The answer is unique. If you'd like a tougher challenge, Fermat made the claim that any number could be represented as the sum of at most 3 triangular numbers. Gauss proved it on July 10, 1796.
Top row can have no zeroes.
Dick Saunders Jr. Hi Ed, Well, this puzzle is made much easier than I at first thought because as it happens the correct final four-digit number is the very first you try if, as I did, you start with the lowest. Generally, with this sort of problem, the number of solutions either quickly climbs, or quickly decreases as the size of the puzzle is increased.