslice7 (talk) 21:10, 16 September 2011 (UTC)

I would like to add that I first read this puzzle, in a space mission scenario featuring 3 computers in lieu of the 3 gods, in an intalian book edited in 1975: "Il grande libro degli enigmi", authors: T. Parmeggiani and C. E. Santelia, editor: RIZZOLI EDITORE, 1975 Milano, page 156, puzzle title: "Da Sol III a Betelgeuse VI". Ppibelfra (talk) 14:58, 20 March 2013 (UTC)

The third clarifying remark does not entail that Random acts as True or False
I would like to make the point that Boolo's third clarifying remark does not entail that Random has a mental state - i.e. acts as either True or False; only that he speaks either truthfully or falsely, when delivering his answer. Thus, the question "If I asked you 'Are you Random?' in your current mental state, would you say ja?", could not be handled by Random as Rabern and Rabern propose (according to the article), actually it cannot be answered. The current mental state of Random is still defined as random and thus it is undecided what he would answer. So no one knows the answer, and hence you cannot get a yes or no. — Preceding unsigned comment added by 94.254.54.146 (talk) 14:07, 13 November 2011 (UTC)


 * Technically, the original problem statement does not specify what happens if you ask a question that does not have clear true and false answers, such as "Is the statement 'this statement is false' true?" or "What's the difference between a duck?". The typical assumption in puzzles of this sort is that you aren't allowed to gain an advantage by violating the implicit assumptions of the problem statement; in other words, whatever the gods do in response to things like that, it will not help you.  (Maybe they'll refuse to answer, or strike you with lightning or something.)  Naturally, if you assume the gods do something helpful in those cases (as in the "exploding heads") version, then that makes the puzzle easier.
 * Rabern & Rabern seem to think a strict reading of the original puzzle implies a certain limited constraint on Random's behavior that is exploitable to gain information--specifically, they interpret "he will either speak truly or speak falsely" to mean "he will either answer as if he were True or answer as if he were False". I personally disagree, and think the original puzzle is simply ambiguous about what would happen in their proposed scenario. But in fairness to Rabern & Rabern, the very next thing they do is fix the ambiguity and then solve the intended version of the puzzle; it's not like they were trying to avoid solving the "real" version. – Antistone (talk) 22:35, 11 May 2021 (UTC)

Seeming unnecessary requirement
In the exploding heads section, the following is suggested:

"Would you answer ja to the question of whether you would answer da to this question?"

But it is stated that this would work only in the modified puzzle, where Random's answers are truly random.

This seems false to me, given Boolos's axiom that

"Random will answer da or ja when asked any yes-no question."

Since neither True nor False can answer this question at all, this is enough for the solution to work in the original puzzle.

--76.121.187.193 (talk) 06:57, 6 December 2011 (UTC)

Page structure problem
The conclusion to the subsection "Random's behaviour" is to look to the end of the section "the solution". But the subsection "Random's behaviour" is technically the end of the section "the solution", so this is self-referential and incorrect. The best solution would appear to be to bump "Random's behaviour" up to a full section, but that seems out of place. Thoughts? — Preceding unsigned comment added by Aannoonn (talk • contribs) 22:24, 7 December 2011 (UTC)

Simple Solution
Just ask "Is a truth true?" to all three of them repeatedly until one of them answers differently to how they answered before. The god who answered differently is Random. This assumes that Random behaves in the "internal coinflip" way and not pseudorandomly seeded by the question. To determine which of the remaining gods is True, you need only to determine which word means "yes". To do this you can ask either of the remaining gods "Would your response to a question be the truth?". Whatever they answer is the word for "yes" and whoever gave that response to the original question is "True" and the other is "False". — Preceding unsigned comment added by Nbrader (talk • contribs) 12:25, 13 February 2016 (UTC) Holy shit, you must be high as fuck! Repititive question still counts as a question.

Missing?
Surely the article is missing the Bush/Rumsfeld solution: waterboard everyone. — Preceding unsigned comment added by 95.150.63.49 (talk • contribs) 12:00, 26 February 2017 (UTC)

Ungrammatical sentence
"If, however, the god is answering randomly." This sentence stands alone. What does it mean? Equinox ◑ 11:17, 11 June 2017 (UTC)


 * It looks like this sentence is setting up for the Rabern and Rabern variant: if the god is answering randomly, as opposed to just randomly deciding whether to answer truthfully or liefully (as in Boolos' clarification), then this lemma doesn't work.

If I'm right, then the point this sentence was setting up for is explained in detail in a separate section, and isn't relevant to this section. I suspect the author was originally writing it with the variant embedded throughout, then later decided to separate the two out, and just left a fragment behind while rewriting. So I'm going to remove it. If I'm wrong, obviously revert and flesh out the sentence with whatever it was supposed to be. --157.131.168.209 (talk) 21:12, 27 February 2018 (UTC)

Wouldn't this be another simple solution [Edit: ISSUE SOLVED]
Wouldn't a potential answer also be the following:

Ask the three gods if they are the random god. Since we know that in this question the answers of the 'True God' and the 'False God' will be the different and that the 'Random God' will either say "da" or "ja", we can claim with certainty that 2 gods will give the same answer and that one of those 2 gods will be the random one. Ergo, the god that answers differently will either be the truthful one or the liar(By no means can it be the random god).

Let's assume that the god that answers differently is god "B"(doesn't make a difference if it is A, B or C). Let's ask only god "B" if it is the Truthful God. Depending on the answer there are 2 solutions:

A1: If god "B" gives the same answer as it did to the previous question(which is "Are you the random god?") then that means that he either answered "Yes, I am the Random god." and "Yes, I am truthful god." respectively which would make him the Lying god since he lied, or he answered "No, I am not the Random god." and "No, I am not truthful god." which creates a contradiction indicating that he answered both questions with a 'Yes' instead of a 'No' making god "B" the lying god.

After that, we ask god "B" if god "A" is the random god. And if he gives the same answer as he gave to the previous two questions( a yes) then that would mean that he said a "yes, A is the random god", but since he is lying in reality A = 'Truthful God' and C = 'Random God'. However, if he gives a different answer to the previous two(No instead of a yes), then A = 'Random God' and C = 'Truthful God'

A2: If it gives a DIFFERENT answer as it did to the previous question(which is "Are you the random god?") then that means that he either answered "No, I am not the Random god." and "Yes, I am truthful god." respectively which would make him the Truthful god since he told the truth, or he answered "Yes, I am the Random god." and "No, I am not truthful god." which doesn't fit the descriptions of either of the gods which means that B answered 'No' and 'Yes' instead of 'Yes' and 'No' making B the Truthful God.

After that, we ask B if A is the random god and if he gives the same answer as he gave to the previous question(which is a 'yes') then that would mean that he is saying that "Yes, A is the random god" which is the truth and therefore A = 'Random God' and C = 'Truthful God'. However, if he gives a different answer to the previous question(or the same answer as first one), then A = 'Truthful God' and C = 'Random God'.

If there are any glaringly obvious mistakes please point them out to me so that they may be corrected

GAPITG (talk) 18:50, 12 September 2017 (UTC)

Comment: As soon as you ask the three gods if they are the random god you have used up your 3 questions, since the problem states "each question must be posed to exactly one god". Rennybosch (talk) 06:26, 17 February 2018 (UTC)

Ok, thank you Rennybosch. I knew I couldn't have come up with a new solution, I just wanted to know exactly what I had done wrong. GAPITG (talk)

Enslaving gods and making wishes seems like OR
Unlike the rest of the article, which is reasonably sourced, the section on "Enslaving gods and and making wishes" has no sources.

I'd normally just add citation tags, but in this case, I'm willing to bet that it can't be sourced, and is pure OR, so I'm going to be bold and delete it. Here's why:


 * It's written as an argument that this solution is correct, rather than reporting on the solution.
 * It's chock full of LessWrong-style jargon, which makes it unlikely that it's sourced to anything Wikipedia would consider a reliable source.
 * It was added by User:Florian Dietz, whose only actions ever were to add this, re-add it after ClueBot reverted it as potential vandalism, and clear the ClueBot warning from his talk page. So, it's plausible that Florian wouldn't know the Wikipedia standards (most importantly: Wikipedia isn't about what's true, only what's verifiable, via reliable sources), and could add OR in purely good faith.

Of course I could be completely wrong about that bet. In which case, please revert the delete and add the citations.

And either way, it's an interesting idea, so if it really is OR, the author really should create at least a blog post or something about it. --157.131.168.209 (talk) 20:58, 27 February 2018 (UTC)

Sorry, that should be User:FlorianDietz. Also see Special:Contributions/FlorianDietz.

More importantly, I noticed that Florian responded to ClueBot as follows:


 * I have no external sources for what I just wrote because I came up with this myself and posted it here directly. I hope this is allowed? Is there a way to mention myself (Florian Dietz) as the author without referring to an external source?

So, all the guesswork above is unnecessary. It is original research, made in good faith by someone who didn't realize that OR is not allowed on Wikipedia. --157.131.168.209 (talk) 21:18, 27 February 2018 (UTC)

I think it would be interesting to note...
(Warning: contains a hint towards solving the puzzle)

One obvious approach someone might take to this puzzle would be to first try to learn the meanings of 'da' and 'ja', and then use that knowledge in their remaining questions. I think it's interesting to note that it's actually impossible to (reliably) obtain BOTH the solution AND the meanings of da/ja in the same 3 questions; correct solutions will identify the gods but will NOT determine the meanings of da/ja, even in retrospect.

This is easily provable, but I do not have a reputable citation for it, so Wikipedia rules on original research prevent me from adding it to the article.

(Proof: There are 6 possible ways to label the gods, and 2 possible ways to label da/ja, producing 6*2 = 12 possible worlds.  But a binary question yields at most 1 bit of information (possibly less, because of Random, but definitely not more than 1 bit).  3 bits from 3 questions can distinguish between at most 2^3 = 8 possible worlds, which is not enough to solve the puzzle AND learn the meanings of 'da' and 'ja' at the same time.)  – Antistone (talk) 22:58, 11 May 2021 (UTC)