Spreading points in a half-disc
Place n points inside a semicircle of radius 1 so that the minimum distance between any two points is as large as possible.
Formal definition
- ContainerAn upper half-disk of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centered at (1, 0), with the circular arc in y ≥ 0
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the half-disk or on its boundary
- Scoreδ(P) is the minimum Euclidean distance over all point pairs (the verifier compares δ(P)² using exact arithmetic internally)
- ObjectiveMaximize the minimum pairwise distance δ(P)
Getting a feel for it
Sector partitions and boundary anchors
Bounded by a diameter of length 2 and an arc of length π, the half-disk offers natural anchor positions at the endpoints (0, 0), (2, 0) and the center (1, 0). Partitioning the half-disk into circular sectors centered at (1, 0) yields tight upper bounds for small n, while larger point sets alternate between the arc, the diameter, and the interior.
Exact optima for four and five points
Using a partition into three 60° sectors, HwaterB #40 proved that the continuous optimal distance for both n = 4 and n = 5 is 1. Four points attain 1 with rational coordinates, whereas the unique five-point optimum requires the irrational coordinate √3/2; finite-decimal certificates can only approach 1 (the displayed 0.999999999 is strictly less than 1 and does not establish grid optimality). All cases n ≥ 6 remain open.
SourceCurrent 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 (1) ↓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-semicircle.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-semicircle.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-a-semicircle-2026-10,
title = {{Spreading points in a half-disc} (P20)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-a-semicircle.json},
license = {CC BY 4.0}
}Discussion
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