Given an octahedron of side length 10 units, what is the maximum number of cubes of side length 1 unit that can be packed inside it.

I have checked it for two orientations, both of them consist of square grids of cubes aligned with the square cross sections of the octahedron, and 327 cubes is the solution I came up with. Another good arrangement could be to align the grids of cubes so that they are parallel to two of the triangular faces.

But is there a good way to prove which packing would be the most efficient without enumerating the orientations which intuitively seem dense?

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