Covering a square with n equal discs

Formal definition

• ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right, boundary included
• SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
• Scorer(P) is the largest, over the square, of the distance to the nearest submitted point. The verifier finds its square exactly, then takes an exact square root rounded upward at 10⁻¹⁸ — upward, so the stored number never claims a tighter cover than the arrangement achieves$$r\left(P\right)=\underset{x\in K}{\max }\underset{i}{\min }\Vert x-p_i\Vert$$
• Why this is finiteInside its own nearest-neighbour region a point is the nearest one, and |x − p|² is convex, so its largest value on a convex polygon is at a corner. The search over a continuous region collapses to finitely many rational corners, in whole numbers throughout, touching no float
• ObjectiveMake r(P) as small as possible over all legal point sets P$$\underset{P}{\min }r\left(P\right)$$