Qube
A Qube is constructed from multiple unit volume cubes each one comprising 3 cuboid elements. All 3 elements have the same area to volume ratio.
The internal perimeter of the unit volume cube is the sum total of the perimeter of all 3 elements.
The external perimeter of the unit volume cube is the sum total of the perimeter of the exterior joints and edges when all 3 elements are mated.
If the difference between internal and external perimeters of the unit volume cube is equal to units, and the maximum sum of the external and internal perimeters of all the unit volume cubes can be 10000 units, find the maximum volume possible of the Qube.
Note: All edge dimensions of the Qube are integers and uniform.
The answer is 216.
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1 solution
Unit Volume Cube = (0.382 x 1.0 + 2 x (0.618 x 0.5)) x 1.0
Area to volume ratio = 7 + 5^0.5
Interior perimeter = 26.472 units
Exterior perimeter = 18.236 units
Internal perimeter - External perimeter = 6 + 5^0.5 units
Maximum volume = 10000/(26.472 + 18.236) = 223.67 units
Maximum integer Volume = 6^3 = 216 units