Spreading points in a quadrant
Place n points in a quarter-disc of radius 1 so as to maximize the minimum distance between any two points.
Formal definition
- ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the container or on its boundary
- Scoreδ(P) is the minimum Euclidean distance over all point pairs; internally, the verifier compares δ(P)² in exact arithmetic
- ObjectiveMaximize δ(P)
Getting a feel for it
Connection to circle packing
Spreading points is equivalent to packing equal circles whose centers are restricted to the container: discs of radius δ/2 around the points have disjoint interiors. Optimal configurations are governed by rigid contacts between point pairs and the container boundary.
Known results
A three-region proof by XSsMC #250 resolves the continuous n = 4 case with optimal distance √(2−√2); that instance is archived with its historical records and precision notes. All larger instances remain open.
Current best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (2) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/spread-points-in-a-quadrant.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/spread-points-in-a-quadrant.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-a-quadrant-2026-10,
title = {{Spreading points in a quadrant} (P31)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-a-quadrant.json},
license = {CC BY 4.0}
}Discussion
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