Spreading points in a plus sign
Place n points inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) so that the minimum distance between any two points is as large as possible.
Formal definition
- ContainerA symmetric cross-shaped region formed by removing the four 1 × 1 corner squares from [0, 3] × [0, 3], with lower-left corner (0, 0) and upper-right corner (3, 3)
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the cross-shaped region 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
Five-square partition and arm vertices
The symmetric cross consists of a central 1 × 1 square and four 1 × 1 arms, providing eight outer corner vertices and four re-entrant corners. Partitioning the cross into its five unit squares or finer subregions yields pigeonhole upper bounds on the minimum distance, while optimal configurations anchor points at the outer arm tips and balance the central core.
Four proven cases
Optimality is proven for n = 4, 5, 6, and 9: NUE_13 #31 supplied the constructions and conjectures, with the upper bounds for n = 4 and n = 5 completed on this site; HwaterB #40 proved the n = 6 optimum √2 via a five-square partition and the n = 9 optimum √5/2 via an eight-region partition. All other sizes 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 (3) ↓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-cross.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-cross.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-a-cross-2026-10,
title = {{Spreading points in a plus sign} (P21)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-a-cross.json},
license = {CC BY 4.0}
}Discussion
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