P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 18

Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.

Instancen = 18

ObjectiveMaximize covered disc radius

Best known, unproven3.446Source-precision interval: 3.446–3.447 (values inside match)exhibited site certificate 3.446282746Compiled by Erich Friedman; n=18 Found by Jeremy Tan in 2018. Source expression: r = 3.446+.

Formal definition

ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
ConstraintsThe n distinct centres lie in the target disc. Covering discs may overlap and protrude, but must cover every target point.
PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
ObjectiveCovered disc radius R = 1/r, larger is better.

Open the full editor ↗

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source

Current leader

3.446282746

covered disc radius

Matches the best known

Answer sourceJeremy Tan

Solution methodPublished reference construction

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Record holder's solver noteNo solver note yet (expand)＋

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ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.161848766",
      "y": "1.372346197"
    },
    {
      "turn": "0.000000000",
      "x": "1.174729055",
      "y": "0.633521508"
    },
    {
      "turn": "0.000000000",
      "x": "1.345116238",
      "y": "1.006016528"
    },
    {
      "turn": "0.000000000",
      "x": "0.694603983",
      "y": "1.337087782"
    },
    {
      "turn": "0.000000000",
      "x": "0.706539167",
      "y": "0.652472022"
    },
    {
      "turn": "0.000000000",
      "x": "0.797535415",
      "y": "0.996470359"
    },
    {
      "turn": "0.000000000",
      "x": "0.259546127",
      "y": "0.987091301"
    },
    {
      "turn": "0.000000000",
      "x": "1.694887681",
      "y": "1.414759953"
    },
    {
      "turn": "0.000000000",
      "x": "1.708922519",
      "y": "0.609713350"
    },
    {
      "turn": "0.000000000",
      "x": "1.000091094",
      "y": "0.189470838"
    },
    {
      "turn": "0.000000000",
      "x": "0.176887719",
      "y": "1.369956558"
    },
    {
      "turn": "0.000000000",
      "x": "0.190283181",
      "y": "0.601577471"
    },
    {
      "turn": "0.000000000",
      "x": "0.971838899",
      "y": "1.810039824"
    },
    {
      "turn": "0.000000000",
      "x": "0.509437275",
      "y": "1.766026483"
    },
    {
      "turn": "0.000000000",
      "x": "0.536436332",
      "y": "0.217339755"
    },
    {
      "turn": "0.000000000",
      "x": "1.864767794",
      "y": "1.015075957"
    },
    {
      "turn": "0.000000000",
      "x": "1.434189068",
      "y": "1.795750217"
    },
    {
      "turn": "0.000000000",
      "x": "1.461662162",
      "y": "0.219867658"
    }
  ],
  "radius": "0.290167718"
}

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

Instance parameters

{
  "n": 18,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.161848766",
      "y": "1.372346197"
    },
    {
      "turn": "0.000000000",
      "x": "1.174729055",
      "y": "0.633521508"
    },
    {
      "turn": "0.000000000",
      "x": "1.345116238",
      "y": "1.006016528"
    },
    {
      "turn": "0.000000000",
      "x": "0.694603983",
      "y": "1.337087782"
    },
    {
      "turn": "0.000000000",
      "x": "0.706539167",
      "y": "0.652472022"
    },
    {
      "turn": "0.000000000",
      "x": "0.797535415",
      "y": "0.996470359"
    },
    {
      "turn": "0.000000000",
      "x": "0.259546127",
      "y": "0.987091301"
    },
    {
      "turn": "0.000000000",
      "x": "1.694887681",
      "y": "1.414759953"
    },
    {
      "turn": "0.000000000",
      "x": "1.708922519",
      "y": "0.609713350"
    },
    {
      "turn": "0.000000000",
      "x": "1.000091094",
      "y": "0.189470838"
    },
    {
      "turn": "0.000000000",
      "x": "0.176887719",
      "y": "1.369956558"
    },
    {
      "turn": "0.000000000",
      "x": "0.190283181",
      "y": "0.601577471"
    },
    {
      "turn": "0.000000000",
      "x": "0.971838899",
      "y": "1.810039824"
    },
    {
      "turn": "0.000000000",
      "x": "0.509437275",
      "y": "1.766026483"
    },
    {
      "turn": "0.000000000",
      "x": "0.536436332",
      "y": "0.217339755"
    },
    {
      "turn": "0.000000000",
      "x": "1.864767794",
      "y": "1.015075957"
    },
    {
      "turn": "0.000000000",
      "x": "1.434189068",
      "y": "1.795750217"
    },
    {
      "turn": "0.000000000",
      "x": "1.461662162",
      "y": "0.219867658"
    }
  ],
  "radius": "0.290167718"
}

DISCUSSION

Discussion

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