A distant planet is inhabited by creatures having more than two arms. The ones with an even number of arms are the knights who always tell the truth; the ones with an odd number of arms are the liars who always speak falsely.

The rules of the planet allow only creatures having eight, nine or ten arms to work as guards. Four guards were conversing amongst themselves:

• The amber one said, "The azure one and I have 19 arms together."
• The azure one said, "The jade one and I have 18 arms together."
• The jade one said, "The ivory one and I have 18 arms together."
• The ivory one said, "The amber one and I have 19 arms together."

How many arms does each guard have?

Just wanted to add this, since I didn't see it mentioned in the other posted solutions.

Consider the statement, "X and I have 18 arms together."  If the speaker is a knight, then X is also a knight (since the speaker would have 8 or 10 arms, leaving 8 or 10 arms for X).  If the speaker is a liar, then he has 9 arms.  This means that X doesn't have 9 arms (otherwise the speaker's statement would be true), so X is a knight.  Thus any time someone makes a statement, "X and I have 18 arms together," we don't necessarily know what the speaker is, but we know that X is a knight.

With that insight, it's even easier to solve IMO, because right off the bat you know that Jade and Ivory are knights.  It immediately follows from this that Ivory has 10 arms, Amber has 9, and Jade has 8, so the only thing left to figure out is Azure, and it's easy to see that Azure must also have 9 arms.

Posted by tomarken on 2014-03-24 14:16:43