MinMax Arena
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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle

Also known ascircles in a circlepacking equal circles in a circlecci

2y0
0x2
VERIFIED CONSTRUCTIONr = 0.333333333
n = 6Current record · open

Place n non-overlapping circles of equal radius inside a circle of radius 1 to maximize their common radius.

Formal definition

  • ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
  • SubmissionExactly n circles: one common radius and n center coordinates
  • ConstraintsEvery circle lies wholly inside the container; no two interiors overlap, and tangency is allowed
  • ObjectiveMaximize the common radius

Getting a feel for it

Concentric rings and contact graphs

Inside a circular container, equal circles typically arrange themselves into an outer ring along the boundary surrounding an inner ring or central core, locked together by tangencies. When the ring counts match hexagonal symmetry (such as n = 7 or n = 19), the packing is especially tight; intermediate sizes often produce asymmetric contact graphs or loose rattlers.

Proven optima and open frontier

Mathematical optimality proofs are known for n = 1..14 and n = 19, while other sizes remain open. This problem imports Packomania's public best-known constructions up to n = 300, grouping the larger instances into a single benchmark table.

Source
ONE LEADERBOARD PER n

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 (0) ↓
n1
CURRENT RECORD1
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n2
CURRENT RECORD1/2
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n3
CURRENT RECORD0.464101615proven optimum 2√3 − 3
Answer sourceReference answer
Solution method—
n4
CURRENT RECORD√2 − 1
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n5
CURRENT RECORD0.370191908proven optimum sin 36° / (1 + sin 36°)
Answer sourceReference answer
Solution method—
n6
CURRENT RECORD1/3
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n7
CURRENT RECORD1/3
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n8
CURRENT RECORD1/(1 + 1/sin(π/7))
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n9
CURRENT RECORD1/(1 + √(2(2 + √2)))
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n10
CURRENT RECORD0.262258924proven optimum 0.262258923
Answer sourceReference answer
Solution method—
n11
CURRENT RECORD1/(1 + 1/sin(π/9))
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n12
CURRENT RECORD0.248163470
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n13
CURRENT RECORD√5 − 2
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n14
CURRENT RECORD0.231030727
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n15
CURRENT RECORD0.221172539best known [0.2211725390863909372643164849255, 0.2211725390863909372643164849265]
Hard
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n16
CURRENT RECORD0.216664742best known [0.2166647429244224210106479369325, 0.2166647429244224210106479369335]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n17
CURRENT RECORD0.208679665best known [0.2086796655704997432000801252635, 0.2086796655704997432000801252645]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n18
CURRENT RECORD0.205604646best known [0.2056046467595682246931939690925, 0.2056046467595682246931939690935]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n19
CURRENT RECORD1/(1 + √2 + √6)
Optimal
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n20
CURRENT RECORD0.195224010best known [0.1952240110187488782913056948325, 0.1952240110187488782913056948335]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n21
CURRENT RECORD0.190392146best known [0.1903921468490535119688089730565, 0.1903921468490535119688089730575]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n22
CURRENT RECORD0.183833026best known [0.1838330265816816772644114805125, 0.1838330265816816772644114805135]
Hard
Record holderἈθηνᾶ
Solution methodAI · Anthropic: Claude Opus 5
n23
CURRENT RECORD0.180336009best known [0.1803360092544365003492521932515, 0.1803360092544365003492521932525]
Hard
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n24
CURRENT RECORD0.176939130best known [0.1769391305959616826957326824525, 0.1769391305959616826957326824535]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n25
CURRENT RECORD0.173827661best known [0.1738276614212222333592595944125, 0.1738276614212222333592595944135]
Hard
Record holderἈθηνᾶ
Solution methodAI · Anthropic: Claude Opus 5
n26
CURRENT RECORD0.171580252best known [0.1715802521871668482839362261715, 0.1715802521871668482839362261725]
Hard
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n27
CURRENT RECORD0.169307930best known [0.1693079311345730465689703176575, 0.1693079311345730465689703176585]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n28
CURRENT RECORD0.166252749best known [0.1662527500386069358149423890575, 0.1662527500386069358149423890585]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n29
CURRENT RECORD0.162903649best known [0.1629036492766443539450697723185, 0.1629036492766443539450697723195]
Hard
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n30
CURRENT RECORD0.161349108best known [0.1613491090646898651673588465355, 0.1613491090646898651673588465365]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
COMPUTATIONAL FRONTIER

N = 31–300 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.

NCURRENT RECORDKNOWN BESTRECORD SOURCESTATUS

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/circle-circle-packing.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/circle-circle-packing.json
BibTeX (click to copy)
@misc{minmaxarena-circle-circle-packing-2026-10,
  title  = {{Equal-circle packing in a unit circle} (P02)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/circle-circle-packing.json},
  license = {CC BY 4.0}
}
DISCUSSION

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