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:
1: One digit is 1.
2: The sum of the digits is 8.
3: Bert's second statement is false.
1: One digit is 3.
2: The difference of the digits is 4.
3: Exactly one of Carl's statements is true.
1: One digit is 6.
2: Alex's first statement is false.
3: The first digit is larger.
What is the secret number?
Assume that Alex is the knight. If so, his statements 1 and 2 imply that the number is either 17 or 71. In this situation, Bert's consecutive statements 1 and 2 are false, so that Bert is a liar.
Carl's first statement is also similarly false, and his statement 2 is false as well since it contradicts Alex's true statement. Thus, both Bert and Carl are liars. This is a contradiction, since we cannot have two liars amongst the three. Accordingly, our origial assumption is erroneous and Alex cannot be the knight.
Assume that Carl is the knight. Then precisely one of Alex and Bert is a liar while the other is a knave. By Carl's true statement 2, it follows that Alex's statement 1 is false. Since, Alex is either a knave or a liar, it follows that his statement 3 must also be false. Thus, Bert's statement is true. Therefore Bert must be a knave in the L-T-L format, so that Alex must be the liar. Hence, by Carl's true statements 1 and 2 in conjunction with Bert's true statement 2, it follows that the number is 62, so that the sum of the digits is 8. Thus, Alex's statement 2 is true, which is a contradiction, since Alex is the liar. Accordingly, it follows that Carl cannot be the knight.
Assume that Bert is the knight. If so, by his statements 1 and 2, it follows that the number is either 37 or 73. His statement 3 implies that Carl is the knave in the L-T-L format, so that Alex must be the liar. We know that the number is either 37 or 73, so that Carl's statement 1 is indeed false. Since Carl's satement 3 must also be false, it follows that the number is 37. Since Alex, is the liar, all his statements are false. So that none of the digits is 1, the sum of the digits is not 8 (it is 10), Bert's statement 2 is true (not false). Also, as a knave in the L-T-L format, Carl's statement is correct in identifying the falsity of Alex's first statement.
Consequently, Alex is the liar, Bert is the knight and Carl is the knave and the secret number is 37.
Edited on June 3, 2008, 4:35 pm