Place n points inside the unit square, minimizing the sum of 1/distance² taken over every pair.
Formal definition
ContainerThe container is a square of side 1, with the origin (0, 0) at its lower-left corner and (1, 1) at its upper-right.
SubmissionExactly n points, no two coinciding
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the sum of 1/distance² over all pairs as small as possible; scored in exact rationals
Getting a feel for it
Where the room for improvement is
1/distance² punishes closeness brutally: points get pushed out to a boundary ring first, then shed inner layers as n grows. Layer counts jump at particular n, and the jumps are where the contest lives.
Where the frontier is
The general theory of pairwise 1/r² energy (asymptotics, separation) is Borodachov, Hardin and Saff, Discrete Energy on Rectifiable Sets (2019); a per-n table of optima in the square does not exist, and every n here is open. The few trivial closed forms are proved on site.