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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 29

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

Instancen = 29
ObjectiveMaximize common radius
Best known, unproven0.162903649value from Eckard Specht's table of best known packings, which does not mark this one as proved

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 full editor ↗
2y0
0x2
VERIFIED CONSTRUCTIONr = 0.162903648

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

0.162903648

common radius

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
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": [
    [
      "1.054795022",
      "0.164698973"
    ],
    [
      "0.727950535",
      "0.208446976"
    ],
    [
      "1.369537674",
      "0.248886022"
    ],
    [
      "0.390213198",
      "0.426510369"
    ],
    [
      "1.641914347",
      "0.462713976"
    ],
    [
      "1.020558335",
      "0.488702437"
    ],
    [
      "0.697876125",
      "0.533720067"
    ],
    [
      "1.335300987",
      "0.572889486"
    ],
    [
      "0.217458885",
      "0.702746397"
    ],
    [
      "1.798413778",
      "0.748473182"
    ],
    [
      "0.525121812",
      "0.809956095"
    ],
    [
      "0.850726386",
      "0.821447704"
    ],
    [
      "1.174659788",
      "0.856341072"
    ],
    [
      "1.50300251",
      "0.886421894"
    ],
    [
      "0.163248111",
      "1.024012"
    ],
    [
      "1.833965194",
      "1.072335024"
    ],
    [
      "0.470911038",
      "1.131221698"
    ],
    [
      "0.796515612",
      "1.142713307"
    ],
    [
      "1.120449013",
      "1.177606675"
    ],
    [
      "1.455030305",
      "1.209664415"
    ],
    [
      "1.743183076",
      "1.385239167"
    ],
    [
      "0.623761299",
      "1.418949335"
    ],
    [
      "0.291779744",
      "1.422703042"
    ],
    [
      "1.260740658",
      "1.471662197"
    ],
    [
      "0.937021897",
      "1.5084937"
    ],
    [
      "1.539819587",
      "1.639785209"
    ],
    [
      "0.617701964",
      "1.744700283"
    ],
    [
      "1.254681324",
      "1.797413145"
    ],
    [
      "0.930962563",
      "1.834244648"
    ]
  ],
  "radius": "0.162903648"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29
}

The current leader's answer

{
  "centers": [
    [
      "1.054795022",
      "0.164698973"
    ],
    [
      "0.727950535",
      "0.208446976"
    ],
    [
      "1.369537674",
      "0.248886022"
    ],
    [
      "0.390213198",
      "0.426510369"
    ],
    [
      "1.641914347",
      "0.462713976"
    ],
    [
      "1.020558335",
      "0.488702437"
    ],
    [
      "0.697876125",
      "0.533720067"
    ],
    [
      "1.335300987",
      "0.572889486"
    ],
    [
      "0.217458885",
      "0.702746397"
    ],
    [
      "1.798413778",
      "0.748473182"
    ],
    [
      "0.525121812",
      "0.809956095"
    ],
    [
      "0.850726386",
      "0.821447704"
    ],
    [
      "1.174659788",
      "0.856341072"
    ],
    [
      "1.50300251",
      "0.886421894"
    ],
    [
      "0.163248111",
      "1.024012"
    ],
    [
      "1.833965194",
      "1.072335024"
    ],
    [
      "0.470911038",
      "1.131221698"
    ],
    [
      "0.796515612",
      "1.142713307"
    ],
    [
      "1.120449013",
      "1.177606675"
    ],
    [
      "1.455030305",
      "1.209664415"
    ],
    [
      "1.743183076",
      "1.385239167"
    ],
    [
      "0.623761299",
      "1.418949335"
    ],
    [
      "0.291779744",
      "1.422703042"
    ],
    [
      "1.260740658",
      "1.471662197"
    ],
    [
      "0.937021897",
      "1.5084937"
    ],
    [
      "1.539819587",
      "1.639785209"
    ],
    [
      "0.617701964",
      "1.744700283"
    ],
    [
      "1.254681324",
      "1.797413145"
    ],
    [
      "0.930962563",
      "1.834244648"
    ]
  ],
  "radius": "0.162903648"
}

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