Place n points in the square of side 1. Every location in the square is served by whichever point is nearest; your score is the average, over the whole square, of the squared distance from a location to the point serving it. Make that average as small as you can.
Formal definition
ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
AssignmentEvery location belongs to the nearest point; the exactly-equidistant locations form a set of zero area, so their assignment cannot change the score
ObjectiveMake E(P) = ∫∫ min‖x − pᵢ‖² dx as small as possible. Scored in exact rationals, rounded up at 10⁻¹⁸
Getting a feel for it
An analogy: respawn points
Read the square as a map and the n points as respawn points. A player appears uniformly at random anywhere on it and is sent to the nearest one — your score is the average squared length of that trip.
Where the room for improvement is
Moving every point to the centre of mass of its region, over and over, is Lloyd's algorithm: it always stops, but where it stops is a stationary point, not the best one. This energy has many local minima, and which one you fall into depends entirely on where you started — that gap is the contest.
Where the frontier is
n ≤ 2 is proven; n = 3, 4 and 5 rest on an unproven symmetry conjecture (Roychowdhury, arXiv:1608.03815); of anything larger the literature says it "is extremely difficult and the answer is not known yet". One floor holds for every n: 5/(18√3·n), the level of regular hexagons — which never tile a square.
ONE LEADERBOARD PER n
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.