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In your quest to escape a Magician's lair, you are greeted by 3 identical looking gatekeepers. You can only proceed after correctly identifying which one is which. The only trouble is, they can magically trade places during the course of the game!

One is a Knight, who always tells the truth if he knows it, and remains silent otherwise.

One is a Knave, who always tells a lie if he's sure it's a lie, and remains silent otherwise.

One is the Magician himself, who can do any of the following actions when asked a question:

1.) Tell the truth and trade places (instantly and magically) with the Knight.

2.) Tell the truth and remain in place.

3.) Tell a lie and trade places (instantly and magically) with the Knave.

4.) Tell a lie and remain in place.

5.) Remain silent and trade places (instantly and magically) with the Knight.

6.) Remain silent and trade places (instantly and magically) with the Knave.

Note: It is impossible to tell whether the Magician has traded places or not using any form of sensory input. At the start of the game, the gatekeepers know who is who. They can all answer questions about who started where, however even the Knight and the Knave cannot tell when the Magician trades places with the other gatekeeper. After they become unsure of an answer, the Knight and the Knave will default to silence.

What is the fastest strategy for identifying the 3 gatekeepers?

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  • $\begingroup$ Do the keepers know who is who? this is important distinction $\endgroup$
    – Destructible Lemon
    Aug 23, 2016 at 5:20

2 Answers 2

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The formatting here is bad but I can't use spoilers well.

EDIT: This answer is not quite right. It's close though

I think there might be a more efficient way by asking meta questions, but IDK (by meta questions, one that imply that the questioned knows if they have been moved) still 4 questions for meta questions. On another note, just realised I have four questions only. found more efficient meta questions, leaving this here for legacy non-meta questions, and also I think that the meta-questions hinge on them being able to know which is which, which I'm not sure of as being allowed

The meta is really spinning my head right now, just realised my second answer wasn't valid. This is quite easily the most strike through I've ever used. consequently, I believe four three (new rules) questions is the minumum

Ask "what number am I thinking of?" to someone in position one

Ask them if 1+1=2

Ask whether the person left of them is mage

Explanation:

1 Ask one what number you are thinking of. They will remain silent. You know that this now either the knight or the knave. 2 Test which it is, by asking 1+1=2 or something. You now know which of the knight or knave they are. 3 Ask if the person to their left is mage. They know this because they either were swapped with them, or not swapped at all. You can now know with this info

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  • $\begingroup$ the magician can remain silent too $\endgroup$
    – Sechiro
    Aug 23, 2016 at 8:05
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    $\begingroup$ @Sechiro but the magician must swap when he remains silent; this is why we ask the unknowable question $\endgroup$
    – Destructible Lemon
    Aug 23, 2016 at 8:39
  • $\begingroup$ I think this is very close and does get the heart of the challenge (forcing the wizard's location). However, to the first question, the wizard can say any number and remain in place (or swap with either other gatekeeper). He is not sure whether it is truth or a lie when he picks a number, but it must be one of the two, and either way he is allowed to remain in place. I think that's the only contingency you're missing. $\endgroup$
    – stevenjackson121
    Aug 23, 2016 at 9:14
  • $\begingroup$ @user1540815 One could interpret the question that way. However having identified the magician is a huge benefit, so the strategy in that case is trivial. $\endgroup$
    – Taemyr
    Aug 23, 2016 at 9:16
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    $\begingroup$ Wait I'm an idiot. If the wizard doesn't know whether he just responded with truth or lie, he can only stay stationary (moving might cause him to violate the rules if he chooses wrong). I apologize to Destructible Watermelon and also to Taemyr, it does trivialize it if the wizard answer an unknowable problem and thus remains stationary. $\endgroup$
    – stevenjackson121
    Aug 23, 2016 at 9:40
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What I think is a proof that it is not possible if the wizard can tell truth and lies and teleport, when he does not know the answer to questions

EDIT: It has been brought to my attention that through paradoxes, this question is solvable, however, that doesn't quite fit. Regard this as a proof of the above, with the law of excluded middle

not spoilered because this is to stop people trying to solve impossible question

In any case, the only one we need to work to find is the wizard; the others are easily distinguished

We can only pin down the wizard to "this guy is not wizard". If we ever get the wizard to remain silent, we win (because we can get him), so we assume wizard will not remain silent. We can force the wizard into two possibilities, by using "Would the Wizard say no to this question?" (wizard must lie or be silent), or its inverse, and we can then find the knave using this, but it doesn't matter, because we can only pin him down to these two, we cannot question the definite non-wizard, because he can't know if the wizard swapped places when we asked "Would the Wizard say no to this question?" to someone, and we cannot ask the possible wizards, because then they could lie OR tell truth, and then swap places, but also not swap places. So, it is not possible to definitely find the identities if the wizard can teleport given an unknowable question

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