Let ~S~ be the total mass of the weights after balancing the scale. Then the total mass of the weights on the beams of the large scale is equal to ~S/2~, and the sum of the weights on the beams of these scales is equal to ~S/4~, and so on. Generally, a beam of a scale at depth ~r~ holds a load of mass ~S/2^r~.

Since some beams (at depth ~r~) already have a weight of mass ~m~, additionally the following inequality must hold: ~S/2^r \ge m~. Finally, we have ~S = \max\{m \times 2^r\}~ for each weight of mass ~m~, that is at depth ~r~.

We are left with efficiently comparing the numbers of the form ~a \times 2^b~, where ~a~ and ~b~ are small enough to fit in a 32-bit data type. Since these numbers in binary notation have at most 32 ones (multiplication with ~2^b~ corresponds to left shift ~b~ times), it is sufficient to just linearly iterate over the ones in the binary notation of these numbers.