### "Flawless Harmony" Puzzle

**Source:**AUSTMS Puzzle Corner 35

**Problem:**

Call a nine-digit number flawless if it has all the digits from 1 to 9 in some order. An unordered pair of flawless numbers is called harmonious if they sum to 987654321. Note that (a, b) and (b, a) are considered to be the same unordered pair.

Without resorting to an exhaustive search, prove that the number of harmonious pairs is odd.

Update (23 Oct 2014):

**Solution:**Posted by me (Pratik Poddar) in comments!