# Which door should you take?

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There are 3 guards, and you can ask one of them a question that can be answered with the guard pointing to one door.

[...]

One guard is insane...

If you ask the truthful guard a question with one "truth-evaluatable atom" or whatever it's properly called, it will tell you the truth.

If you ask the liar one atom it will tell a lie, but if you get it to make a statement regarding that lie it will tell another lie and turn it into a truth.

So if you ask it a question with two atoms you can get it to tell a truth. The same question asked to the truth-teller would also be the truth.

If you ask the insane guard a question with two atoms, it may make the first a lie and then treat the second as truthful (maintaining the lie) or it may make the first true and turn it into a lie with the second. Either way it will make a two-atom question into a lie. As with the liar, if you ask it two of these (so 4 atoms in total) the second lie can turn the first lie into a truth.

Meanwhile if you ask the liar a 4-atom question it should give you 2 truths; if you ask the truth-teller it should give you 4 truths.

So without knowing which guard you're asking, any of them should respond with the truth.

Such a question with 4 evaluatable truth "atoms" might be:

Which door could you point to if I were to ask you "which door could you point to if I were to ask you (which door could you point to if I were to ask you which door is good)"?

Electronically this is like saying:

The truth-teller is like a Buffer gate.

The liar is like a NOT gate.

The insane guard is alternately like a Buffer gate or NOT gate (or vice versa).

2 NOT gates in series is equivalent to a Buffer gate.

A NOT gate and Buffer gate in series is equivalent to a NOT gate.

Therefore 2 copies of (a NOT gate and Buffer gate in series) in series is equivalent to a Buffer gate.

I don't know if both the question and the answer are logically and semantically "bullet proof" but if not, I think it should be possible to fix it so that it is. ???

In case that last variation made sense, then continuing the theme...

5-door insane variation:

Assume a similar set up to the last variation, but now there are 5 guards (you don't know which is which) and 5 doors.

One guard always tells the truth.

One guard always lies.

One guard is insane and consistently alternates (as described above).

One guard is a duplicate of one of the above 3 guards (but you don't know which).

One guard alternates between acting like each of the above 4 guards (alternating for each evaluatable atom in your question), in an unknown but consistent order. Assume that it remembers whether the insane guard was last truthful or dishonest, and acts appropriately.

The addition of the 4th guard doesn't immediately complicate the problem. The same 3-door question would work on this guard, whatever the guard is.

In case you end up asking the 5th guard a question, you don't know which it will start acting as. If you ask it a 4-atom question, it will act like each of the others exactly once. But you don't know if, when it acts like the insane guard, if it will start being truthful or dishonest. Same goes with the 4th guard, which may be another insane guard.

If you ask it a question with 16 atoms, it will end up evaluating 4 atoms as each of the other guards (in each case ending up with a truth).

So if you construct a question similar to the 4-atom question above, but this time with 16 atoms, this guard should tell the truth. Similarly, if you ask the same question to any of the other guards each multiple of 4 atoms should result in a truthful evaluation, so the final result should also be truthful.

Edited by md65536

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I always thought of it as "if I asked you if this was the right door, would you say yes?" forcing the double negative out of the liar. If either guardian answers yes then this is the right door, otherwise they both answer no.

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• 1 month later...

This is an old puzzle, but if you haven't seen it before it should make you think.

You stand before two identical doors. On door leads to a long and rewarding life, the other door leads to sudden death.

Before each door is a guardian who, if you ask him the correct question will tell you which door to take.

However one guardian always lies and the other always tells the truth. You don't know which one lies.

You are only allowed one question. You walk up to a guardian and ask him the one question you are allowed and he tells you which door to take.

EVERYONE LOOK AT MINE, IT CAN BE DONE! ***SPOILER ALERT***

*edited (forgot to make it a spoiler answer)

I would ask the first guard what the other guard would say because it would have to be a lie. Then I would go through the opposite door.

Edited by morgsboi
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EVERYONE LOOK AT MINE, IT CAN BE DONE! ***SPOILER ALERT***

*edited (forgot to make it a spoiler answer)

I would ask the first guard what the other guard would say because it would have to be a lie. Then I would go through the opposite door.

You have got it. But strictly speaking if you are going to walk through the door indicated by the guard your question should be "Which door would the other guard say leads to sudden death?".

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You have got it. But strictly speaking if you are going to walk through the door indicated by the guard your question should be "Which door would the other guard say leads to sudden death?".

Still works

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Still works

Yep. Enjoy your long and rewarding life!

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Yep. Enjoy your long and rewarding life!

Sarcasm?

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Sarcasm?

Absolutely not. You are young and I am quite old and so it seemed a suitable reply when you consider the words of the puzzle. I really do hope you have a long and rewarding life - I haven't done so bad myself up to now.

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Absolutely not. You are young and I am quite old and so it seemed a suitable reply when you consider the words of the puzzle. I really do hope you have a long and rewarding life - I haven't done so bad myself up to now.

Oh okay, thank you. Just that at my age (14), normally if someone says that it's sarcastic. But thanks

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not bad at all md65536, at any level .. I'd start by simplifying the problem until it's solvable to me

I'd try to identify the insane, the alternating, the clone, .. then I knows who to ask for a truthful answer

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I know it's being a bit literal, but how do you know that one guard tells the truth and the other lies? Because if one of the guards told you that then you wouldn't know if its the truth or a lie so therefore you wouldn't be able to know which door to take. Because for all you know is they both lie.

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For a slightly stricter question:

Assume you in fact only have one guard, not two. This guard either always tells the truth or always lies. What question could you ask him to find out which door to take?

=Uncool-

Surely this is the same only you need to specify any of the doors. If I asked you if the left door was the correct answer, would you say "yes"? You still get the lie about the lie, so he says yes if it's right and no if it's wrong.

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Surely this is the same only you need to specify any of the doors. If I asked you if the left door was the correct answer, would you say "yes"? You still get the lie about the lie, so he says yes if it's right and no if it's wrong.

I would say, assuming there's another guard......... (then the same answer)

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I know it's being a bit literal, but how do you know that one guard tells the truth and the other lies? Because if one of the guards told you that then you wouldn't know if its the truth or a lie so therefore you wouldn't be able to know which door to take. Because for all you know is they both lie.

We're given that one guard lies and the other tells the truth.

In a case where you can't assume that, Marqq's post (bottom of http://www.scienceforums.net/topic/53927-which-door-should-you-take/page__view__findpost__p__623410) gives a question that is self-referential for the guard, and only depends on the one guard you're asking, so it doesn't matter how many other guards there are and whether those other guards lie.

I'd try to identify the insane, the alternating, the clone, .. then I knows who to ask for a truthful answer

You don't know which guard tells the truth and which lies. The solution to the puzzle involves treating them generically, so that you get the same answer whether they're honest or lying.

In the original puzzle, the guard that you ask will point to one of two doors. There are two possible answers (you could also make the guard's head explode but that won't give you useful information).

If you also want to know whether the guard you ask is lying or not, there are 4 possibilities:

Door 1 is good, guard is honest

Door 1 is good, guard is lying

Door 2 is good, guard is honest

Door 2 is good, guard is lying

The "pigeonhole principle" tells you that more than 1 of these 4 possibilities will correspond to a single possible answer (of 2 possible) given by the guard.

The trick is to get "Door 1, honest" to overlap with "Door 1, lying" etc, so you know which door to take without knowing if the guard is honest or not.

If you construct a question that tells you whether the guard is lying or not, it will be impossible to always* also know which door is good. There aren't enough possible answers (just 2, or 3 including head explosion) to specify all that information.

* I suppose it should be possible to get the guard's answer to tell you something like "The guard is telling the truth AND pointing to the good door... or not" so that you can know both in only 1 of 4 cases. So you should be able to increase the number of "bits" of useful information you get from the guard while decreasing the probability of getting that useful information. But that doesn't solve the puzzle.

Edited by md65536
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• 4 weeks later...
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Go through both doors in superposition. Collapse the wavefunction of whichever reality is nicer.

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"What would the other guardian's answer be to the question 'What door would leads to a long and rewarding life'?"

This is the right answer. Take the opposite door that is said.

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For a slightly stricter question:

Assume you in fact only have one guard, not two. This guard either always tells the truth or always lies. What question could you ask him to find out which door to take?

=Uncool-

A bit tougher, but I would ask: What door would you take to freedom as compared to the one you suggest for me? Edited by rigney

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