P02 · Packing and covering · Classic · Weak baseline · Hard

Equal-circle packing in a unit circle · n = 40

Also known ascircles in a circlepacking equal circles in a circlecci

Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.

Instancen = 40

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.1403736042027147047559231794295, 0.1403736042027147047559231794305]	See source attribution	packomania.com ↗
Site record	0.140373604	邱仲普	Record details ↓

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

Formal definition

ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
SubmissionExactly n circles: one shared radius and n centres
ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
ObjectiveMake the common radius as large as possible

Challenge this record ↗

Layout shown: site record

2y0

12345678910111213141516171819202122232425262728293031323334353637383940

0x2

VERIFIED CONSTRUCTIONr = 0.140373604

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

The optima for n = 1..14 and 19 are proved; optimality remains open for the other sizes. This site imports Packomania's public constructions through n = 300 and presents the algorithm-scale cases in one 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 idea in the discussion. Accepted contributions can earn proof points.

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

取 Packomania cci40 的公开高精度圆心及 r=0.140373604202714704755923179430，目标设为 T=0.140373604。先将所有中心向量乘 s=(1-T)/(1-r)=1.0000000002358171435901113，再整体旋转 theta=2*pi*frac(234*0.6180339887498948482045868343656)=3.8952818896579631，最后平移到圆心 (1,1)。每个坐标只在 floor/ceil 两个 1e-9 格点间选择，用 0-1 线性模型联合满足分离与容器约束；直接四舍五入及未缩放的旋转扫描均只能平纪录。最终严格整数复算：radius_int=140373604，最小外边界平方余量 63134190，最小圆对平方余量 1705941。

Read and reply in the discussion →

Record history (1 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.140373603→0.140373604
2026-09-07

ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2. Coordinates and radii share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string, for example "0.2".

The current leader's answer

{
  "centers": [
    [
      "0.442892221",
      "1.65466668"
    ],
    [
      "0.261664405",
      "1.440247759"
    ],
    [
      "0.683542497",
      "1.799257273"
    ],
    [
      "0.159189246",
      "1.178870949"
    ],
    [
      "0.957946894",
      "1.858597156"
    ],
    [
      "0.688436486",
      "1.518552724"
    ],
    [
      "0.509411731",
      "1.297915845"
    ],
    [
      "0.146396992",
      "0.898415332"
    ],
    [
      "1.236836777",
      "1.826356994"
    ],
    [
      "0.401405513",
      "1.036919285"
    ],
    [
      "0.972319833",
      "1.571664129"
    ],
    [
      "0.839117012",
      "1.281667997"
    ],
    [
      "0.224652095",
      "0.628794981"
    ],
    [
      "0.681889343",
      "1.049077251"
    ],
    [
      "1.259397582",
      "1.546517745"
    ],
    [
      "0.470283126",
      "0.764752297"
    ],
    [
      "1.490465078",
      "1.7059756"
    ],
    [
      "1.119023082",
      "1.303384048"
    ],
    [
      "0.765754654",
      "0.781148894"
    ],
    [
      "1",
      "1.000000001"
    ],
    [
      "0.385607677",
      "0.398768253"
    ],
    [
      "1.460711694",
      "1.350835289"
    ],
    [
      "0.631238708",
      "0.534725569"
    ],
    [
      "1.69177919",
      "1.510293144"
    ],
    [
      "1.307543238",
      "1.095347643"
    ],
    [
      "1.025958908",
      "0.675731907"
    ],
    [
      "0.612095828",
      "0.232870354"
    ],
    [
      "1.268372408",
      "0.81734649"
    ],
    [
      "1.588238968",
      "1.100723734"
    ],
    [
      "0.891442962",
      "0.429308581"
    ],
    [
      "1.819306463",
      "1.26018159"
    ],
    [
      "1.552229517",
      "0.822295441"
    ],
    [
      "1.172109795",
      "0.436026018"
    ],
    [
      "0.879958774",
      "0.148796356"
    ],
    [
      "1.414523295",
      "0.5776406"
    ],
    [
      "1.859444531",
      "0.982318439"
    ],
    [
      "1.160625607",
      "0.155513793"
    ],
    [
      "1.807912171",
      "0.706341246"
    ],
    [
      "1.438490093",
      "0.2715151"
    ],
    [
      "1.670205949",
      "0.461686405"
    ]
  ],
  "radius": "0.140373604"
}

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

Instance parameters

{
  "n": 40
}

The current leader's answer

{
  "centers": [
    [
      "0.442892221",
      "1.65466668"
    ],
    [
      "0.261664405",
      "1.440247759"
    ],
    [
      "0.683542497",
      "1.799257273"
    ],
    [
      "0.159189246",
      "1.178870949"
    ],
    [
      "0.957946894",
      "1.858597156"
    ],
    [
      "0.688436486",
      "1.518552724"
    ],
    [
      "0.509411731",
      "1.297915845"
    ],
    [
      "0.146396992",
      "0.898415332"
    ],
    [
      "1.236836777",
      "1.826356994"
    ],
    [
      "0.401405513",
      "1.036919285"
    ],
    [
      "0.972319833",
      "1.571664129"
    ],
    [
      "0.839117012",
      "1.281667997"
    ],
    [
      "0.224652095",
      "0.628794981"
    ],
    [
      "0.681889343",
      "1.049077251"
    ],
    [
      "1.259397582",
      "1.546517745"
    ],
    [
      "0.470283126",
      "0.764752297"
    ],
    [
      "1.490465078",
      "1.7059756"
    ],
    [
      "1.119023082",
      "1.303384048"
    ],
    [
      "0.765754654",
      "0.781148894"
    ],
    [
      "1",
      "1.000000001"
    ],
    [
      "0.385607677",
      "0.398768253"
    ],
    [
      "1.460711694",
      "1.350835289"
    ],
    [
      "0.631238708",
      "0.534725569"
    ],
    [
      "1.69177919",
      "1.510293144"
    ],
    [
      "1.307543238",
      "1.095347643"
    ],
    [
      "1.025958908",
      "0.675731907"
    ],
    [
      "0.612095828",
      "0.232870354"
    ],
    [
      "1.268372408",
      "0.81734649"
    ],
    [
      "1.588238968",
      "1.100723734"
    ],
    [
      "0.891442962",
      "0.429308581"
    ],
    [
      "1.819306463",
      "1.26018159"
    ],
    [
      "1.552229517",
      "0.822295441"
    ],
    [
      "1.172109795",
      "0.436026018"
    ],
    [
      "0.879958774",
      "0.148796356"
    ],
    [
      "1.414523295",
      "0.5776406"
    ],
    [
      "1.859444531",
      "0.982318439"
    ],
    [
      "1.160625607",
      "0.155513793"
    ],
    [
      "1.807912171",
      "0.706341246"
    ],
    [
      "1.438490093",
      "0.2715151"
    ],
    [
      "1.670205949",
      "0.461686405"
    ]
  ],
  "radius": "0.140373604"
}

Submit radius and centers. Write every number as a decimal string, for example "0.2". · Verifier v1.0.0

DISCUSSION

Discussion

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