P06 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a circle · n = 48

Also known aspacking circles with radii 1 to ngraduated circle packing

Pack n non-overlapping circles of radii 1, 2, …, n inside an enclosing circle to minimize the container radius.

Instancen = 48

ObjectiveMinimize container radius

Record comparison

container radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[207.80084340725, 207.80084340735]	See source attribution	packomania.com ↗
Site record	207.800843409	邱仲普	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerA circle of radius radius centered at (radius, radius), where the smallest circle has unit radius 1 and radius is the objective value to minimize
SubmissionContainer radius and centers, listed in order of radii 1, 2, …, n
ConstraintsCircle i has radius i; no two interiors overlap; every circle lies wholly inside the container circle
ObjectiveMinimize the container radius

Challenge this record ↗

Layout shown: site record

415.601686818y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748

0x415.601686818

VERIFIED CONSTRUCTIONr = 207.800843409

Getting a feel for it

Large circles and interstitial gaps

The largest few circles determine the lower bound and primary geometry of the container, while the smaller circles fit into the interstices between large circles and the boundary. Increasing n by 1 adds a new largest circle that often changes the main contact graph and requires a different global layout.

Proven optima and open frontier

Packing circles of radii 1..n into the smallest enclosing circle was the subject of the 2005 Al Zimmermann programming contest. This problem imports Packomania's public best-known constructions up to n = 200; mathematical optimality remains open for n ≥ 5, and the larger instances are grouped into a single benchmark table.

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)＋

从站内公开的 207.800843413 构型出发，把半径和坐标乘以 10^9 转成整数。固定目标半径 207800843409，每个坐标仅允许在原值的 [-8,8] 内移动；线性化圆内包含和两圆分离约束，包含约束预留 2·8² 的最坏二次余量，分离约束则利用非负二次余量。整数 MILP 找到 48 个圆的偏移后，按原始整数平方式重新核验全部包含约束和 1128 个圆对，得到本证书。继续方向：更低目标 207800843408 在 Packomania 原始构型的 96 个旋转取整相位中没有找到修整；这不是不可能性证明，可扩大相位、局部偏移或采用非线性整数搜索。

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
207.800843413→207.800843409
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
207.800843414→207.800843413
2026-09-05

ANSWER FORMAT

How to write your answer

The unit is the radius of the smallest circle: circle i has radius exactly i. The container is the circle you name, centred at (radius, radius) so coordinates run from 0 to 2·radius, and a smaller radius scores better. Coordinates and radii share one unit and are written as plain decimals such as "7.5", to at most nine decimal places.

Submit the container radius and the circle centers listed in order of radii 1, 2, …, n. The container is centered at (radius, radius). Write each number as a decimal string such as "7.5".

The current leader's answer

{
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      "195.976366910"
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      "142.539512152",
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      "5.534525169"
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    [
      "77.637112554",
      "361.948530055"
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    [
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      "343.440886627"
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    [
      "9.832815095",
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    [
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      "398.746909730"
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    [
      "402.975234385",
      "175.688128485"
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    [
      "18.755530249",
      "262.353212162"
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    [
      "394.081217798",
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    [
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    [
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    [
      "255.636463698",
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    [
      "285.925708852",
      "33.727677054"
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    [
      "315.902849532",
      "51.793419128"
    ],
    [
      "73.451220411",
      "130.496484639"
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    [
      "198.070579813",
      "20.252239694"
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    [
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    [
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    [
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    [
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    [
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    [
      "34.765017725",
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    [
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    [
      "376.086065465",
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    [
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    [
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    [
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    [
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    [
      "199.814578807",
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    [
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    [
      "131.045747812",
      "347.961340176"
    ]
  ],
  "radius": "207.800843409"
}

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

Instance parameters

{
  "n": 48
}

The current leader's answer

{
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      "206.991830566",
      "195.976366910"
    ],
    [
      "11.666110416",
      "249.017059532"
    ],
    [
      "406.691445279",
      "159.141464940"
    ],
    [
      "142.539512152",
      "400.773554755"
    ],
    [
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      "5.534525169"
    ],
    [
      "77.637112554",
      "361.948530055"
    ],
    [
      "355.723711092",
      "343.440886627"
    ],
    [
      "9.832815095",
      "181.463737725"
    ],
    [
      "263.130537936",
      "398.746909730"
    ],
    [
      "402.975234385",
      "175.688128485"
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    [
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      "262.353212162"
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    [
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      "268.113299810"
    ],
    [
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    [
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    [
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    [
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    [
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  ],
  "radius": "207.800843409"
}

Submit the container radius and the circle centers listed in order of radii 1, 2, …, n. The container is centered at (radius, radius). Write each number as a decimal string such as "7.5". · Verifier v1.0.0

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

N=5和6的最优性今天#841 replies41d ago