MinMax Arena
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P06 · Packing and covering · Classic · Hard

Packing circles of radius 1,2,…,n into a circle · n = 84

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 = 84
ObjectiveMinimize container radius

Record comparison

container radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[475.70280956825, 475.70280956835]See source attributionpackomania.com ↗
Site record475.702809576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

951.405619152y0
0x951.405619152
VERIFIED CONSTRUCTIONr = 475.702809576

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
The record-holding arrangement for Packing circles of radius 1,2,…,n into a circle n = 84, 475.702809576
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)＋

从实时页面的 84 圆九位整数证书出发，将容器半径目标降至 475.702809576，在每个圆心 ±32 个整数单位窗口内建立线性化修复模型并由 HiGHS 求解；独立整数检查确认全部 84 个包含约束与 3486 个成对分离约束。该解严格领先 1e-9；更低目标可扩大窗口或进行全局重启。

Read and reply in the discussion →
Record history (2 changes)
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    475.702809577→475.702809576
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    475.702809578→475.702809577
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": "475.702809576"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 84
}

The current leader's answer

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}

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

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N=5和6的最优性今天#841 replies