How many balloons do I need in each corner?
Jelly, 14 bytes

3Ḷ×þ`ZU$Ƭ×µ§§‘


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Takes input as a 3x3 matrix and outputs a list going clockwise starting from the bottom-right.

Explanation

3Ḷ×þ`ZU$Ƭ×µ§§‘     Main Link
3Ḷ                 [0, 1, 2]
  ×þ`              outer product by multiplication with itself ([0, 0, 0], [0, 1, 2], [0, 2, 4])
        Ƭ          (include the original, plus) rotate clockwise until results are no longer unique (4 results total)
     ZU$           zip and upend (clockwise rotation)
         ×         vectorized multiply with original (gets the workload distributions to each balloon)
           §§      sum results for each balloon (vectorized-sum rows in the matrix; sum the results)
              ‘    increment