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.

The most intiguing statement is the last one. If we suppose it's true, then the only possibility is that Bert must have made all true statements and Alex must have made all false statements.

So "The number is not 1, 2, 8, nor 9" is false, "The number is 2, 4, 7 or 9" is true, and the statement "The number is odd" is true. This means the number must be 9.

To check if the other statements match up, "The number is less than or equal to 5" and "Bert's first statement is false" are false while "The number is even", "Alex's first statemet is true" are false while "The number is at least 5" is true. So the number could be 9.

If we wanted to make sure the solution is unique, we assume the last statement is false. Then notice "Alex's first statement is true" would be inconsistent if one made all false statements and one made all true statements. So one of them made an unknown number of true/false statements, and by assumption, his last statement is false, so he must have made all false statements.

So we know "The number is 2, 4, 7, or 9" is false, and "The number is odd" is false. So the number is either 6 or 8, but that makes "The number is less than or equal to 5" false and "Alex's first statement is true" false. Since one of them must have all true statements, a contradiction is reached, so there are no other solutions.

Posted by Gamer on 2007-03-16 10:54:34