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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 28

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

Instancen = 28
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.39890485NUE_13Record details ↓

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 ↗

Layout shown: site record

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 = 28, 4.39890485
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13
Solution methodHuman
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.

Record history (1 changes)
  1. NUE_13Human
    4.398829404→4.39890485
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": "1.070138483",
      "y": "1.210769211"
    },
    {
      "turn": "0.000000000",
      "x": "0.808088102",
      "y": "0.913633303"
    },
    {
      "turn": "0.000000000",
      "x": "1.173553855",
      "y": "0.846722997"
    },
    {
      "turn": "0.000000000",
      "x": "0.686160855",
      "y": "1.287544932"
    },
    {
      "turn": "0.000000000",
      "x": "1.461437739",
      "y": "1.125885666"
    },
    {
      "turn": "0.000000000",
      "x": "0.917049614",
      "y": "0.529668390"
    },
    {
      "turn": "0.000000000",
      "x": "0.957891916",
      "y": "1.573629264"
    },
    {
      "turn": "0.000000000",
      "x": "0.427241545",
      "y": "0.985749982"
    },
    {
      "turn": "0.000000000",
      "x": "1.356103747",
      "y": "1.474227314"
    },
    {
      "turn": "0.000000000",
      "x": "0.566348324",
      "y": "0.607583946"
    },
    {
      "turn": "0.000000000",
      "x": "1.545798154",
      "y": "0.786773284"
    },
    {
      "turn": "0.000000000",
      "x": "1.272062908",
      "y": "0.474348320"
    },
    {
      "turn": "0.000000000",
      "x": "0.596648522",
      "y": "1.684704051"
    },
    {
      "turn": "0.000000000",
      "x": "0.277511462",
      "y": "1.324338824"
    },
    {
      "turn": "0.000000000",
      "x": "1.758120111",
      "y": "1.400619712"
    },
    {
      "turn": "0.000000000",
      "x": "0.682590958",
      "y": "0.207152051"
    },
    {
      "turn": "0.000000000",
      "x": "1.202492337",
      "y": "1.845052393"
    },
    {
      "turn": "0.000000000",
      "x": "0.176994995",
      "y": "0.707375109"
    },
    {
      "turn": "0.000000000",
      "x": "1.849435917",
      "y": "1.090204555"
    },
    {
      "turn": "0.000000000",
      "x": "0.993245243",
      "y": "0.151582112"
    },
    {
      "turn": "0.000000000",
      "x": "0.812858564",
      "y": "1.954292913"
    },
    {
      "turn": "0.000000000",
      "x": "0.038834111",
      "y": "1.086687665"
    },
    {
      "turn": "0.000000000",
      "x": "1.596051240",
      "y": "1.761742811"
    },
    {
      "turn": "0.000000000",
      "x": "0.315510889",
      "y": "0.332842176"
    },
    {
      "turn": "0.000000000",
      "x": "1.644286493",
      "y": "0.414330612"
    },
    {
      "turn": "0.000000000",
      "x": "0.347597162",
      "y": "1.580181008"
    },
    {
      "turn": "0.000000000",
      "x": "1.897690521",
      "y": "0.712342145"
    },
    {
      "turn": "0.000000000",
      "x": "1.374231010",
      "y": "0.125412667"
    }
  ],
  "radius": "0.227329309"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 28,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.070138483",
      "y": "1.210769211"
    },
    {
      "turn": "0.000000000",
      "x": "0.808088102",
      "y": "0.913633303"
    },
    {
      "turn": "0.000000000",
      "x": "1.173553855",
      "y": "0.846722997"
    },
    {
      "turn": "0.000000000",
      "x": "0.686160855",
      "y": "1.287544932"
    },
    {
      "turn": "0.000000000",
      "x": "1.461437739",
      "y": "1.125885666"
    },
    {
      "turn": "0.000000000",
      "x": "0.917049614",
      "y": "0.529668390"
    },
    {
      "turn": "0.000000000",
      "x": "0.957891916",
      "y": "1.573629264"
    },
    {
      "turn": "0.000000000",
      "x": "0.427241545",
      "y": "0.985749982"
    },
    {
      "turn": "0.000000000",
      "x": "1.356103747",
      "y": "1.474227314"
    },
    {
      "turn": "0.000000000",
      "x": "0.566348324",
      "y": "0.607583946"
    },
    {
      "turn": "0.000000000",
      "x": "1.545798154",
      "y": "0.786773284"
    },
    {
      "turn": "0.000000000",
      "x": "1.272062908",
      "y": "0.474348320"
    },
    {
      "turn": "0.000000000",
      "x": "0.596648522",
      "y": "1.684704051"
    },
    {
      "turn": "0.000000000",
      "x": "0.277511462",
      "y": "1.324338824"
    },
    {
      "turn": "0.000000000",
      "x": "1.758120111",
      "y": "1.400619712"
    },
    {
      "turn": "0.000000000",
      "x": "0.682590958",
      "y": "0.207152051"
    },
    {
      "turn": "0.000000000",
      "x": "1.202492337",
      "y": "1.845052393"
    },
    {
      "turn": "0.000000000",
      "x": "0.176994995",
      "y": "0.707375109"
    },
    {
      "turn": "0.000000000",
      "x": "1.849435917",
      "y": "1.090204555"
    },
    {
      "turn": "0.000000000",
      "x": "0.993245243",
      "y": "0.151582112"
    },
    {
      "turn": "0.000000000",
      "x": "0.812858564",
      "y": "1.954292913"
    },
    {
      "turn": "0.000000000",
      "x": "0.038834111",
      "y": "1.086687665"
    },
    {
      "turn": "0.000000000",
      "x": "1.596051240",
      "y": "1.761742811"
    },
    {
      "turn": "0.000000000",
      "x": "0.315510889",
      "y": "0.332842176"
    },
    {
      "turn": "0.000000000",
      "x": "1.644286493",
      "y": "0.414330612"
    },
    {
      "turn": "0.000000000",
      "x": "0.347597162",
      "y": "1.580181008"
    },
    {
      "turn": "0.000000000",
      "x": "1.897690521",
      "y": "0.712342145"
    },
    {
      "turn": "0.000000000",
      "x": "1.374231010",
      "y": "0.125412667"
    }
  ],
  "radius": "0.227329309"
}

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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