Spreading points in a rectangle
Place n points inside a 2 × 1 rectangle to maximize the minimum Euclidean distance between any two points.
Formal definition
- ContainerA 2 × 1 rectangle [0, 2] × [0, 1], with lower-left corner (0, 0) and upper-right corner (2, 1)
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the rectangle 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
Strip partitions and staggered chains
Inside a 2 × 1 rectangle, the four corners and the two long horizontal edges provide natural anchor positions. Partitioning the rectangle into vertical strips or unit squares yields tight pigeonhole upper bounds on the minimum distance, while optimal configurations stagger points across the top and bottom edges.
Four to six points solved; remaining cases
HwaterB #40's three-strip argument proves the continuous optimum √13/3 for n = 4, and XSsMC #250's two-square reduction with the classical three-point theorem proves √6 − √2 for n = 5; both cases retain their history and close competition without claiming nine-decimal grid optimality. NUE_13 #31's tightening-strip argument proves the exact optimum 1 for n = 6, attained by the 2 × 3 grid. Optimality proofs have not yet been adopted for n = 7…20, which 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 (4) ↓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-rectangle.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-rectangle.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-rectangle-2026-10,
title = {{Spreading points in a rectangle} (P17)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-rectangle.json},
license = {CC BY 4.0}
}Discussion
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