In an island peopled by Knights (who always say the truth) and Liars (who obviously always lie), two persons spoke about each other.
A said: "B would say I lie."
B said: "That is true."
Can anything be deduced as to what kind is each of them?
(In reply to
Solution by Brian Wainscott)
Well Brian... to nit pick myself...
You write that A's statement is clearly false. But that's not necessarily the case. A's statement is false ONLY if one of the two following conditions is true:
A is a Knight and B is a Knight
A is a Liar and B is a Liar.
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To analyze this problem (just like all other knight and liar problems... *sigh*), I would suggest one considers the four possibilities:
___A_B
1 ) K K
2 ) K L
3 ) L K
4 ) L L
(where K = Knight, and L = Liar)
And if we do so, we realize that cases 1, 2, and 3 all lead to inconsistency.
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Now, to be annoyingly thorough about this:
case 1:
If A and B were both Knights, then A wouldn't say that "B would say I lie."
case 2:
If A were a Knight and B were a Liar, then A WOULD say that "B would say I lie" because it is true. But B, being a Liar would, not agree with the truth. Therefore, this is inconsistent.
case 3:
If A were a Liar and B were a Knight, then A wouldn't say "B would say I lie", because that is the truth. Another inconsistency... so this can't be the case.
in case 4, however:
If A and B are both Liars, then the statement "B would say I lie" is false (because B would say that A tells the truth), so A can say this falsehood. Then B, realizing this is a false statement will claim it is true by saying "That is true.".
This is consistent, and is the only case that is consistent. Therefore A and B are both Liars.
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Wow... this has actually brought tears to my eyes.
--- SK
re: Solution
