P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 13

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

Instancen = 13

ObjectiveMaximize common radius

Proven optimum√5 − 2the best a certificate can hold 0.236067977r = √5 − 2; optimality proved by Fodor (2003)

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

12345678910111213

0x2

VERIFIED CONSTRUCTIONr = 0.236067977

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

0.236067977

common radius

OptimalThe optimum is proven; there is nothing left to gain

Record holderReference answer

Solution method—

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": [
    [
      "1.763932023",
      "1.000000000"
    ],
    [
      "1.236067977",
      "1.726542528"
    ],
    [
      "0.763932023",
      "1.726542528"
    ],
    [
      "0.236067977",
      "1.000000000"
    ],
    [
      "0.763932023",
      "0.273457472"
    ],
    [
      "1.236067977",
      "0.273457472"
    ],
    [
      "1.618033989",
      "1.449027977"
    ],
    [
      "1.090169944",
      "1.277514551"
    ],
    [
      "0.381966011",
      "1.449027977"
    ],
    [
      "0.708203932",
      "1.000000000"
    ],
    [
      "0.381966011",
      "0.550972023"
    ],
    [
      "1.090169944",
      "0.722485449"
    ],
    [
      "1.618033989",
      "0.550972023"
    ]
  ],
  "radius": "0.236067977"
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "centers": [
    [
      "1.763932023",
      "1.000000000"
    ],
    [
      "1.236067977",
      "1.726542528"
    ],
    [
      "0.763932023",
      "1.726542528"
    ],
    [
      "0.236067977",
      "1.000000000"
    ],
    [
      "0.763932023",
      "0.273457472"
    ],
    [
      "1.236067977",
      "0.273457472"
    ],
    [
      "1.618033989",
      "1.449027977"
    ],
    [
      "1.090169944",
      "1.277514551"
    ],
    [
      "0.381966011",
      "1.449027977"
    ],
    [
      "0.708203932",
      "1.000000000"
    ],
    [
      "0.381966011",
      "0.550972023"
    ],
    [
      "1.090169944",
      "0.722485449"
    ],
    [
      "1.618033989",
      "0.550972023"
    ]
  ],
  "radius": "0.236067977"
}

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