Spreading points in an annulus
Place n points inside an annulus with outer radius 1 and inner radius 0.5 so that the minimum distance between any two points is as large as possible.
Formal definition
- ContainerAn annulus centered at (1, 1) with outer radius 1 and inner radius 0.5 (i.e. 0.5 ≤ ‖p − (1, 1)‖ ≤ 1, with the central disk of radius 0.5 removed)
- SubmissionExactly n distinct points
- ConstraintsEvery point must lie inside the container or on its boundary
- Scoreδ(P) is the minimum Euclidean distance over all pairs of points; internally, the verifier compares δ(P)² using exact arithmetic
- ObjectiveMaximize δ(P)
Getting a feel for it
Geometric intuition
The annulus has a band width of 0.5 (outer radius 1, inner radius 0.5): for small n, points space out roughly evenly around the outer rim, but as n grows a single ring is no longer enough and points must stagger between the inner and outer boundaries while the central hole blocks shortcuts across the middle.
Current status
Spreading points inside an annulus with outer radius 1 and inner radius 0.5 is an original variant introduced on this site, with no prior published literature. Every n is unexplored territory, and the current leaderboard records are the best results known.
Current 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) ↓This problem is frozen
This narrow annulus may degenerate over a wide range of n, making evenly spaced single-ring configurations provably optimal. Frozen pending a degeneracy audit.
Closed to new records since 2026-08-27. Existing records remain in the history and on their holders' profiles, and this URL stays active; since records can no longer change hands, they no longer count toward contribution points as of 2026-10-04.
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-annulus.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-annulus.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-an-annulus-2026-10,
title = {{Spreading points in an annulus} (P27)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-an-annulus.json},
license = {CC BY 4.0}
}Discussion
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