Answer:

Explanation:
You want a 3×3 square consisting of the digits 0–8 such that the rows in order have sums in the ratio 1:2:3, and the columns in order have the same sums.
Sums
The digits 0-8 have the sum 8·9/2 = 36. For the rows or columns to have the required ratios, their sums must be 6, 12, 18.
The only ways to make sums of 6 from the given digits are ...
0 +1 +5 = 0 +2 +4 = 1 +2 +3
The only ways to make sums of 18 from the given digits are ...
3 +7 +8 = 4 +6 +8 = 5 +6 +7
Arrangements
There are only 6 ways to choose one of the 6-sums and one of the 18-sums such that all the digits are different. Those 6 ways give rise to 6 solutions to the problem.
Each solution has a transpose (interchanged rows and columns) that is also a solution. So, there are only 3 unique solutions to the puzzle.
The solutions are listed in the Answer section, above.
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