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?

If the azure one has 8 arms, then the jade one has 10 arms and is a knight. If the azure one has 10 arms, then the jade one has 8 arms and is a knight. Suppose the azure one has 9 arms. Then, the azure one is a liar, so the jade one cannot have 9 arms. Therefore, the jade one has either 8 or 10 arms, so the jade one is a knight. That proves that the jade one is a knight. If the jade one has 8 arms, then the ivory one has 10 arms and is a knight. If the jade one has 10 arms, then the ivory one has 8 arms and is a knight. Therefore, the ivory one has to be a knight. That means that the ivory one and the amber one have 19 arms together. Since the ivory one is a knight, the ivory one has 10 arms and the amber one has 9 arms. Since the jade one is a knight, the jade one has 8 arms. The amber one is a liar, so the amber one and the azure one cannot have 19 arms together. Therefore, the azure one cannot have 10 arms. If the azure one has 8 arms, then the azure one and the jade one have 16 arms together. Then, the azure one would be lying, which is impossible. Therefore, the azure one has 9 arms.

Posted by Math Man on 2012-09-17 22:53:23