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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 21

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

Instancen = 21
ObjectiveMaximize covered disc radius
Best known, unproven3.804Source-precision interval: 3.804–3.805 (values inside match)exhibited site certificate 3.804522521Compiled by Erich Friedman; n=21 Found by Jeremy Tan in 2018. Source expression: r = 3.804+.

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
The record-holding arrangement for Equal discs covering a disc n = 21, 3.804522521
Current leader

3.804522521

covered disc radius

Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

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": "1.015833608",
      "y": "1.000039053"
    },
    {
      "turn": "0.000000000",
      "x": "0.944682883",
      "y": "1.455480708"
    },
    {
      "turn": "0.000000000",
      "x": "0.946930804",
      "y": "0.544251953"
    },
    {
      "turn": "0.000000000",
      "x": "0.596155108",
      "y": "1.210318416"
    },
    {
      "turn": "0.000000000",
      "x": "0.597197650",
      "y": "0.787691530"
    },
    {
      "turn": "0.000000000",
      "x": "1.351138240",
      "y": "0.657605254"
    },
    {
      "turn": "0.000000000",
      "x": "1.349444605",
      "y": "1.344123047"
    },
    {
      "turn": "0.000000000",
      "x": "1.488949076",
      "y": "0.994201623"
    },
    {
      "turn": "0.000000000",
      "x": "1.840006037",
      "y": "1.211010535"
    },
    {
      "turn": "0.000000000",
      "x": "1.841036940",
      "y": "0.793136456"
    },
    {
      "turn": "0.000000000",
      "x": "0.177519149",
      "y": "0.997971059"
    },
    {
      "turn": "0.000000000",
      "x": "0.525381961",
      "y": "0.330143084"
    },
    {
      "turn": "0.000000000",
      "x": "0.522082871",
      "y": "1.667507039"
    },
    {
      "turn": "0.000000000",
      "x": "0.869164897",
      "y": "0.102171375"
    },
    {
      "turn": "0.000000000",
      "x": "1.283679243",
      "y": "1.797553401"
    },
    {
      "turn": "0.000000000",
      "x": "1.287610749",
      "y": "0.203855977"
    },
    {
      "turn": "0.000000000",
      "x": "0.864736744",
      "y": "1.897172169"
    },
    {
      "turn": "0.000000000",
      "x": "1.654727209",
      "y": "1.601668486"
    },
    {
      "turn": "0.000000000",
      "x": "1.657687761",
      "y": "0.401569103"
    },
    {
      "turn": "0.000000000",
      "x": "0.180204082",
      "y": "0.574145285"
    },
    {
      "turn": "0.000000000",
      "x": "0.178113046",
      "y": "1.421804909"
    }
  ],
  "radius": "0.262845073"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.015833608",
      "y": "1.000039053"
    },
    {
      "turn": "0.000000000",
      "x": "0.944682883",
      "y": "1.455480708"
    },
    {
      "turn": "0.000000000",
      "x": "0.946930804",
      "y": "0.544251953"
    },
    {
      "turn": "0.000000000",
      "x": "0.596155108",
      "y": "1.210318416"
    },
    {
      "turn": "0.000000000",
      "x": "0.597197650",
      "y": "0.787691530"
    },
    {
      "turn": "0.000000000",
      "x": "1.351138240",
      "y": "0.657605254"
    },
    {
      "turn": "0.000000000",
      "x": "1.349444605",
      "y": "1.344123047"
    },
    {
      "turn": "0.000000000",
      "x": "1.488949076",
      "y": "0.994201623"
    },
    {
      "turn": "0.000000000",
      "x": "1.840006037",
      "y": "1.211010535"
    },
    {
      "turn": "0.000000000",
      "x": "1.841036940",
      "y": "0.793136456"
    },
    {
      "turn": "0.000000000",
      "x": "0.177519149",
      "y": "0.997971059"
    },
    {
      "turn": "0.000000000",
      "x": "0.525381961",
      "y": "0.330143084"
    },
    {
      "turn": "0.000000000",
      "x": "0.522082871",
      "y": "1.667507039"
    },
    {
      "turn": "0.000000000",
      "x": "0.869164897",
      "y": "0.102171375"
    },
    {
      "turn": "0.000000000",
      "x": "1.283679243",
      "y": "1.797553401"
    },
    {
      "turn": "0.000000000",
      "x": "1.287610749",
      "y": "0.203855977"
    },
    {
      "turn": "0.000000000",
      "x": "0.864736744",
      "y": "1.897172169"
    },
    {
      "turn": "0.000000000",
      "x": "1.654727209",
      "y": "1.601668486"
    },
    {
      "turn": "0.000000000",
      "x": "1.657687761",
      "y": "0.401569103"
    },
    {
      "turn": "0.000000000",
      "x": "0.180204082",
      "y": "0.574145285"
    },
    {
      "turn": "0.000000000",
      "x": "0.178113046",
      "y": "1.421804909"
    }
  ],
  "radius": "0.262845073"
}

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. · Verifier v1.0.0

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