Packing octahedron with cubes

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?

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$