You are in a popular tourist town in the land of Liars and Knights. You happen to overhear a conversation another tourist is having with three of the locals: Alex, Bert and Carl. Each of the three could be a knight, a knave or a liar. You know that for each question the tourist asks, Alex, Bert and Carl each give one response, but you don't know who said what. The conversation is as follows:

1. What type are each of you?
   • I am a knight.
   • I am a knave.
   • I am a liar.
2. How many of you are the same type?
   • We are all the same type.
   • We are all different types.
   • Exactly two of us are the same type.
3. What type is Alex?
   • A knave.
   • A liar.
   • Different from Bert.

Can you determine which type Alex, Bert and Carl are?

Whoever said that they were a liar in the first question is a knave because a knight would not lie and say that, and a liar would not tell the truth and say that. The knave lied, so their next answer is true. Since at most one of the answers to the second question can be true, there cannot be any knights among them. Suppose the one who said that they were a knight was a knave. Then, they would be lying, so their second answer would be true, but at most one of the answers to the second question is true. Therefore, they are a liar. Since there are no knights among them, Alex is either a knave or a liar, so at least one answer to Question 3 is true. The one who told the truth in Question 3 must be a knave, since none of them are knights. We also know that the knave who claimed to be a liar and the liar who claimed to be a knight in Question 1 both lied. Therefore, they both lied in Question 3, so there are two knaves and a liar. Also, the answers to the third question are two truths and a lie. Since one of the first two answers is true, the third answer is false, so Alex and Bert are of the same type. Therefore, Alex and Bert are both knaves and Carl is a liar.

Posted by Math Man on 2011-01-02 12:51:08