challenges / unit-cube-packing
Unit cubes in the smallest cube
Erich Friedman's cubes-in-cubes problem (AlphaEvolve repository problem 35, packing in a dilate): place n unit cubes, freely rotated, with disjoint interiors so that their axis-aligned bounding cube is as small as possible, for n in {9, 10, 11, 12, 13, 14, 28}. Scored against the best-known side lengths.
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# Unit cubes in the smallest cube
## Goal
`pack.py` exposes `pack(n: int, time_budget: float, seed: int) -> list[tuple[float, float, float, float, float, float]]`:
`n` unit cubes `(x, y, z, ax, ay, az)`. `(x, y, z)` is the centre; `ax, ay, az` are Euler angles in
**degrees**, applied as `R = Rz(az) @ Ry(ay) @ Rx(ax)` to the axis-aligned cube `[-1/2, 1/2]^3`
before translating (the AlphaEvolve convention, so its constructions port unchanged). Interiors
must be pairwise disjoint (touching is fine). The container is the smallest axis-aligned cube
around all `8n` vertices, so `side(n) = max over the three coordinate axes of (max - min)`.
Minimise `side(n)`. Rotating the whole configuration is free, so an axis-aligned container loses
nothing.
`s(n)`, the smallest such side, is Erich Friedman's "Cubes in Cubes" problem (Packing Center),
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