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You arrive on the island with knights and knaves. Like usual, knights can only tell the truth and knaves can only tell lies. You wish to determine the truth of a rumor that one of the inhabitants has recently proved Goldbach’s Conjecture. What single question—answerable by “Yes” or “No”—can you ask of an arbitrary inhabitant in order to determine the truth?

I am stuck because there is only one inhabitant to ask and thus you have no basis for comparison of whether the one you are asking is a knight or a knave.

Asaf Karagila
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UserX
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  • Consider asking the inhabitant what they would say if you were to ask a particular Question. – hardmath Aug 31 '14 at 20:40
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    Removed the tag goldbachs-conjecture, since the question doesn't really have anything to do with that. –  Aug 31 '14 at 20:40
  • @hardmath: How would that help determine the truth of the rumor? Wouldn't that only help to determine whether they are a knight or a knave? – UserX Aug 31 '14 at 20:45
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    No. If you ask a liar/knave what he would say to "is 1+1 = 2", he knows he should say no, but he lies about that, so says yes, as would a knight too. So this "reflection" make the difference irrelevant. – Henno Brandsma Aug 31 '14 at 20:50
  • You may want to look at [Puzzling.SE] since this is really not about mathematics. – AlexR Aug 31 '14 at 20:54
  • @HennoBrandsma Using what you have said, I think that I should ask something along the lines of "If I asked you whether one of the inhabitants has proved Goldbach's Conjecture, would you say 'yes?'"

    If it has been proved, then the knight would say yes and thus answers me yes. The knave would say no but must lie about that and thus answers me yes. Likewise if it has not been proved.

    Am I on the right track?

     – UserX Aug 31 '14 at 20:56
  • @MikeMiller Off-topic tip: [tag:goldbachs-conjecture] creates a link to the tag like this: [tag:goldbachs-conjecture] – AlexR Aug 31 '14 at 20:58
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    @AlexR I have been trying to figure out how to do that for months! Thank you! –  Aug 31 '14 at 20:59
  • This is exactly the same as the standard Knight/Knave puzzle involving two doors. The proposition "the left door is the correct one" is replaced by "someone has proven GC", that's all. – Jack M Aug 31 '14 at 21:18

1 Answers1

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Q: What would you answer, if I asked you if someone had actually proven Goldbachs' conjecture?

Truth table: $$\begin{array}{ccc} \mathrm{Answer}\backslash \mathrm{Fact} & y & n \\ \mathrm{Knave} & y & n \\ \mathrm{Knight} & y & n \end{array}$$

Since the table for the straight-forward question is $$\begin{array}{ccc} \mathrm{Answer}\backslash \mathrm{Fact} & y & n \\ \mathrm{Knave} & n & y \\ \mathrm{Knight} & y & n \end{array}$$ and the Knave lies when answering, "double inversion" occurs, giving you the true answer to the straight-forward question from both kinds of people.

AlexR
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