Welcome to the Spring 2008 Barge Team Competition. This is the first problem of the semester. The problem is about positive four digit integers with the following properties:

a) All the digits are different.

b) Reversing the order of the digits produces a new number that is four times as large as the original number.

By reversing the order of the digits, I mean that the digits now appear in the opposite order. So reversing the order of the digits of 5304 produces the number 4035. Also, the leftmost digit of the original number is not allowed to be 0 - something like 0315 is not allowed.

Your job is to:

1) find all such four digit numbers, and

2) prove that there are no others.

A computer programs that searches all possibilities will not be considered a valid proof - you should be able to produce by hand a relatively short argument that shows you have found all of the possibilities. Partial credit is available if you can find such numbers but can't prove that you have found them all.