The first thing I thought was if we would have to carry numbers around. I also was unsure about how we would get all of the 180 solutions.
The biggest pattern I found was that the units column would have to equal 19. The tens 18 and the hundreds 8. Another pattern I found was that you could swap a column of three around in 6 different ways. One could not be used in the ones column and the numbers 6, 7, 8 and 9 could not be used in the hundreds column.
I broke the problem down by first trying to find some solutions and then comparing my solutions to see if I could find some patterns. I found the 19, 18, 8 pattern this way. Then I focused on the rows. There were 5 combinations that would equal 19. Then I focused on how this fitted together and how we would find all of the solutions.
Well first I found three individual combos that would add up to 999. 168 + 327 + 694 = 999. 158 + 327 + 694 = 999. 326 + 175 + 498 = 999. This way we figured out what the rows equal. Then we found the 5 unique combos that add up to 19. The combos were, 8+2+9, 8+7+4, 6+5+8, 6+4+9, 7+3+9. We could swap these numbers around 6x. So 180 divided be 5 is equal to 36. Which gives us the 36 different solutions per 1 unique 5 solution. And we can reverse this 36 x 5 = 180. For the 180 solutions.
The challenging part of this was forming a plan that would give me 180 solutions. But once I found all the solutions that would give me 19 it became a lot easier.