P67 · Packing and covering · Classic · Applied frontier · Hard

Variable-radius circles in a fixed-perimeter rectangle · n = 19

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

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.

Instancen = 19

ObjectiveMaximize sum of radii

Record comparison

sum of radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[2.241, 2.242)	See source attribution	Erich Friedman ↗
Site record	2.241565272	邱仲普	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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

Challenge this record ↗

Layout shown: site record

1.033357305y0

12345678910111213141516171819

0x0.966642695

VERIFIED CONSTRUCTIONscored by the sum of the 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.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver notePublic method and reproducibility notes (expand)＋

从当前纪录的公开 SVG 重建圆心、半径和矩形尺寸；只保留接触邻域做连续 SLSQP 精修，并加入约 2e-10 的安全间隔。随后在 1e-9 网格上枚举半径的上下取整组合，用整数线性化的接触约束调整圆心和矩形宽度，最后对所有边界及圆对使用整数平方距离逐项复核。n=19 的两个半径上取整组合可将可验证和提高一个网格单位；其余组合要么不超过纪录，要么在精确边界复核时失败。

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.241565271→2.241565272
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.24156527→2.241565271
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.24156519→2.24156527
2026-09-01

ANSWER FORMAT

How to write your answer

The rectangle has lower-left corner (0,0), width width and height height, satisfying 2(width + height) = 4.

Submit width, height, and exactly n triples [x, y, r]. The sum width + height must equal 2 exactly; write every number as a decimal string with at most nine places.

The current leader's answer

{
  "circles": [
    [
      "0.135340212",
      "0.135340213",
      "0.135340212"
    ],
    [
      "0.161372037",
      "0.678545524",
      "0.161372036"
    ],
    [
      "0.355526040",
      "0.475032819",
      "0.119898648"
    ],
    [
      "0.614902780",
      "0.499198897",
      "0.140601432"
    ],
    [
      "0.100329265",
      "0.933028040",
      "0.100329265"
    ],
    [
      "0.673483773",
      "0.282186042",
      "0.084179153"
    ],
    [
      "0.376960242",
      "0.107839786",
      "0.107839786"
    ],
    [
      "0.484708806",
      "0.291169959",
      "0.104809468"
    ],
    [
      "0.592708112",
      "0.107908094",
      "0.107908094"
    ],
    [
      "0.844685341",
      "0.626230748",
      "0.121957354"
    ],
    [
      "0.445052913",
      "0.701045566",
      "0.123199732"
    ],
    [
      "0.124549130",
      "0.395005426",
      "0.124549130"
    ],
    [
      "0.847974356",
      "0.385627533",
      "0.118668339"
    ],
    [
      "0.297535561",
      "0.280972089",
      "0.082641379"
    ],
    [
      "0.562090434",
      "0.913765697",
      "0.119591608"
    ],
    [
      "0.823639515",
      "0.890354125",
      "0.143003180"
    ],
    [
      "0.321079244",
      "0.911930738",
      "0.121426567"
    ],
    [
      "0.657286422",
      "0.726427491",
      "0.090546155"
    ],
    [
      "0.832938961",
      "0.133703734",
      "0.133703734"
    ]
  ],
  "height": "1.033357305",
  "width": "0.966642695"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19
}

The current leader's answer

{
  "circles": [
    [
      "0.135340212",
      "0.135340213",
      "0.135340212"
    ],
    [
      "0.161372037",
      "0.678545524",
      "0.161372036"
    ],
    [
      "0.355526040",
      "0.475032819",
      "0.119898648"
    ],
    [
      "0.614902780",
      "0.499198897",
      "0.140601432"
    ],
    [
      "0.100329265",
      "0.933028040",
      "0.100329265"
    ],
    [
      "0.673483773",
      "0.282186042",
      "0.084179153"
    ],
    [
      "0.376960242",
      "0.107839786",
      "0.107839786"
    ],
    [
      "0.484708806",
      "0.291169959",
      "0.104809468"
    ],
    [
      "0.592708112",
      "0.107908094",
      "0.107908094"
    ],
    [
      "0.844685341",
      "0.626230748",
      "0.121957354"
    ],
    [
      "0.445052913",
      "0.701045566",
      "0.123199732"
    ],
    [
      "0.124549130",
      "0.395005426",
      "0.124549130"
    ],
    [
      "0.847974356",
      "0.385627533",
      "0.118668339"
    ],
    [
      "0.297535561",
      "0.280972089",
      "0.082641379"
    ],
    [
      "0.562090434",
      "0.913765697",
      "0.119591608"
    ],
    [
      "0.823639515",
      "0.890354125",
      "0.143003180"
    ],
    [
      "0.321079244",
      "0.911930738",
      "0.121426567"
    ],
    [
      "0.657286422",
      "0.726427491",
      "0.090546155"
    ],
    [
      "0.832938961",
      "0.133703734",
      "0.133703734"
    ]
  ],
  "height": "1.033357305",
  "width": "0.966642695"
}

Submit width, height, and exactly n triples [x, y, r]. The sum width + height must equal 2 exactly; write every number as a decimal string with at most nine places. · Verifier v1.0.0

DISCUSSION

Discussion

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