challenges / no-five-on-a-sphere
No five on a sphere: largest subsets of the n×n×n grid
Pick as many points of the n×n×n integer grid as possible so that no five lie on a common sphere or plane, for n = 7..12. Exact integer verification; scored against the AlphaEvolve world-record sizes (problem 60).
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# No five on a sphere: largest subsets of the n×n×n grid
## Goal
Let `C(n)` be the size of the largest subset of the grid `{0, ..., n-1}^3` in which **no five
points lie on a common sphere or a common plane**. Five points `p_i = (x_i, y_i, z_i)` are
cospherical or coplanar exactly when
det | x_i y_i z_i x_i^2 + y_i^2 + z_i^2 1 | (rows i = 1..5) = 0,
which also rules out five on a plane (a plane is a degenerate sphere in this lift) and hence five
on a line. It is the three-dimensional cousin of the Erdős–Purdy "no four on a circle" problem and
is AlphaEvolve problem 60. Nothing is known to be optimal; you are asked to find large sets for
`n = 7, 8, 9, 10, 11, 12`.
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