Spreading points in an L
Place n points inside an L-shape (a 2 × 2 square missing its top-right 1 × 1 corner) so that the minimum distance between any two points is as large as possible.
Formal definition
- ContainerAn L-shaped region formed by removing the top-right 1 × 1 square (1, 2] × (1, 2] from [0, 2] × [0, 2], with the origin (0, 0) at the lower-left corner
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the L-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
Three unit squares and the re-entrant corner
The L-shaped region partitions naturally into three 1 × 1 unit squares: the corner square and two arms. Because points may sit directly on the six outer vertices and the re-entrant corner (1, 1), pigeonhole arguments over the three unit squares give tight upper bounds for small n, while optimal configurations lock points along the boundary.
Proven cases and open frontier
Proofs by HwaterB #40 settle n = 4 and n = 7 with continuous optima √2 and √6 − √2, retaining historical records (for n = 7, the irrational continuous optimum is not attained with exact equality on the nine-decimal grid). All other instances 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-an-l.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-an-l.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-an-l-2026-10,
title = {{Spreading points in an L} (P19)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-an-l.json},
license = {CC BY 4.0}
}Discussion
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