P30 · Packing and covering · Classic · Hard

Equal circles in a quadrant · n = 15

Place n equal circles inside a quarter-disc of radius 1 so as to maximize their common radius.

Instancen = 15

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.1102120235, 0.1102120245]	See source attribution	web.archive.org ↗
Site record	0.110212024	Sigmoid with Codex	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
SubmissionExactly n circles, specified by one common radius and n centers
ConstraintsEvery circle lies completely inside the container; no two overlap in their interiors, and tangency is allowed
ObjectiveMaximize the common radius

Open full editor ↗

Layout shown: site record

1y0

123456789101112131415

0x1

VERIFIED CONSTRUCTIONr = 0.110212024

Getting a feel for it

Why it is hard

Optimal packings form rigid contact networks where circles brace against one another, the straight edges, and the curved arc, often with tilted rows and occasional rattlers that touch nothing. Regular grid-like layouts are rarely optimal.

Known results

The best known values come from Specht's Packomania (ccq) table, and none have been proven optimal. All 14 instances here remain open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.110212023→0.110212024
2026-09-02

ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant. Coordinates and radii share the same unit and are written as decimal strings such as "0.25", with at most nine decimal places.

Submit radius and centers. Write each number as a decimal string such as "0.25". All circles share the same radius.

The current leader's answer

{
  "centers": [
    [
      "0.303505230",
      "0.616001212"
    ],
    [
      "0.322354962",
      "0.350215437"
    ],
    [
      "0.499801482",
      "0.515728870"
    ],
    [
      "0.608327555",
      "0.323872537"
    ],
    [
      "0.828751603",
      "0.323872537"
    ],
    [
      "0.723708054",
      "0.517657505"
    ],
    [
      "0.110212026",
      "0.289631433"
    ],
    [
      "0.662511935",
      "0.110212026"
    ],
    [
      "0.110212026",
      "0.730479529"
    ],
    [
      "0.450384314",
      "0.170117232"
    ],
    [
      "0.882935983",
      "0.110212026"
    ],
    [
      "0.238256694",
      "0.110212025"
    ],
    [
      "0.499801483",
      "0.736152918"
    ],
    [
      "0.110212025",
      "0.510055481"
    ],
    [
      "0.303505231",
      "0.836425260"
    ]
  ],
  "radius": "0.110212024"
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "centers": [
    [
      "0.303505230",
      "0.616001212"
    ],
    [
      "0.322354962",
      "0.350215437"
    ],
    [
      "0.499801482",
      "0.515728870"
    ],
    [
      "0.608327555",
      "0.323872537"
    ],
    [
      "0.828751603",
      "0.323872537"
    ],
    [
      "0.723708054",
      "0.517657505"
    ],
    [
      "0.110212026",
      "0.289631433"
    ],
    [
      "0.662511935",
      "0.110212026"
    ],
    [
      "0.110212026",
      "0.730479529"
    ],
    [
      "0.450384314",
      "0.170117232"
    ],
    [
      "0.882935983",
      "0.110212026"
    ],
    [
      "0.238256694",
      "0.110212025"
    ],
    [
      "0.499801483",
      "0.736152918"
    ],
    [
      "0.110212025",
      "0.510055481"
    ],
    [
      "0.303505231",
      "0.836425260"
    ]
  ],
  "radius": "0.110212024"
}

Submit radius and centers. Write each number as a decimal string such as "0.25". All circles share the same radius. · Verifier v1.0.0

DISCUSSION

Discussion

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