P09 · Packing and covering · Classic · Hard

Equal circles in a half-disc · n = 9

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

Instancen = 9

ObjectiveMaximize common radius

Record comparison

common radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	0.205604646	See source attribution	web.archive.org ↗
Site record	No player record yet	—	—

Formal definition

ContainerAn upper half-disk of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centered at (1, 0), with the circular arc in y ≥ 0
SubmissionExactly n circles: one common radius and n center coordinates
ConstraintsEvery circle lies wholly inside the half-disk; no two interiors overlap, and tangency is allowed
ObjectiveMaximize the common radius

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Layout shown: starter layout

1y0

123456789

0x2

VERIFIED CONSTRUCTIONr = 0.205604646

Getting a feel for it

Mixed straight and curved boundary

A half-disk combines a flat diameter, a curved upper arc, and two 90° corners at the ends of the base. Strong packings balance linear rows along the diameter against arched layers along the upper boundary, locking circles into a rigid contact graph while occasionally leaving a rattler in an interior pocket.

Literature benchmarks and open frontier

Best-known values for circle packing in a half-disk are taken from Eckard Specht's csc survey table on Packomania, which marks none of these cases as mathematically proven optimal; all fourteen instances remain open.

Source

LAYOUT & HISTORY

Starter layout

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ANSWER FORMAT

How to write your answer

The container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above. Coordinates and radii share one unit and are written as plain decimals such as "0.25", to at most nine decimal places.

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.25". All circles share a single radius.

Starter answer

{
  "centers": [
    [
      "0.232673012",
      "0.205604647"
    ],
    [
      "0.643882306",
      "0.205604647"
    ],
    [
      "1.356117694",
      "0.205604647"
    ],
    [
      "1.767326988",
      "0.205604647"
    ],
    [
      "1",
      "0.411209294"
    ],
    [
      "0.438277659",
      "0.561722341"
    ],
    [
      "1.561722341",
      "0.561722341"
    ],
    [
      "0.794395353",
      "0.767326988"
    ],
    [
      "1.205604647",
      "0.767326988"
    ]
  ],
  "radius": "0.205604646"
}

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

Instance parameters

{
  "n": 9
}

Starter answer

{
  "centers": [
    [
      "0.232673012",
      "0.205604647"
    ],
    [
      "0.643882306",
      "0.205604647"
    ],
    [
      "1.356117694",
      "0.205604647"
    ],
    [
      "1.767326988",
      "0.205604647"
    ],
    [
      "1",
      "0.411209294"
    ],
    [
      "0.438277659",
      "0.561722341"
    ],
    [
      "1.561722341",
      "0.561722341"
    ],
    [
      "0.794395353",
      "0.767326988"
    ],
    [
      "1.205604647",
      "0.767326988"
    ]
  ],
  "radius": "0.205604646"
}

Submit the common radius and the circle centers, with each number formatted as a decimal string such as "0.25". All circles share a single radius. · Verifier v1.0.0

DISCUSSION

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