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

Variable-radius circles in a fixed-perimeter rectangle · n = 13

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 = 13

ObjectiveMaximize sum of radii

Record comparison

sum of radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[1.835, 1.836)	See source attribution	Erich Friedman ↗
Site record	1.835020096	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

0.974726413y0

12345678910111213

0x1.025273587

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)＋

从公开纪录图重建 n=13 的矩形与圆构型；以接触邻域为初值做带安全间隔的连续局部优化，再在 1e-9 整数网格上枚举高分半径取整组合。对每个组合，用线性化接触约束调整圆心和矩形宽度，并对所有边界与圆对做整数平方距离复核。一个可行取整组合在不破坏约束的情况下把半径和提高了一个网格单位；服务器精确验证通过。

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
1.835020095→1.835020096
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
1.835020094→1.835020095
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V3
1.835020065→1.835020094
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": [
    [
      "0.189900105",
      "0.189900105",
      "0.189900105"
    ],
    [
      "0.660731532",
      "0.370634171",
      "0.103601596"
    ],
    [
      "0.879789388",
      "0.489198034",
      "0.145484199"
    ],
    [
      "0.151515764",
      "0.529151396",
      "0.151515764"
    ],
    [
      "0.565757117",
      "0.852276913",
      "0.122449500"
    ],
    [
      "0.852844718",
      "0.172428869",
      "0.172428869"
    ],
    [
      "0.626596846",
      "0.604472070",
      "0.132714590"
    ],
    [
      "0.422983341",
      "0.423211751",
      "0.139890917"
    ],
    [
      "0.365975634",
      "0.893238451",
      "0.081487962"
    ],
    [
      "0.529368738",
      "0.151709963",
      "0.151709963"
    ],
    [
      "0.854754894",
      "0.804207720",
      "0.170518693"
    ],
    [
      "0.380780067",
      "0.686007036",
      "0.126271586"
    ],
    [
      "0.147046352",
      "0.827680061",
      "0.147046352"
    ]
  ],
  "height": "0.974726413",
  "width": "1.025273587"
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "circles": [
    [
      "0.189900105",
      "0.189900105",
      "0.189900105"
    ],
    [
      "0.660731532",
      "0.370634171",
      "0.103601596"
    ],
    [
      "0.879789388",
      "0.489198034",
      "0.145484199"
    ],
    [
      "0.151515764",
      "0.529151396",
      "0.151515764"
    ],
    [
      "0.565757117",
      "0.852276913",
      "0.122449500"
    ],
    [
      "0.852844718",
      "0.172428869",
      "0.172428869"
    ],
    [
      "0.626596846",
      "0.604472070",
      "0.132714590"
    ],
    [
      "0.422983341",
      "0.423211751",
      "0.139890917"
    ],
    [
      "0.365975634",
      "0.893238451",
      "0.081487962"
    ],
    [
      "0.529368738",
      "0.151709963",
      "0.151709963"
    ],
    [
      "0.854754894",
      "0.804207720",
      "0.170518693"
    ],
    [
      "0.380780067",
      "0.686007036",
      "0.126271586"
    ],
    [
      "0.147046352",
      "0.827680061",
      "0.147046352"
    ]
  ],
  "height": "0.974726413",
  "width": "1.025273587"
}

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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