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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 25

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

Instancen = 25
ObjectiveMaximize covered disc radius
Best known, unproven4.181Source-precision interval: 4.181–4.182 (values inside match)exhibited site certificate 4.181540528Compiled by Erich Friedman; n=25 Found by Jeremy Tan in 2018. Source expression: r = √6 + √3 = 4.181+.

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.
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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 = 25, 4.181540528
Current leader

4.181540528

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": "1.408025335",
      "y": "0.830360738"
    },
    {
      "turn": "0.000000000",
      "x": "1.168564404",
      "y": "0.591529452"
    },
    {
      "turn": "0.000000000",
      "x": "0.831435608",
      "y": "1.408470536"
    },
    {
      "turn": "0.000000000",
      "x": "0.830360729",
      "y": "0.591974659"
    },
    {
      "turn": "0.000000000",
      "x": "1.408470534",
      "y": "1.168564399"
    },
    {
      "turn": "0.000000000",
      "x": "0.591529451",
      "y": "0.831435604"
    },
    {
      "turn": "0.000000000",
      "x": "0.591974671",
      "y": "1.169639271"
    },
    {
      "turn": "0.000000000",
      "x": "1.169639262",
      "y": "1.408025322"
    },
    {
      "turn": "0.000000000",
      "x": "0.999999996",
      "y": "0.999999987"
    },
    {
      "turn": "0.000000000",
      "x": "0.423410286",
      "y": "1.578109822"
    },
    {
      "turn": "0.000000000",
      "x": "0.421890179",
      "y": "0.423410280"
    },
    {
      "turn": "0.000000000",
      "x": "1.578109808",
      "y": "1.576589727"
    },
    {
      "turn": "0.000000000",
      "x": "1.576589734",
      "y": "0.421890194"
    },
    {
      "turn": "0.000000000",
      "x": "0.183504126",
      "y": "1.001074875"
    },
    {
      "turn": "0.000000000",
      "x": "1.001074865",
      "y": "1.816495871"
    },
    {
      "turn": "0.000000000",
      "x": "0.998925144",
      "y": "0.183504124"
    },
    {
      "turn": "0.000000000",
      "x": "1.816495873",
      "y": "0.998925153"
    },
    {
      "turn": "0.000000000",
      "x": "1.816941080",
      "y": "1.337128807"
    },
    {
      "turn": "0.000000000",
      "x": "0.183058901",
      "y": "0.662871211"
    },
    {
      "turn": "0.000000000",
      "x": "1.339278526",
      "y": "1.816050650"
    },
    {
      "turn": "0.000000000",
      "x": "0.660721464",
      "y": "0.183949335"
    },
    {
      "turn": "0.000000000",
      "x": "1.816050664",
      "y": "0.660721483"
    },
    {
      "turn": "0.000000000",
      "x": "1.337128810",
      "y": "0.183058908"
    },
    {
      "turn": "0.000000000",
      "x": "0.662871214",
      "y": "1.816941087"
    },
    {
      "turn": "0.000000000",
      "x": "0.183949349",
      "y": "1.339278545"
    }
  ],
  "radius": "0.239146313"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.408025335",
      "y": "0.830360738"
    },
    {
      "turn": "0.000000000",
      "x": "1.168564404",
      "y": "0.591529452"
    },
    {
      "turn": "0.000000000",
      "x": "0.831435608",
      "y": "1.408470536"
    },
    {
      "turn": "0.000000000",
      "x": "0.830360729",
      "y": "0.591974659"
    },
    {
      "turn": "0.000000000",
      "x": "1.408470534",
      "y": "1.168564399"
    },
    {
      "turn": "0.000000000",
      "x": "0.591529451",
      "y": "0.831435604"
    },
    {
      "turn": "0.000000000",
      "x": "0.591974671",
      "y": "1.169639271"
    },
    {
      "turn": "0.000000000",
      "x": "1.169639262",
      "y": "1.408025322"
    },
    {
      "turn": "0.000000000",
      "x": "0.999999996",
      "y": "0.999999987"
    },
    {
      "turn": "0.000000000",
      "x": "0.423410286",
      "y": "1.578109822"
    },
    {
      "turn": "0.000000000",
      "x": "0.421890179",
      "y": "0.423410280"
    },
    {
      "turn": "0.000000000",
      "x": "1.578109808",
      "y": "1.576589727"
    },
    {
      "turn": "0.000000000",
      "x": "1.576589734",
      "y": "0.421890194"
    },
    {
      "turn": "0.000000000",
      "x": "0.183504126",
      "y": "1.001074875"
    },
    {
      "turn": "0.000000000",
      "x": "1.001074865",
      "y": "1.816495871"
    },
    {
      "turn": "0.000000000",
      "x": "0.998925144",
      "y": "0.183504124"
    },
    {
      "turn": "0.000000000",
      "x": "1.816495873",
      "y": "0.998925153"
    },
    {
      "turn": "0.000000000",
      "x": "1.816941080",
      "y": "1.337128807"
    },
    {
      "turn": "0.000000000",
      "x": "0.183058901",
      "y": "0.662871211"
    },
    {
      "turn": "0.000000000",
      "x": "1.339278526",
      "y": "1.816050650"
    },
    {
      "turn": "0.000000000",
      "x": "0.660721464",
      "y": "0.183949335"
    },
    {
      "turn": "0.000000000",
      "x": "1.816050664",
      "y": "0.660721483"
    },
    {
      "turn": "0.000000000",
      "x": "1.337128810",
      "y": "0.183058908"
    },
    {
      "turn": "0.000000000",
      "x": "0.662871214",
      "y": "1.816941087"
    },
    {
      "turn": "0.000000000",
      "x": "0.183949349",
      "y": "1.339278545"
    }
  ],
  "radius": "0.239146313"
}

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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