MinMax Arena
中/EN
P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 23

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

Instancen = 23
ObjectiveMaximize common radius
Best known, unproven0.180336009value 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 = 0.180336008

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

0.180336008

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": [
    [
      "0.839651315",
      "0.196173304"
    ],
    [
      "1.200211262",
      "0.205163848"
    ],
    [
      "0.510138293",
      "0.34282075"
    ],
    [
      "1.522006038",
      "0.368051621"
    ],
    [
      "0.824903057",
      "0.556543661"
    ],
    [
      "1.220043701",
      "0.565290186"
    ],
    [
      "0.275472945",
      "0.616712114"
    ],
    [
      "1.742729314",
      "0.653298077"
    ],
    [
      "0.575416933",
      "0.817006669"
    ],
    [
      "1.440766976",
      "0.850536642"
    ],
    [
      "1.015796515",
      "0.86255672"
    ],
    [
      "0.181091484",
      "0.964816186"
    ],
    [
      "1.819644356",
      "1.005673418"
    ],
    [
      "0.576786281",
      "1.177676088"
    ],
    [
      "1.408583443",
      "1.209769888"
    ],
    [
      "1.048111383",
      "1.221778176"
    ],
    [
      "0.245268176",
      "1.319732594"
    ],
    [
      "1.737858771",
      "1.356950265"
    ],
    [
      "0.787095149",
      "1.470685448"
    ],
    [
      "1.160327902",
      "1.564548883"
    ],
    [
      "0.455577044",
      "1.612741954"
    ],
    [
      "1.513207993",
      "1.639113928"
    ],
    [
      "0.899311667",
      "1.813456156"
    ]
  ],
  "radius": "0.180336008"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23
}

The current leader's answer

{
  "centers": [
    [
      "0.839651315",
      "0.196173304"
    ],
    [
      "1.200211262",
      "0.205163848"
    ],
    [
      "0.510138293",
      "0.34282075"
    ],
    [
      "1.522006038",
      "0.368051621"
    ],
    [
      "0.824903057",
      "0.556543661"
    ],
    [
      "1.220043701",
      "0.565290186"
    ],
    [
      "0.275472945",
      "0.616712114"
    ],
    [
      "1.742729314",
      "0.653298077"
    ],
    [
      "0.575416933",
      "0.817006669"
    ],
    [
      "1.440766976",
      "0.850536642"
    ],
    [
      "1.015796515",
      "0.86255672"
    ],
    [
      "0.181091484",
      "0.964816186"
    ],
    [
      "1.819644356",
      "1.005673418"
    ],
    [
      "0.576786281",
      "1.177676088"
    ],
    [
      "1.408583443",
      "1.209769888"
    ],
    [
      "1.048111383",
      "1.221778176"
    ],
    [
      "0.245268176",
      "1.319732594"
    ],
    [
      "1.737858771",
      "1.356950265"
    ],
    [
      "0.787095149",
      "1.470685448"
    ],
    [
      "1.160327902",
      "1.564548883"
    ],
    [
      "0.455577044",
      "1.612741954"
    ],
    [
      "1.513207993",
      "1.639113928"
    ],
    [
      "0.899311667",
      "1.813456156"
    ]
  ],
  "radius": "0.180336008"
}

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