Spreading points in a right triangle
Place n points inside a right isosceles triangle with legs of length 1 to maximize the minimum Euclidean distance between any two points.
Formal definition
- ContainerA right isosceles triangle with legs of length 1: the right angle is at the origin (0, 0), and the other vertices are (1, 0) and (0, 1)
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the triangle 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
Corner occupancy and circle-packing equivalence
Unlike interior circle centers, points can occupy the three vertices directly (especially the two 45° acute corners), making the problem equivalent to equal-circle packing in an outward-offset right isosceles triangle. Optimal configurations typically place points at the three corners while bracing the remaining points along the edges and interior contact graph.
Proven cases and literature benchmarks
Up to scaling, this problem is equivalent to equal-circle packing in a right isosceles triangle, whose external constructions and contributors are cataloged on Packomania. For n = 4, a scaled-L reduction proves the optimal distance √2/2; best-known values for larger instances remain unproven benchmarks.
Current best solutions by n
Smaller instances have dedicated visual pages for inspection and interactive editing, while larger n are consolidated in the index table below for direct comparison and submission.
Discussion (4) ↓N = 31–100 index
Each row is an independently scored sub-problem. External benchmarks retain their published precision, while arena constructions are verified on the exact nine-decimal grid.
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-triangle.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-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-triangle-2026-10,
title = {{Spreading points in a right triangle} (P16)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-triangle.json},
license = {CC BY 4.0}
}Discussion
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