Here is a numbered list of statements, some true, some false, which refer to a specific number (unique positive integer, base 10).

It just so happens that if a statement is true then its index number appears among the number's digits, and if a statement is false then its index number does not appear among the number's digits.

0. The sum of the number's digits is a prime.
1. The product of the number's digits is odd.
2. Each of the number's digits is less than the next digit (if there is one).
3. No two of the number's digits are equal.
4. None of the number's digits is greater than 4.
5. The number has fewer than 6 digits.
6. The product of the number's digits is not divisible by 6.
7. The number is even.
8. No two of the number's digits differ by 1.
9. At least one of the number's digits is equal to the sum of two other digits. (Any of the digits may be equal, as long as all 3 digits are distinct... for example: {2, 2, 4} or {2, 3, 5} )

Find the number.

(In reply to Solution by Federico Kereki)

Good solution, FK, with one exception: 2359 does not make statement (1) true.  However, it does make statement (0) true, requiring 0 to be in the number.  Since the number is given to be an integer, which by convention does not start with a 0, 0 cannot be in the number if rule (2) holds.  Since 2 cannot be eliminated without making (1) true, the initial premise is false and 8 must be in the number.

Posted by Bryan on 2004-07-20 13:51:00