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Hmmm
You don't need calculus because you are dealing with integer values only. This is more of a pattern/series problem to me.
The quadratic formula merely approximates the maximum score based upon how many boxes you have to work with, but even this is irrelevant.
The maximum score will ALWAYS occur when N_boxes/2 are in the stack if N_boxes is even and it will ALWAYS occur when (N_boxes-1)/2 are in the stack if N_boxes is odd. The reason I say N_boxes-1 is illustrated by this example:
Say you have seven boxes to work with. It is easier to have a stack of three than a stack of four based upon the laws of physics. You will get the same score if you have three stacked and four on the ground or vice versa, but it is probably easier to "maintain" a stack of three.
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