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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 9

Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.

Instancen = 9
ObjectiveMaximize common radius
Proven optimum1/(1 + √(2(2 + √2)))the best a certificate can hold 0.276768653r = 1/(1 + √(2(2+√2))); optimality proved by Pirl (1969)

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

Formal definition

  • ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
  • SubmissionExactly n circles: one shared radius and n centres
  • ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
  • ObjectiveMake the common radius as large as possible
Open the editor and look ↗
2y0
0x2
VERIFIED CONSTRUCTIONr = 0.276768653

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

Proven for n = 2..14 and 19 (the bold marks in Specht's cci survey); 15..18 and everything from 20 up are open, including all n offered here.

Source
Current leader

0.276768653

common radius

OptimalThe optimum is proven; there is nothing left to gain
Record holderReference answer
Solution method—
Challenge this record ↗
Record history (6 changes)
  1. AnonymousHuman
    0.276700000→0.276768653
  2. Looker142857Human
    0.274276281→0.276700000
  3. xxxHuman
    0.271091346→0.274276281
  4. xxxHuman
    0.270141611→0.271091346
  5. xxxHuman
    0.268764262→0.270141611
  6. 送到Human
    0.136808057→0.268764262
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 radius and centers. Write every number as a decimal string, for example "0.2".

The current leader's answer

{
  "centers": [
    [
      "1.723231346",
      "1.000000000"
    ],
    [
      "1.511401789",
      "1.511401789"
    ],
    [
      "1.000000000",
      "1.723231346"
    ],
    [
      "0.488598211",
      "1.511401789"
    ],
    [
      "0.276768654",
      "1.000000000"
    ],
    [
      "0.488598211",
      "0.488598211"
    ],
    [
      "1.000000000",
      "0.276768654"
    ],
    [
      "1.511401789",
      "0.488598211"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ],
  "radius": "0.276768653"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "centers": [
    [
      "1.723231346",
      "1.000000000"
    ],
    [
      "1.511401789",
      "1.511401789"
    ],
    [
      "1.000000000",
      "1.723231346"
    ],
    [
      "0.488598211",
      "1.511401789"
    ],
    [
      "0.276768654",
      "1.000000000"
    ],
    [
      "0.488598211",
      "0.488598211"
    ],
    [
      "1.000000000",
      "0.276768654"
    ],
    [
      "1.511401789",
      "0.488598211"
    ],
    [
      "1.000000000",
      "1.000000000"
    ]
  ],
  "radius": "0.276768653"
}

Submit radius and centers. Write every number as a decimal string, for example "0.2". · Verifier v1.0.0