Covering a square with n equal discs

Formal definition

• ContainerThe closed unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
• SubmissionExactly n distinct points, each coordinate a decimal string with at most nine places
• Scorer(P) is the maximum over the square of the distance to the nearest submitted point. The verifier computes its exact rational square and rounds the square root upward at 10⁻¹⁸, guaranteeing a valid covering radius$$r\left(P\right)=\underset{x\in K}{\max }\underset{i}{\min }\Vert x-p_i\Vert$$
• Why this is finiteWithin each point's Voronoi cell, the convex function |x − p|² attains its maximum on a convex polygon at a vertex. The continuous maximum therefore reduces to checking finitely many rational Voronoi vertices in exact arithmetic
• ObjectiveMinimize r(P) over all valid point sets P$$\underset{P}{\min }r\left(P\right)$$