P67 · Packing and covering · Classic · Applied frontier

Variable-radius circles in a fixed-perimeter rectangle

Also known ascircle packing in a rectangle of fixed perimetervariable radius circle packingmaximum sum of radii

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VERIFIED CONSTRUCTIONscored by the sum of the radii

n = 26Current record · open

Pack n circles with disjoint interiors into a rectangle of fixed perimeter 4, choosing both the rectangle's dimensions and each circle's radius to maximize the sum of the radii.

Formal definition

Containerwidth > 0, height > 0, and width + height = 2; the lower-left corner is fixed at (0,0)
SubmissionExactly n circles, each written as [x,y,r] with strictly positive radius
ConstraintsEvery circle lies wholly inside the rectangle; interiors are pairwise disjoint and tangency is allowed
ObjectiveMaximize the exact sum of all circle radii

Getting a feel for it

Coupled aspect ratio and packing

Because the aspect ratio is free, shifting perimeter from height to width alters wall contacts and interstitial gaps throughout the box. The rectangle dimensions and the circle arrangement must be optimized together.

Combining large and small circles

Equal radii leave large gaps between curved boundaries. Good configurations use a few large circles to set the rectangle's aspect ratio and smaller circles to fill the remaining pockets; adding a single circle can shift the optimal aspect ratio to a different contact topology.

Published frontier

Dominik Kamp's 2026 computational project provides complete coordinates up to n = 60. The benchmark starts at the first non-trivial case n = 3 and includes verifiable nine-decimal certificates. In October 2026, a computer-assisted proof by zzzcy, independently verified by the platform, established the exact nine-decimal grid optima for n = 3–9; n ≥ 10 remains open.

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) ↓

CURRENT RECORD8(7 − 2√2) / 41

Record holder邱仲普

Solution methodAI · MoonshotAI: Kimi K3

CURRENT RECORD6 − 2√2 − 2√(4 − 2√2)

Record holder邱仲普

Solution methodAI · MoonshotAI: Kimi K3

CURRENT RECORD(407 + 920√2 − 307√(1 + 4√2) − 34√(2 + 8√2))/712

Record holder邱仲普

Solution methodAI · DeepSeek: DeepSeek V4 Pro 0423

CURRENT RECORD1.212458617

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD6/(3 + 2√(2 − √2))

Record holder邱仲普

Solution methodAI · Z.ai: GLM 5.3

CURRENT RECORD(3 + 13√2 + √(1 + 4√2) − 5√(2 + 8√2))/4

Record holder邱仲普

Solution methodAI · DeepSeek: DeepSeek V3

CURRENT RECORD1.528312766

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD1.610004233best known [1.610, 1.611)

Matches the best known

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD1.68520647best known [1.685, 1.686)

Matches the best known

Record holder邱仲普

Solution methodAI · MoonshotAI: Kimi K3

CURRENT RECORD1.765978316best known [1.765, 1.766)

Matches the best known

123456789101112

Record holder邱仲普

Solution methodAI · MoonshotAI: Kimi K3

CURRENT RECORD1.835020096best known [1.835, 1.836)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD1.916657222best known [1.916, 1.917)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD1.985786491best known [1.985, 1.986)

Matches the best known

123456789101112131415

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.059275747best known [2.059, 2.060)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.117334953best known [2.117, 2.118)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.180914322best known [2.180, 2.181)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.241565272best known [2.241, 2.242)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.30571386best known [2.305, 2.306)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.36583237best known [2.365, 2.366)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.425137269best known [2.425, 2.426)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.484239542best known [2.484, 2.485)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.535696914best known [2.535, 2.536)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.592537687best known [2.592, 2.593)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.639320556best known [2.63932, 2.63933)

Matches the best known

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Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.69152336best known [2.69152, 2.69153)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.740975598best known [2.740, 2.741)

Matches the best known

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Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

CURRENT RECORD2.792639723best known [2.792, 2.793)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD2.844493962best known [2.844, 2.845)

Matches the best known

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Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

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/variable-circles-in-fixed-perimeter-rectangle.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/variable-circles-in-fixed-perimeter-rectangle.json

BibTeX (click to copy)

@misc{minmaxarena-variable-circles-in-fixed-perimeter-rectangle-2026-10,
  title  = {{Variable-radius circles in a fixed-perimeter rectangle} (P67)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/variable-circles-in-fixed-perimeter-rectangle.json},
  license = {CC BY 4.0}
}

DISCUSSION

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