Alex, Bert, and Carl know a specific number 1 to 9. From the statements below can you determine the number knowing one of them made three true statements and one of them made three false statements?

Alex: The number is less than or equal to 5.
Bert: The number is 2, 4, 7, or 9.
Carl: The number is even.
Alex: Bert's first statement is false.
Carl: The number is at least 5.
Bert: The number is odd.
Carl: Alex's first statement is true.
Alex: The number is not 1, 2, 8, nor 9.
Bert: Carl made exactly one true statement.

If the number is less than or equal to 5, then that makes Alex's first statement true. This makes Carl's third statement true, and so all of Bert's statements would be false. But Bert's first and second can't both be false, because 2 and 4 are the only evens below 5, and Bert's second statement says that the number is odd. So Alex's first statement can't be true.

This makes Alex's first, and Carl's third statements both false. So, naturally all of Bert's must be true. So the possible values are 2, 4, 7, and 9. Then, Bert says the number is odd, so we are down to 7, or 9. Carl's first statement would be made false, because the number is odd. But, 7 and 9 are both at least 5. So Carl's second statement is true, holding to Bert's last statement that Carl made only one true statement.

Since all of Bert's statements must be true, Alex's second stating that Bert's first is false, is false as well. Alex's third statement that the number is not 1, 2, 8, nor 9, is false, because Alex must lie all 3 times to fit the first part of the problem. This means that one of those values is the real value, and the only one that fits in with Bert's statements is 9.

Based on these, it would appear that the solution to this problem is 9.

Edited on March 16, 2007, 2:37 pm

Posted by Justin on 2007-03-16 14:36:00