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

Equal-circle packing in a unit circle · n = 25

Also known ascircles in a circlepacking equal circles in a circlecci

Place n non-overlapping circles of equal radius inside a circle of radius 1 to maximize their common radius.

Instancen = 25

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.1738276614212222333592595944125, 0.1738276614212222333592595944135]	See source attribution	packomania.com ↗
Site record	0.173827661	Ἀθηνᾶ	Record details ↓

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

Formal definition

ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
SubmissionExactly n circles: one common radius and n center coordinates
ConstraintsEvery circle lies wholly inside the container; no two interiors overlap, and tangency is allowed
ObjectiveMaximize the common radius

Open full editor ↗

Layout shown: site record

2y0

12345678910111213141516171819202122232425

0x2

VERIFIED CONSTRUCTIONr = 0.173827661

Getting a feel for it

Concentric rings and contact graphs

Inside a circular container, equal circles typically arrange themselves into an outer ring along the boundary surrounding an inner ring or central core, locked together by tangencies. When the ring counts match hexagonal symmetry (such as n = 7 or n = 19), the packing is especially tight; intermediate sizes often produce asymmetric contact graphs or loose rattlers.

Proven optima and open frontier

Mathematical optimality proofs are known for n = 1..14 and n = 19, while other sizes remain open. This problem imports Packomania's public best-known constructions up to n = 300, grouping the larger instances into a single benchmark table.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderἈθηνᾶ

Solution methodAI · Anthropic: Claude Opus 5

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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)

ἈθηνᾶAI · Anthropic: Claude Opus 5
0.173827660→0.173827661
2026-09-04

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 the common radius and the circle centers, with each number formatted as a decimal string such as "0.2".

The current leader's answer

{
  "centers": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.465791530",
      "0.369776196"
    ],
    [
      "1.572688850",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468234",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.854920950",
      "1.457916100"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}

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

Instance parameters

{
  "n": 25
}

The current leader's answer

{
  "centers": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.465791530",
      "0.369776196"
    ],
    [
      "1.572688850",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468234",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.854920950",
      "1.457916100"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.2". · Verifier v1.0.0

DISCUSSION

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