Carmela has forgotten the password of her safe. However she remembers that:

1. The number is a ten digit positive integer using each of the digits from 0 to 9 exactly once.
2. The sum of the numbers in the 3rd and 4th positions is equal to the product of the numbers in the 6th and 9th positions.
3. The product of the numbers in the 5th and 10th positions is odd.
4. The digit in the 2nd position is larger than the digit in the 5th position, which is larger than the digit in the 7th position.
5. The digit in the 8th position is larger than the digit in the 6th position, which is larger than the digit in the 2nd position.
6. The sum of the digits in the 2nd, 6th, and 9th positions is a perfect square.
7. The sum of the digits in the 4th and 9th positions is equal to the digit in the 10th position.
8. The sum of the digits in the 1st and 5th positions is equal to the sum of the digits in the 7th and 10th positions.
9. The digit in the 9th position is even.

From the above clues, can you help Carmela to crack the password to open the safe?

I started out solving this in a different order, but at some point realized there was a quicker way and started over.  Here's the quicker way:

From 4 and 5, we know that 8th > 6th > 2nd > 5th > 7th.  Specifically, we know that the 6th digit is greater than 2. 

From 9 we know that the 9th digit is even.

From 2, we know that the 9th digit must be 2 or 4 (since if the 9th digit is >= 6, then 9th x 6th >= 18, which exceeds the possible sum of the 3rd and 4th digits).

Assume the 9th digit is 4.  Then:

From 2, the 6th digit must be 3.  The 3rd and 4th digits must sum to 12, so they must be 5 and 7 in some order.

From 4 and 5, the 2nd digit must be 2, the 5th digit must be 1, and the 7th digit must be 0.

From 7, the 4th digit must be less than 6, so the 4th digit is 5, the 3rd digit is 7, and the 10th digit is 9.

From 8, the 1st digit must be 8.

There are no condtradictions, so our assumption that the 9th digit = 4 produces the valid solution 8275130649.

Edited on July 2, 2014, 3:33 pm

Posted by tomarken on 2014-07-02 14:32:27