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I guess some people are familiar with The hardest logical puzzle. The rules in this original question however allow us to ask one god more than one question.

But what if we change the rules to asking each god exactly one question?

To specify the variant, the random god answers ja or da regardless of the question. Basically imagine him as a deaf god who only indicates you are talking and you are done talking, and then says something.

Also questions must be determinable, meaning no asking about predictions of random processes or using logical paradoxes as suggested in the wikipedia article. We can react to obtained answer and modify our question and which god are we going to ask next.

After many attempts I failed to find any solution, I started with some analysis. At first it seems impossible, because we are required to obtain enough of information about 12 scenarios using only three bits, therefore 8 possible combinations. Because to assign identities, there are 6 possibilities, and to determine whether Random's answer was lie or not, we need another bit, resulting in 12 different scenarios. However the same goes for the original version, and there this "entropy/information contradiction" does not apply.

Maybe because we might never get to ask the Random any question. Here however we are going to do that here for sure. But on the other, the information about his "honesty" is a piece of information directly linked to his position once we know where the other one with the same honesty is located. Basically I am trying to say, honesty of his answer might be redundant information. And, during asking, we can get from Random only whether he is lying or not. I attempted again using this time some strategy learned from this analysis.

I tried the way of reflecting somehow validity of Random's answer into other questions (that's something I did not try in the first run), so I try to enrich the set of considered statements used to build the questions with this: "If I have already asked Random, was his answer true/lie" in order to reflect this piece of information in latter question. But no matter what I tried, never could I determine their identities.

Ideas were a lot: Making sure I asked the Random as first or second, asking the first a question determining honesty of the answer upfront and updating later questions with dependence on the previous questions. Never any strategy led to success.

With failing to provide any solution, I tried therefore to somehow convince myself this version does not have any solution, but I do not know absolutely where to start reasoning about this.

Could it be solved or can we prove there is no solution? Or is answering such a question completely out of reach?

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After your first question, there will be four possibilities remaining. This is because the two orderings where the god you talked to is the random god cannot be eliminated, and in the best case, only half of the remaining 4 orderings can be removed.

Since there are 4 orderings to be distinguished in two questions, your last two questions must be perfect, meaning neither of them can be directed at the random god. But this is impossible in your variant, because whenever the first god you talked to is nonrandom, one of your later questions must be to a random god.

The italicized sentence is the only handwavy part, but it can be made rigorous. Suppose your second question was to the random god. Then of the remaining 4 possible orderings, at least one of them involves you talking to the random god, so this cannot be eliminated. Of the remaining 3, in the worst case, only one can be eliminated. This leaves 3 remaining scenarios in the worst case, which cannot be distinguished with one remaining question.

The same logic shows you cannot succeed if your last question is to a random god.

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  • $\begingroup$ For now I only upvote, when I convince using this knowledge myself as well, I accept it. $\endgroup$ – TStancek Sep 27 '17 at 14:47

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