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"What will happen next?" asked the old man. His voice was friendly, his eyes bright, as always. His hair and beard seemed to be even whiter than ever before.

"The standard procedure is as follows: New arrivals are subject of a test after they are guided to the big hall with two doors and two guards. One door leads to heaven, the other to hell. One guard always tells the truth, the other is lying all the time. They are simple guys, and can only answer with yes or no. The test subject has to ask a single question that makes him able to identify the door to heaven."

"Oh, I've heard about that."

"Sure you have. We know you are familiar with this kind of situation. But let me clarify: The guards cannot answer hypothetical questions about their own answers. Those just break their mind. They are simple guys, as I mentioned. They can answer hypothetical questions about each other's answers though, as those do not need that much abstraction. They both know if the other one is a truth-teller or a liar, so have all the information that allows them to answer these kind of questions."

"That shouldn't be a problem."

"Sure. We know your skills in the subject. That's why we made it a little bit different for you. It should be a serious test, shouldn't it? There is a third guard in the hall this time. He is either a truth-teller or a liar, but I won't tell you which. However, the other guys know his type, and he is aware of theirs. He will be dressed like the other two guards, so they are all indistinguishable. You are allowed to ask two questions, but they cannot be addressed to the same guy. All you have to do is to identify the door to heaven. Shall we start?"

"I'm looking forward to it."

There they were. The famous hall. The old man and the guide were facing the guards. They both enjoyed the moment - it was something special for both of them after all.
The old man finally went up to one of the guards and asked something. The guard answered. The old man turned to his guide.

"I think it's all just the standard procedure from now, once we send out that guy," he said, pointing to one of the guards.

"Sure, Raymond. You have passed the test," answered the guide, and opened the door to heaven.

What was the question the old man asked the guard, and what conclusion did he reach from the answer?

Raymond Smullyan, the brilliant mathematician, who was well known for his 'Knights and Knaves' extensions of the classic puzzle, passed away at the age of 97 on 6th February 2017.
This simple puzzle is my modest personal tribute.

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  • $\begingroup$ My English is still broken, sorry. And this is my first self-made puzzle here. I'm happy to hear any feedback in both subjects. $\endgroup$
    – elias
    Commented Feb 16, 2017 at 23:02
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    $\begingroup$ Smullyan was, like, my first dive in logic. Respects. $\endgroup$
    – Lilian Delaveau
    Commented Feb 16, 2017 at 23:04
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    $\begingroup$ Please correct me if I'm wrong, but if one always tells the truth, another one always lies, and the last one is either lying or tells the truth, wouldn't it be easier to just ask something like "Is my beard white?" (or something that is true ie. is 1+1=2) and see what each of them respond. That way the truth tellers would say yes, and the liars would say no. Maybe I didn't understand the puzzle, but this would be much easier. $\endgroup$
    – bitcell
    Commented Feb 17, 2017 at 9:51
  • $\begingroup$ @bitcell, indeed I did not clearly state that the old man's task is not to identify the truth-tellers and liars, but to identify the door to heaven. I will clarify that, thanks for the input. However, even identifying the truth-tellers with your question would not work, because he is allowed to ask only two guys. If the two guards answer differently for that question, he has no idea if the third guard is a truth-teller or a liar. $\endgroup$
    – elias
    Commented Feb 17, 2017 at 10:06
  • $\begingroup$ @elias, thanks for clarification, now i understand it $\endgroup$
    – bitcell
    Commented Feb 17, 2017 at 10:11

3 Answers 3

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My initial answer was somewhat overcomplicated. The following modified version is basically equivalent but simpler. Note that some of the comments refer to the original answer; sorry for any confusion.

Call the guards A,B,C. (I assume that each knows the other's type.) Raymond asks A, "Does B tell the truth?". A will say yes iff A and B are of the same type.

Now

if A says yes then A and B are of the same type and can send A away, giving us the standard situation. And if A says no then A and B are of different types and can send C away, giving us the standard situation. (In that case we need to address the next question to B rather than A.)

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    $\begingroup$ Correct answer and clear explanation, bravo! (I do have a much simpler question though.) $\endgroup$
    – elias
    Commented Feb 16, 2017 at 23:27
  • $\begingroup$ Nice puzzle and a quick solve! +1/+1 $\endgroup$
    – Techidiot
    Commented Feb 16, 2017 at 23:33
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    $\begingroup$ I'm not at all surprised if there's a simple solution. The above was just the first thing I tried; it worked, so I posted it :-). One obvious improvement is that since what C says is fixed we don't really need to bring C into the question at all; we can ask A "If I asked B whether that's a door, what would he say?" Then B's actual answer would match his truthiness, so A's would be yes iff A~B, etc. $\endgroup$
    – Gareth McCaughan ♦
    Commented Feb 17, 2017 at 12:03
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    $\begingroup$ See my last spoilered paragraph. My solution doesn't really require them to be able to say anything other than "yes" or "no", and what I said about types should be understood as shorthand for something more cumbersome with only yes/no answers. $\endgroup$
    – Gareth McCaughan ♦
    Commented Feb 17, 2017 at 12:57
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    $\begingroup$ Further simplification of my simplified version above: just ask A, "Does B tell the truth?". A will say yes iff A and B are of the same type. Now proceed as before. $\endgroup$
    – Gareth McCaughan ♦
    Commented Feb 17, 2017 at 14:07
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I can do it in one question!

"If I asked one of the other guards who is of the opposite type as you whether the left door was the correct way to go, what would they say?"

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  • $\begingroup$ Although this does not match the scenario I described above (reducing the 3-guard question to the 2-guard one), I have to agree this is a nice find. I hoped I excluded solutions with one question with the criteria about hypothetical questions, but your answer clearly shows that I did not. Thanks! $\endgroup$
    – elias
    Commented Feb 17, 2017 at 5:38
  • $\begingroup$ Arguably, asking a guard about their own type is a hypothetical question about their own answers. I think you can do it in one question even without that, though. $\endgroup$
    – histocrat
    Commented Feb 17, 2017 at 16:20
  • $\begingroup$ @histocrat , " think you can do it in one question even without that, though." What will be that one question ? $\endgroup$
    – Hemant Agarwal
    Commented Oct 3, 2021 at 8:50
  • $\begingroup$ I think I must've been wrong four years ago, you need some kind of direct or indirect self-reference to get a liar and truthteller to say the same thing in the same situation. $\endgroup$
    – histocrat
    Commented Oct 3, 2021 at 18:35
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Ask a guard: "Which door will the OTHER guard tell me is heaven?" If you asked the truth-teller, then he knows that the liar will say that the door to hell is the door to heaven. If you asked the liar, the liar knows the truth-teller will say that the door to heaven is the door to heaven. However, since he always lies, he will say the opposite. Therefore, Ask a guard: "Which door will the OTHER guard tell me is heaven?" and do the opposite.

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    $\begingroup$ Please read the question. There are 3 guards. $\endgroup$
    – greenturtle3141
    Commented Feb 17, 2017 at 19:49
  • $\begingroup$ @boboquack This isn't a new question, it's an (albeit incorrect) answer. $\endgroup$
    – Rand al'Thor
    Commented Feb 17, 2017 at 22:38

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