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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 20

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

Instancen = 20
ObjectiveMaximize covered disc radius
Best known, unproven3.692Source-precision interval: 3.692–3.693 (values inside match)exhibited site certificate 3.69210531Compiled by Erich Friedman; n=20 Found by Jeremy Tan in 2018. Source expression: r = 3.692+.

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 = 20, 3.69210531
Current leader

3.69210531

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": "0.746562324",
      "y": "0.592839512"
    },
    {
      "turn": "0.000000000",
      "x": "0.741716268",
      "y": "1.404103824"
    },
    {
      "turn": "0.000000000",
      "x": "0.517096317",
      "y": "0.997115435"
    },
    {
      "turn": "0.000000000",
      "x": "1.212389086",
      "y": "0.589005845"
    },
    {
      "turn": "0.000000000",
      "x": "1.448670835",
      "y": "1.002680105"
    },
    {
      "turn": "0.000000000",
      "x": "1.207464024",
      "y": "1.413502162"
    },
    {
      "turn": "0.000000000",
      "x": "0.986809809",
      "y": "0.999921252"
    },
    {
      "turn": "0.000000000",
      "x": "1.590879903",
      "y": "1.497783371"
    },
    {
      "turn": "0.000000000",
      "x": "1.596784434",
      "y": "0.509311023"
    },
    {
      "turn": "0.000000000",
      "x": "0.270911507",
      "y": "1.399352982"
    },
    {
      "turn": "0.000000000",
      "x": "0.275734426",
      "y": "0.591965507"
    },
    {
      "turn": "0.000000000",
      "x": "1.858363990",
      "y": "1.236847818"
    },
    {
      "turn": "0.000000000",
      "x": "1.861132352",
      "y": "0.773423406"
    },
    {
      "turn": "0.000000000",
      "x": "1.391633634",
      "y": "1.854262988"
    },
    {
      "turn": "0.000000000",
      "x": "1.401811025",
      "y": "0.150476607"
    },
    {
      "turn": "0.000000000",
      "x": "0.955802655",
      "y": "0.182792357"
    },
    {
      "turn": "0.000000000",
      "x": "0.946043111",
      "y": "1.816621354"
    },
    {
      "turn": "0.000000000",
      "x": "0.045177150",
      "y": "0.994296405"
    },
    {
      "turn": "0.000000000",
      "x": "0.501239693",
      "y": "0.187809217"
    },
    {
      "turn": "0.000000000",
      "x": "0.491572491",
      "y": "1.806174434"
    }
  ],
  "radius": "0.270848179"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.746562324",
      "y": "0.592839512"
    },
    {
      "turn": "0.000000000",
      "x": "0.741716268",
      "y": "1.404103824"
    },
    {
      "turn": "0.000000000",
      "x": "0.517096317",
      "y": "0.997115435"
    },
    {
      "turn": "0.000000000",
      "x": "1.212389086",
      "y": "0.589005845"
    },
    {
      "turn": "0.000000000",
      "x": "1.448670835",
      "y": "1.002680105"
    },
    {
      "turn": "0.000000000",
      "x": "1.207464024",
      "y": "1.413502162"
    },
    {
      "turn": "0.000000000",
      "x": "0.986809809",
      "y": "0.999921252"
    },
    {
      "turn": "0.000000000",
      "x": "1.590879903",
      "y": "1.497783371"
    },
    {
      "turn": "0.000000000",
      "x": "1.596784434",
      "y": "0.509311023"
    },
    {
      "turn": "0.000000000",
      "x": "0.270911507",
      "y": "1.399352982"
    },
    {
      "turn": "0.000000000",
      "x": "0.275734426",
      "y": "0.591965507"
    },
    {
      "turn": "0.000000000",
      "x": "1.858363990",
      "y": "1.236847818"
    },
    {
      "turn": "0.000000000",
      "x": "1.861132352",
      "y": "0.773423406"
    },
    {
      "turn": "0.000000000",
      "x": "1.391633634",
      "y": "1.854262988"
    },
    {
      "turn": "0.000000000",
      "x": "1.401811025",
      "y": "0.150476607"
    },
    {
      "turn": "0.000000000",
      "x": "0.955802655",
      "y": "0.182792357"
    },
    {
      "turn": "0.000000000",
      "x": "0.946043111",
      "y": "1.816621354"
    },
    {
      "turn": "0.000000000",
      "x": "0.045177150",
      "y": "0.994296405"
    },
    {
      "turn": "0.000000000",
      "x": "0.501239693",
      "y": "0.187809217"
    },
    {
      "turn": "0.000000000",
      "x": "0.491572491",
      "y": "1.806174434"
    }
  ],
  "radius": "0.270848179"
}

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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