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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 30

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

Instancen = 30
ObjectiveMaximize common radius

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 = 0.150000000

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

0.150000000

common radius

Record holderAnonymous
Solution methodHuman
Challenge this record ↗
Record history (2 changes)
  1. AnonymousHuman
    0.126458327→0.150000000
  2. NUE_13Human
    0.041811385→0.126458327
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": [
    [
      "1.167646081",
      "0.166696459"
    ],
    [
      "1.229198878",
      "0.758777541"
    ],
    [
      "1.388413460",
      "1.066524801"
    ],
    [
      "1.203965992",
      "1.303123896"
    ],
    [
      "1.706454082",
      "1.443922787"
    ],
    [
      "1.512088746",
      "1.678428416"
    ],
    [
      "1.244506497",
      "1.814074058"
    ],
    [
      "0.884133168",
      "1.358958697"
    ],
    [
      "1.033752961",
      "1.024459887"
    ],
    [
      "0.999856643",
      "1.635740339"
    ],
    [
      "0.745833784",
      "1.811110063"
    ],
    [
      "0.450919170",
      "1.648853019"
    ],
    [
      "0.259705082",
      "1.417688202"
    ],
    [
      "0.169831583",
      "0.817461240"
    ],
    [
      "0.547061869",
      "1.312053320"
    ],
    [
      "0.515038049",
      "0.301922708"
    ],
    [
      "0.158123228",
      "1.117232677"
    ],
    [
      "0.478111336",
      "0.934070138"
    ],
    [
      "0.861039031",
      "0.467836644"
    ],
    [
      "0.739266533",
      "1.081711471"
    ],
    [
      "0.570717552",
      "0.648721073"
    ],
    [
      "0.787756801",
      "0.176924776"
    ],
    [
      "0.846214661",
      "0.767470151"
    ],
    [
      "1.462699121",
      "0.342738441"
    ],
    [
      "1.718147539",
      "0.545286781"
    ],
    [
      "1.836188800",
      "0.847394985"
    ],
    [
      "1.814565409",
      "1.164080079"
    ],
    [
      "1.161036579",
      "0.466623641"
    ],
    [
      "1.526337947",
      "0.800110019"
    ],
    [
      "0.302729783",
      "0.513878365"
    ]
  ],
  "radius": "0.150000000"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "centers": [
    [
      "1.167646081",
      "0.166696459"
    ],
    [
      "1.229198878",
      "0.758777541"
    ],
    [
      "1.388413460",
      "1.066524801"
    ],
    [
      "1.203965992",
      "1.303123896"
    ],
    [
      "1.706454082",
      "1.443922787"
    ],
    [
      "1.512088746",
      "1.678428416"
    ],
    [
      "1.244506497",
      "1.814074058"
    ],
    [
      "0.884133168",
      "1.358958697"
    ],
    [
      "1.033752961",
      "1.024459887"
    ],
    [
      "0.999856643",
      "1.635740339"
    ],
    [
      "0.745833784",
      "1.811110063"
    ],
    [
      "0.450919170",
      "1.648853019"
    ],
    [
      "0.259705082",
      "1.417688202"
    ],
    [
      "0.169831583",
      "0.817461240"
    ],
    [
      "0.547061869",
      "1.312053320"
    ],
    [
      "0.515038049",
      "0.301922708"
    ],
    [
      "0.158123228",
      "1.117232677"
    ],
    [
      "0.478111336",
      "0.934070138"
    ],
    [
      "0.861039031",
      "0.467836644"
    ],
    [
      "0.739266533",
      "1.081711471"
    ],
    [
      "0.570717552",
      "0.648721073"
    ],
    [
      "0.787756801",
      "0.176924776"
    ],
    [
      "0.846214661",
      "0.767470151"
    ],
    [
      "1.462699121",
      "0.342738441"
    ],
    [
      "1.718147539",
      "0.545286781"
    ],
    [
      "1.836188800",
      "0.847394985"
    ],
    [
      "1.814565409",
      "1.164080079"
    ],
    [
      "1.161036579",
      "0.466623641"
    ],
    [
      "1.526337947",
      "0.800110019"
    ],
    [
      "0.302729783",
      "0.513878365"
    ]
  ],
  "radius": "0.150000000"
}

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