Riesz 2-energy in a square

Formal definition

• ContainerThe unit square of side 1, with lower-left corner at (0, 0) and upper-right corner at (1, 1)
• SubmissionExactly n distinct points
• ConstraintsEvery point lies inside the container or on its boundary
• EnergyE₂(P) is the Riesz 2-energy of P$$E_2\left(P\right)=\sum _{1\le i<j\le n}\frac{1}{{\Vert p_i-p_j\Vert }^2}$$
• ScoreThe verifier rounds each pair's energy upward to the 10⁻⁹ grid before summing, and displays score(P)/10⁹$$\mathrm{score}\left(P\right)=\sum _{1\le i<j\le n}\lceil \frac{{10}^9}{{\Vert p_i-p_j\Vert }^2}\rceil$$
• ObjectiveMinimize score(P), which approximates E₂(P) with an upward rounding error of at most 10⁻⁹ per pair