One Question, Foreign Words

Question

At a mountain checkpoint two trails diverge: one reaches the open border crossing, the other dead-ends at a ravine. The lone sentry belongs to a clan that either always lies or always tells the truth, and he answers only in his dialect: "bal" or "dor", one means yes and one means no, and you don't know which. With one yes/no question, identify the trail to the crossing. Derive the question; don't

Accepted Answer

Point at a trail and ask: "If I asked you whether this trail leads to the open crossing, would you say 'bal'?" Take the trail if and only if he answers "bal." Why both unknowns cancel. Work in bits. Let p = "this trail is the crossing," ell = "he lies," m = "bal means yes." - To the inner question, he would assert the proposition p oplus ell; the word he'd utter is "bal" exactly when that asserted bit matches the convention: (p oplus ell) leftrightarrow m. - The outer question asks whether that word is "bal," whose true answer is the bit above. He asserts it flipped by ell again, and converts to a word via m again. So the uttered word is "bal" iff big(((p oplus ell) leftrightarrow m) oplus ellbig) leftrightarrow m. The ell appears twice (a lie about his own lie) and the m appears twice (the convention decoding its own mention): both double-flips are identities, leaving exactly p. Check the honest case ell = 0: word is "bal" iff (p leftrightarrow m) leftrightarrow m = p. Liar case ell = 1: iff ((lnot p leftrightarrow m) oplus 1) leftrightarrow m = (lnot p leftrightarrow lnot m) leftrightarrow m = p. All 8 cases (p, ell, m) collapse to: "bal" iff the trail is the crossing. The trap:

One Question, Foreign Words

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