P30 · Packing and covering · Classic · Hard

Equal circles in a quadrant · n = 5

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

Instancen = 5

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	0.180960531	See source attribution	web.archive.org ↗
Site record	No player record yet	—	—

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: starter layout

1y0

12345

0x1

VERIFIED CONSTRUCTIONr = 0.180960531

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

Starter layout

Inspect the layout, solver notes and record history.

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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.

Starter answer

{
  "centers": [
    [
      "0.43687737",
      "0.180960532"
    ],
    [
      "0.798798433",
      "0.180960532"
    ],
    [
      "0.180960532",
      "0.43687737"
    ],
    [
      "0.521069153",
      "0.628243039"
    ],
    [
      "0.180960532",
      "0.798798433"
    ]
  ],
  "radius": "0.180960531"
}

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

Instance parameters

{
  "n": 5
}

Starter answer

{
  "centers": [
    [
      "0.43687737",
      "0.180960532"
    ],
    [
      "0.798798433",
      "0.180960532"
    ],
    [
      "0.180960532",
      "0.43687737"
    ],
    [
      "0.521069153",
      "0.628243039"
    ],
    [
      "0.180960532",
      "0.798798433"
    ]
  ],
  "radius": "0.180960531"
}

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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