Alex, Bert, and Carl know a secret two digit number. It is known that one of them is a knight who always tells the truth, one is a liar who makes all false statements, and one is a knave who alternates between true and false statements. Each one of them makes statements about the number as follows:

Alex:
1: One digit is 1.
2: The sum of the digits is 8.
3: Bert's second statement is false.

Bert:
1: One digit is 3.
2: The difference of the digits is 4.
3: Exactly one of Carl's statements is true.

Carl:
1: One digit is 6.
2: Alex's first statement is false.
3: The first digit is larger.

What is the secret number?

One quick solution involves analyzing Bert's 3rd question.

First, assume it is true.  Therefore, Carl is a Knave whose 1st and 3rd statements are false and whose 2nd statement is true.  By Carl-the-Knave's second (TRUE) statement, Alex must be a Liar, therefore Bert is the Knight.  By Bert's statements, we know the two digits to be 3 & 7.  Since Carl's 3rd statement is false, the number must be 37.

Now, consider that Bert's 3rd statement is FALSE.  This makes Bert either a Knave or Liar, and it makes Carl either a Knight or Liar.

Consider if Bert is a Knave.  This would make Alex a Liar (3rd Statement) since Bert-the-Knave's second statement must be true, and Carl would be a Knight.  By Carl's 1st and Bert's 2nd Statements, the digits must be 6 & 2.  However, this conflicts with Alex's 2nd Statement which is presumed to be false.  Thus, Bert cannot be a Knave, and instead must be a Liar.

If Bert is a Liar, then Carl must be a Knight, making Alex the Knave.  Yet this cannot be since Carl claims Alex's 1st Statement to be false while his 3rd statement remains true.  Thus, Bert cannot be a Liar.

-----------------

Alex - Liar

Bert - Knight

Carl - Knave

Code = 37

Posted by hoodat on 2008-06-03 16:23:39