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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 19

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

Instancen = 19
ObjectiveMaximize covered disc radius
Best known, unproven3.605Source-precision interval: 3.605–3.606 (values inside match)exhibited site certificate 3.605551263Compiled by Erich Friedman; n=19 Found by Jeremy Tan in 2018. Source expression: r = √13 = 3.605+.

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 = 19, 3.605551263
Current leader

3.605551263

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": "1.000000007",
      "y": "1.000000977"
    },
    {
      "turn": "0.000000000",
      "x": "1.480381993",
      "y": "1.001521979"
    },
    {
      "turn": "0.000000000",
      "x": "0.761127101",
      "y": "0.583216343"
    },
    {
      "turn": "0.000000000",
      "x": "0.519618075",
      "y": "0.998477980"
    },
    {
      "turn": "0.000000000",
      "x": "0.758490560",
      "y": "1.415261870"
    },
    {
      "turn": "0.000000000",
      "x": "1.238873196",
      "y": "1.416784597"
    },
    {
      "turn": "0.000000000",
      "x": "1.241508518",
      "y": "0.584738878"
    },
    {
      "turn": "0.000000000",
      "x": "0.280744910",
      "y": "0.581693745"
    },
    {
      "turn": "0.000000000",
      "x": "1.721891179",
      "y": "0.586260013"
    },
    {
      "turn": "0.000000000",
      "x": "0.278108625",
      "y": "1.413740407"
    },
    {
      "turn": "0.000000000",
      "x": "1.719254923",
      "y": "1.418306045"
    },
    {
      "turn": "0.000000000",
      "x": "1.002636335",
      "y": "0.167954006"
    },
    {
      "turn": "0.000000000",
      "x": "0.997364029",
      "y": "1.832045784"
    },
    {
      "turn": "0.000000000",
      "x": "0.039236715",
      "y": "0.996955994"
    },
    {
      "turn": "0.000000000",
      "x": "1.960764056",
      "y": "1.003044063"
    },
    {
      "turn": "0.000000000",
      "x": "1.483017744",
      "y": "0.169476912"
    },
    {
      "turn": "0.000000000",
      "x": "0.516982394",
      "y": "1.830523965"
    },
    {
      "turn": "0.000000000",
      "x": "0.522254871",
      "y": "0.166432648"
    },
    {
      "turn": "0.000000000",
      "x": "1.477746202",
      "y": "1.833568663"
    }
  ],
  "radius": "0.277350099"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.000000007",
      "y": "1.000000977"
    },
    {
      "turn": "0.000000000",
      "x": "1.480381993",
      "y": "1.001521979"
    },
    {
      "turn": "0.000000000",
      "x": "0.761127101",
      "y": "0.583216343"
    },
    {
      "turn": "0.000000000",
      "x": "0.519618075",
      "y": "0.998477980"
    },
    {
      "turn": "0.000000000",
      "x": "0.758490560",
      "y": "1.415261870"
    },
    {
      "turn": "0.000000000",
      "x": "1.238873196",
      "y": "1.416784597"
    },
    {
      "turn": "0.000000000",
      "x": "1.241508518",
      "y": "0.584738878"
    },
    {
      "turn": "0.000000000",
      "x": "0.280744910",
      "y": "0.581693745"
    },
    {
      "turn": "0.000000000",
      "x": "1.721891179",
      "y": "0.586260013"
    },
    {
      "turn": "0.000000000",
      "x": "0.278108625",
      "y": "1.413740407"
    },
    {
      "turn": "0.000000000",
      "x": "1.719254923",
      "y": "1.418306045"
    },
    {
      "turn": "0.000000000",
      "x": "1.002636335",
      "y": "0.167954006"
    },
    {
      "turn": "0.000000000",
      "x": "0.997364029",
      "y": "1.832045784"
    },
    {
      "turn": "0.000000000",
      "x": "0.039236715",
      "y": "0.996955994"
    },
    {
      "turn": "0.000000000",
      "x": "1.960764056",
      "y": "1.003044063"
    },
    {
      "turn": "0.000000000",
      "x": "1.483017744",
      "y": "0.169476912"
    },
    {
      "turn": "0.000000000",
      "x": "0.516982394",
      "y": "1.830523965"
    },
    {
      "turn": "0.000000000",
      "x": "0.522254871",
      "y": "0.166432648"
    },
    {
      "turn": "0.000000000",
      "x": "1.477746202",
      "y": "1.833568663"
    }
  ],
  "radius": "0.277350099"
}

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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