P06 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a circle · n = 51

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 = 51

ObjectiveMinimize container radius

Record comparison

container radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[227.61188890825, 227.61188890835]	See source attribution	packomania.com ↗
Site record	227.61188891	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

455.22377782y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

0x455.22377782

VERIFIED CONSTRUCTIONr = 227.61188891

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)＋

使用 Packomania ccin51 的单位圆归一化坐标与公开半径比 227.6118889083。固定九位目标 R=227.611888910，取 golden-angle 序列第 393 个全局旋转，按 (R,R)+R*Rot(theta)*(x,y) 映射并舍入到 1e-9 网格。直接最近舍入剩 2 个约束违例；随后允许每个坐标移动 ±2 个整数单位，解整数可行性模型。容器线性约束预留最坏二次余量，圆间约束利用位移差平方项非负，所以模型解对原二次不等式安全。最终用整数平方检查 51 个容器约束和 1275 个圆对约束，全部通过。目标 227.611888909 对当前十二位公开坐标已有轻微连续约束亏损，后续可从更高精度坐标或非线性重解继续。

Read and reply in the discussion →

Record history (2 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
227.611888913→227.61188891
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
227.611888914→227.611888913
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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  "radius": "227.611888910"
}

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

Instance parameters

{
  "n": 51
}

The current leader's answer

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  "radius": "227.611888910"
}

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