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Let's say there are three doors $1, 2, 3$ with labels: enter image description here

Only one label is true.

Which door do you need to open to get the clue?

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    $\begingroup$ What does it mean by next door? $\endgroup$ – Prince Deepthinker Aug 27 '20 at 2:49
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    $\begingroup$ next = neighboring $\endgroup$ – Nick Aug 27 '20 at 2:53
  • $\begingroup$ I just answered your question $\endgroup$ – Prince Deepthinker Aug 27 '20 at 2:58
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    $\begingroup$ That confused me too, usually 'next' means following and I assumed it meant door 1 as if wrapping around to the front. I think it would be better to use 'neighboring' or 'previous' $\endgroup$ – Jason Goemaat Aug 27 '20 at 16:44
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    $\begingroup$ “Next” in English means “following”, so this puzzle is very ambiguous as it stands. $\endgroup$ – Colin 't Hart Aug 28 '20 at 3:51
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If

1 is not true then the clue is in 1 and label on door 2 or 3 is true. But 2 can't be true since clue is in 1 and 3 also can be true for the same reason.

So

1 is true and 2 and 3 are falls so the clue is in 3.

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  • $\begingroup$ Answer is short and clear. $\endgroup$ – Nick Aug 27 '20 at 15:56
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Since

Doors 2 and 3 are saying the same thing, so they are both true or both false.

Since only one door is true,

Door 1 is true.

Because

Door 1 is true, there cannot be a clue behind it.

Since

Door 2 is false, there cannot be a clue behind it.

Therefore,

if there is a clue, it is behind Door 3, which satisfies all conditions.

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    $\begingroup$ Technically speaking, there is nothing in the puzzle that allows us to conclude that there is a clue behind any door. $\endgroup$ – Acccumulation Aug 27 '20 at 20:34
  • $\begingroup$ @Acccumulation Yes, it is only implied. But the logic still holds that if there is a clue, there is only one door it could behind. I've edited my answer to account for that. $\endgroup$ – asg Aug 27 '20 at 20:41
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If next door means what I think it means then

3. The clue lies with that door. 1 is the true door . This is because if 1 is true then 2 is false so neither have the clue and 3 would be false that 2 has the clue leaving it as the clue door.

Because

the truth then would be that the clue is not next door and so it would open up the possibility that it has the clue. And since neither door previously would have the clue and it is true that one door has the clue it must be 3.

Another explanation:

if it was 2 that was true it would lead to the fact that 1 is false and would also contain a clue and as two doors would contain clues when only one should.

If

3 was true then 2 would both contain the clue and not contain the clue and that would be a contradiction.

So

1 must be true.

If

1 contained the clue it would mean that 2 or 3 is true so one cannot contain the clue

If

2 contained the clue then 2 is true so two cannot contain the clue

If

both 1 and 2 doesn't contain the clue then if one of the doors contains the clue then 3 must contain the clue.

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  • $\begingroup$ Or in half as many words: rot13("2 naq 3 fnl gur fnzr guvat, naq bayl bar ynory vf gehr, fb gurl zhfg obgu or snyfr. Gur gehr ynory zhfg gura or N.") $\endgroup$ – Bass Aug 27 '20 at 4:15
  • $\begingroup$ @Bass, did you mean N=1 in your comment? $\endgroup$ – Nick Aug 27 '20 at 4:38
  • $\begingroup$ I got it right? $\endgroup$ – Prince Deepthinker Aug 27 '20 at 4:46
  • $\begingroup$ Can you please confirm $\endgroup$ – Prince Deepthinker Aug 27 '20 at 4:46
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Assume

the 3rd statement is true, then the clue should be on 2️⃣. But the 2nd statement contradicts that.

Assume

the 2nd statement is true, then the clue should be on 2️⃣. But the 1st statement contradicts that.

Assume

the 1st statement is true, then the clue must not be on 1️⃣. And the 2nd statement is false, then the clue must not be on 2️⃣. The clue can only be in 3️⃣ and this does not contradict the 3rd statement.

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If you're

in the intuitionist world, then the fact 1 is not true doesn't necessarily mean not 1 is. So, given the "only one label is true", the only thing we can do is to compare the evidence that the three separate hypotheses give. If (1) is true, then the clue is in either (2) or (3), if (2) is true, then the clue is in (2), and finally if (3) is true, then the clue is in (1). So, without being able to solve, we guess (2) since it is confirmed by two possible variants out of three.

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    $\begingroup$ I've never really understood inuitionism. Even in intuitionistic logic, if the clue is in (2), that would make label (2) is true and label (3) is true as well, contrary to the puzzle's premise. And yet intuitionistic logic doesn't allow me to conclude from this that the clue is not in (2) because that is the law of the excluded middle, rejected by intuitionists. Intuitionistic logic only allows me to conclude that there can never be (intuitionist) proof that the clue is in (2), and that to conclude it is not in (2) requires direct proof. $\endgroup$ – Jaap Scherphuis Aug 27 '20 at 9:56

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