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The 3 foot square box contains 46,656 cubic inches. The 3 inch square cube has 27 cubic inches. The division of 46,656 by 27 yields 1728 3x3x3 cubes inside the larger one.
64 of them
The question is somewhat puzzling. The surface area of a box is not sufficient information to determine its shape and, therefore, its size. Furthermore, surface area would not be given as 480 cubic inches since that is not a measure of area (but of volume). On the other hand, if the box has a volume of 480 cubic inches, the answer, becomes trivial. 480 inch cubes = 480 cubic inches. Rocket science!
1,728
12 cubes in total.