MinMax Arena
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P07 · Extremal configurations · Classic

Spreading points in a circle

2y0
0x2
VERIFIED CONSTRUCTIONthe closest pair
n = 8Current record · open

Place n points inside a circle of radius 1 to maximize the minimum Euclidean distance between any two points.

Formal definition

  • ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
  • SubmissionExactly n distinct points
  • ConstraintsEvery point lies inside the circle 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)δ(P)=min1≤i<j≤n‖pi−pj‖
  • ObjectiveMaximize the minimum pairwise distance δ(P)

Getting a feel for it

Duality with equal-circle packing

Placing points in a unit disk with minimum pairwise distance d is equivalent to packing non-overlapping circles of radius d/2 inside a concentric disk of radius 1 + d/2 (or radius r = d / (2 + d) after rescaling to the unit disk). At optimality, pairs at distance d form a rigid contact graph, with many points lying on the boundary.

Proven optima and open frontier

Through the exact scaling between point spreading in a disk and equal-circle packing in a disk (cci), proofs and best-known constructions from the cci literature transfer directly. Optimality is proven here for n = 4, while the remaining sizes follow the open cci frontier.

Source
ONE LEADERBOARD PER n

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 (0) ↓
n4
CURRENT RECORD1.414213562
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n5
CURRENT RECORD2 sin 36°
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n6
CURRENT RECORD0.999999999
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n7
CURRENT RECORD0.999999999
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n8
CURRENT RECORD2 sin(π/7)
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n9
CURRENT RECORD2/√(2(2 + √2))
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n10
CURRENT RECORD0.710978235
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n11
CURRENT RECORD2 sin(π/9)
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n12
CURRENT RECORD0.660152734
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n13
CURRENT RECORD(√5 − 1)/2
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n14
CURRENT RECORD0.60088416
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n15
CURRENT RECORD0.20791169best known 0.567962867
Hard
Record holderFounding benchmark
Solution methodHuman
n16
CURRENT RECORD0.195090321best known 0.553185219
Hard
Record holderFounding benchmark
Solution methodHuman
n17
CURRENT RECORD0.183749517best known 0.527421466
Hard
Record holderFounding benchmark
Solution methodHuman
n18
CURRENT RECORD0.173648177best known (√6 − √2)/2
Hard
Record holderFounding benchmark
Solution methodHuman
n19
CURRENT RECORD(√6 − √2)/2
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n20
CURRENT RECORD0.421738225best known 0.485163607
Hard
Record holderNUE_13
Solution methodHuman

This problem is frozen

Mathematically equivalent to P02 (equal circles in a disc): the packing radius r and minimum point separation d are related by the strictly monotone bijection d = 2r/(1 − r), so any solution to one converts directly into a solution to the other. The catalogue retains P02, the standard formulation in the literature.

Closed to new records since 2026-08-28. 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-circle.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-circle.json
BibTeX (click to copy)
@misc{minmaxarena-spread-points-in-circle-2026-10,
  title  = {{Spreading points in a circle} (P07)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/spread-points-in-circle.json},
  license = {CC BY 4.0}
}
DISCUSSION

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