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Omni-MATH / [i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having n inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always t…
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[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that if ?
Plain-text mathematical notation (without MathML)
[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having n inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability 1/2 independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let p_(n) denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that p_(n)→1 if n→∞?
Original LaTeX notation
[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having $n$ inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability $1/2$ independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let $p_n$ denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that $p_n\to 1$ if $n\to\infty$?
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