P01 · Packing and covering · Classic · Hard

Equal-circle packing in a unit square · n = 51

Also known ascircle packing in a squarepacking equal circles in a unit squarecsq

Place n non-overlapping circles of one common radius inside the unit square, making that radius as large as possible.

Instancen = 51

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.0710431381155, 0.0710431381165]	See source attribution	packomania.com ↗
Site record	0.071043138	邱仲普	Record details ↓

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

Formal definition

ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
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

1y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

0x1

VERIFIED CONSTRUCTIONr = 0.071043138

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 = 1–30 are proved and make a compact gallery of classic contact structures. For n = 31–300 the public Packomania frontier is imported; optimality remains open except at n = 36. The large instances share one dense table for algorithmic challenges.

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)＋

Start from the public p01-n51 leader on the 1e-9 grid and target radius 0.071043138. Solve bounded integer center displacements with exact wall bounds and conservative linearized pair-separation constraints, then recompute all constraints using raw integer squares. A displacement bound of 16 grid units yields a layout whose exact bottleneck is 71043138; all 51 wall checks and 1275 pair checks pass in two independent verifiers. The same search on P02-n51 and local polishing on P57/P67-n17 gave no gain.

Read and reply in the discussion →

Record history (1 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.071043137→0.071043138
2026-09-09

ANSWER FORMAT

How to write your answer

The container is a square of side 1. Its lower-left corner is the origin (0, 0) and its upper-right corner is (1, 1). Coordinates and lengths 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 "0.25".

The current leader's answer

{
  "centers": [
    [
      "0.313865334",
      "0.071043138"
    ],
    [
      "0.559965936",
      "0.071043138"
    ],
    [
      "0.806065672",
      "0.071043138"
    ],
    [
      "0.071235438",
      "0.071259278"
    ],
    [
      "0.436915635",
      "0.142086317"
    ],
    [
      "0.683015804",
      "0.142087067"
    ],
    [
      "0.928956862",
      "0.142361195"
    ],
    [
      "0.194093461",
      "0.147484019"
    ],
    [
      "0.559965503",
      "0.213130246"
    ],
    [
      "0.805906994",
      "0.213405124"
    ],
    [
      "0.071043138",
      "0.21852716"
    ],
    [
      "0.317144245",
      "0.218527955"
    ],
    [
      "0.682856693",
      "0.284448303"
    ],
    [
      "0.928956862",
      "0.284449053"
    ],
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    ],
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    ],
    [
      "0.56518337",
      "0.364081832"
    ],
    [
      "0.928956862",
      "0.426535411"
    ],
    [
      "0.194093464",
      "0.431656579"
    ],
    [
      "0.440203659",
      "0.431673116"
    ],
    [
      "0.688233238",
      "0.435125761"
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    [
      "0.071043138",
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    ],
    [
      "0.317144247",
      "0.502700512"
    ],
    [
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    ],
    [
      "0.811283539",
      "0.50616894"
    ],
    [
      "0.194093464",
      "0.573742856"
    ],
    [
      "0.440202746",
      "0.573759392"
    ],
    [
      "0.68823234",
      "0.577212037"
    ],
    [
      "0.928956862",
      "0.585802469"
    ],
    [
      "0.071043138",
      "0.644785992"
    ],
    [
      "0.317143334",
      "0.644786788"
    ],
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      "0.715845668"
    ],
    [
      "0.194260755",
      "0.716119681"
    ],
    [
      "0.682855797",
      "0.727888027"
    ],
    [
      "0.928956862",
      "0.727888746"
    ],
    [
      "0.071043138",
      "0.786872269"
    ],
    [
      "0.31730199",
      "0.787178562"
    ],
    [
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    ],
    [
      "0.805906122",
      "0.798931165"
    ],
    [
      "0.19919341",
      "0.866165105"
    ],
    [
      "0.466592318",
      "0.904355374"
    ],
    [
      "0.920789515",
      "0.921696002"
    ],
    [
      "0.071734769",
      "0.928956862"
    ],
    [
      "0.326652052",
      "0.928956862"
    ],
    [
      "0.606532604",
      "0.928956862"
    ],
    [
      "0.748618881",
      "0.928956862"
    ]
  ],
  "radius": "0.071043138"
}

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

Instance parameters

{
  "n": 51
}

The current leader's answer

{
  "centers": [
    [
      "0.313865334",
      "0.071043138"
    ],
    [
      "0.559965936",
      "0.071043138"
    ],
    [
      "0.806065672",
      "0.071043138"
    ],
    [
      "0.071235438",
      "0.071259278"
    ],
    [
      "0.436915635",
      "0.142086317"
    ],
    [
      "0.683015804",
      "0.142087067"
    ],
    [
      "0.928956862",
      "0.142361195"
    ],
    [
      "0.194093461",
      "0.147484019"
    ],
    [
      "0.559965503",
      "0.213130246"
    ],
    [
      "0.805906994",
      "0.213405124"
    ],
    [
      "0.071043138",
      "0.21852716"
    ],
    [
      "0.317144245",
      "0.218527955"
    ],
    [
      "0.682856693",
      "0.284448303"
    ],
    [
      "0.928956862",
      "0.284449053"
    ],
    [
      "0.194093462",
      "0.289570299"
    ],
    [
      "0.440203659",
      "0.289586839"
    ],
    [
      "0.805906561",
      "0.355492232"
    ],
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      "0.071043138",
      "0.360613438"
    ],
    [
      "0.317144247",
      "0.360614235"
    ],
    [
      "0.56518337",
      "0.364081832"
    ],
    [
      "0.928956862",
      "0.426535411"
    ],
    [
      "0.194093464",
      "0.431656579"
    ],
    [
      "0.440203659",
      "0.431673116"
    ],
    [
      "0.688233238",
      "0.435125761"
    ],
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      "0.502699715"
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    [
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      "0.502700512"
    ],
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      "0.506168119"
    ],
    [
      "0.811283539",
      "0.50616894"
    ],
    [
      "0.194093464",
      "0.573742856"
    ],
    [
      "0.440202746",
      "0.573759392"
    ],
    [
      "0.68823234",
      "0.577212037"
    ],
    [
      "0.928956862",
      "0.585802469"
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    [
      "0.071043138",
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    [
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    [
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    [
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      "0.787178562"
    ],
    [
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    [
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      "0.466592318",
      "0.904355374"
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    [
      "0.920789515",
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    [
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      "0.928956862"
    ],
    [
      "0.326652052",
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    ],
    [
      "0.606532604",
      "0.928956862"
    ],
    [
      "0.748618881",
      "0.928956862"
    ]
  ],
  "radius": "0.071043138"
}

Submit radius and centers. Write every number as a decimal string, for example "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

啊？？！NUE_13#310 replies20h ago 空悲切/.lird#282 replies15d ago