P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 19

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 = 19

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	1/(1 + √2 + √6)	See source attribution	packomania.com ↗
Site record	No player record yet	—	—

FinishedA construction reaching the theoretical ceiling is already published, leaving nothing further to improve

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

Inspect in editor ↗

Layout shown: starter layout

2y0

12345678910111213141516171819

0x2

VERIFIED CONSTRUCTIONr = 0.205604646

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

Starter layout

Inspect the layout, solver notes and record history.

The optimum is proven; no further improvement is possibleSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record history (2 changes)

Packomania · Eckard SpechtHuman
0.149239636→0.205604646
2026-08-24
Jev LiHuman
0.065837836→0.149239636
2026-08-24

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".

Starter answer

{
  "centers": [
    [
      "0.794395353",
      "0.232673012"
    ],
    [
      "1.205604647",
      "0.232673012"
    ],
    [
      "0.438277659",
      "0.438277659"
    ],
    [
      "1.561722341",
      "0.438277659"
    ],
    [
      "0.794395353",
      "0.643882306"
    ],
    [
      "1.205604647",
      "0.643882306"
    ],
    [
      "0.232673012",
      "0.794395353"
    ],
    [
      "1.767326988",
      "0.794395353"
    ],
    [
      "0.588790706",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.411209294",
      "1"
    ],
    [
      "0.232673012",
      "1.205604647"
    ],
    [
      "1.767326988",
      "1.205604647"
    ],
    [
      "0.794395353",
      "1.356117694"
    ],
    [
      "1.205604647",
      "1.356117694"
    ],
    [
      "0.438277659",
      "1.561722341"
    ],
    [
      "1.561722341",
      "1.561722341"
    ],
    [
      "0.794395353",
      "1.767326988"
    ],
    [
      "1.205604647",
      "1.767326988"
    ]
  ],
  "radius": "0.205604646"
}

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

Instance parameters

{
  "n": 19
}

Starter answer

{
  "centers": [
    [
      "0.794395353",
      "0.232673012"
    ],
    [
      "1.205604647",
      "0.232673012"
    ],
    [
      "0.438277659",
      "0.438277659"
    ],
    [
      "1.561722341",
      "0.438277659"
    ],
    [
      "0.794395353",
      "0.643882306"
    ],
    [
      "1.205604647",
      "0.643882306"
    ],
    [
      "0.232673012",
      "0.794395353"
    ],
    [
      "1.767326988",
      "0.794395353"
    ],
    [
      "0.588790706",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.411209294",
      "1"
    ],
    [
      "0.232673012",
      "1.205604647"
    ],
    [
      "1.767326988",
      "1.205604647"
    ],
    [
      "0.794395353",
      "1.356117694"
    ],
    [
      "1.205604647",
      "1.356117694"
    ],
    [
      "0.438277659",
      "1.561722341"
    ],
    [
      "1.561722341",
      "1.561722341"
    ],
    [
      "0.794395353",
      "1.767326988"
    ],
    [
      "1.205604647",
      "1.767326988"
    ]
  ],
  "radius": "0.205604646"
}

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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