Find a four-digit number with four different digits, that is equal to the number formed by its digits in descending order minus the number formed by its digits in ascending order.

There's an interesting rule connected to this problem.

All of the numbers in those sequences (except the number you start with) must be divisble by 9, because the result of the substraction of a number and its inverse is always divisible by 9 (as long as it isn't symmetrical, like 1221 cause that results in 0).

This even seems to work with rational numbers e.g. 7524.871 - 1784.257 = 5740,614 which is divisible by 9 (you can add up the digits to be divisible by 9, it's not within the common bounds of numbers - integers - divisible by 9, but it doesn't produce more digits behind the period - i can't describe it any better as non-native, sorry). I can't give any more proof, but I assume this rule is substantial to basic math (I didn't know that before, though).

After all, this proves that a "magic number" (by our definition, a number of which the difference between the number constructed by its digits in decreasing order and the number constructed by its (the starting number) digits in increasing order result back in the starting number) must be divisible by 9.
This might have helped when we first started to find such a number, without any of the knowledge we gained in the process. ;)
Edited on September 12, 2003, 9:48 am

Posted by abc on 2003-09-12 09:47:03