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?

Suppose Alex is the knight. Then the two digits are 1 and 7 to meet his first two statements. Bert's 2nd statement is false as is his first, so Bert must be a liar. But Carl's first statement is also false, as is his second, so Carl must also be a liar. Since there can be only one liar, the initial supposition is wrong. Alex is NOT a knight.

Suppose Carl is the knight. Then one digit is a 6, Alex's first statement is false, and so must be Alex's 3rd statement (whether he's a knave or a liar, the truth of the 1st and 3rd statements must match.) Bert's 2nd statement is then false, and since his 3rd statement is also false, Bert is the liar, and Alex the knave. That makes Alex's 2nd statement true so the sum of the digits is 8 and the 2nd digit is therefore 2. Carl's 3rd statement is true so the secret number would be 62, except that this number makes the 2nd statement by Bert--a liar--true. That contraditction means Carl is not the knight.

Bert, then, must be the knight, and the digits must be 3 and 7. Since exactly one of Carl's statements is true it must be that the 1st and 3rd of his statements are false (since Carl must be a knave to have a mixture of true and false statements) and so the 2nd digit of the secret number is larger, making it 37. (Alex is a liar.)

The secret number is 37

Posted by Paul on 2008-05-30 16:32:47