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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 24

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

Instancen = 24
ObjectiveMaximize covered disc radius
Best known, unproven4.076Source-precision interval: 4.076–4.077 (values inside match)exhibited site certificate 4.07661341Compiled by Erich Friedman; n=24 Found by Jeremy Tan in 2018. Source expression: r = 4.076+.

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 = 24, 4.07661341
Current leader

4.07661341

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": "1.004640496",
      "y": "1.000006265"
    },
    {
      "turn": "0.000000000",
      "x": "1.368866587",
      "y": "0.717927172"
    },
    {
      "turn": "0.000000000",
      "x": "1.452588531",
      "y": "1.020136835"
    },
    {
      "turn": "0.000000000",
      "x": "1.368101994",
      "y": "1.283069840"
    },
    {
      "turn": "0.000000000",
      "x": "0.699147545",
      "y": "0.662881511"
    },
    {
      "turn": "0.000000000",
      "x": "0.698236494",
      "y": "1.336303198"
    },
    {
      "turn": "0.000000000",
      "x": "1.049200415",
      "y": "1.438068226"
    },
    {
      "turn": "0.000000000",
      "x": "0.562801310",
      "y": "0.999408508"
    },
    {
      "turn": "0.000000000",
      "x": "1.050385557",
      "y": "0.562066495"
    },
    {
      "turn": "0.000000000",
      "x": "0.251860387",
      "y": "1.325704887"
    },
    {
      "turn": "0.000000000",
      "x": "0.252744381",
      "y": "0.672272028"
    },
    {
      "turn": "0.000000000",
      "x": "1.831180227",
      "y": "1.179322265"
    },
    {
      "turn": "0.000000000",
      "x": "1.831662393",
      "y": "0.822927410"
    },
    {
      "turn": "0.000000000",
      "x": "1.404570789",
      "y": "0.274010055"
    },
    {
      "turn": "0.000000000",
      "x": "1.402604934",
      "y": "1.727081946"
    },
    {
      "turn": "0.000000000",
      "x": "0.738264145",
      "y": "1.772632137"
    },
    {
      "turn": "0.000000000",
      "x": "0.390053518",
      "y": "1.663604263"
    },
    {
      "turn": "0.000000000",
      "x": "0.391851305",
      "y": "0.334747799"
    },
    {
      "turn": "0.000000000",
      "x": "0.740355660",
      "y": "0.226662495"
    },
    {
      "turn": "0.000000000",
      "x": "0.111861225",
      "y": "0.998798471"
    },
    {
      "turn": "0.000000000",
      "x": "1.695084496",
      "y": "0.478222915"
    },
    {
      "turn": "0.000000000",
      "x": "1.071735063",
      "y": "1.877277575"
    },
    {
      "turn": "0.000000000",
      "x": "1.074108500",
      "y": "0.122919743"
    },
    {
      "turn": "0.000000000",
      "x": "1.693670110",
      "y": "1.523655925"
    }
  ],
  "radius": "0.245301651"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.004640496",
      "y": "1.000006265"
    },
    {
      "turn": "0.000000000",
      "x": "1.368866587",
      "y": "0.717927172"
    },
    {
      "turn": "0.000000000",
      "x": "1.452588531",
      "y": "1.020136835"
    },
    {
      "turn": "0.000000000",
      "x": "1.368101994",
      "y": "1.283069840"
    },
    {
      "turn": "0.000000000",
      "x": "0.699147545",
      "y": "0.662881511"
    },
    {
      "turn": "0.000000000",
      "x": "0.698236494",
      "y": "1.336303198"
    },
    {
      "turn": "0.000000000",
      "x": "1.049200415",
      "y": "1.438068226"
    },
    {
      "turn": "0.000000000",
      "x": "0.562801310",
      "y": "0.999408508"
    },
    {
      "turn": "0.000000000",
      "x": "1.050385557",
      "y": "0.562066495"
    },
    {
      "turn": "0.000000000",
      "x": "0.251860387",
      "y": "1.325704887"
    },
    {
      "turn": "0.000000000",
      "x": "0.252744381",
      "y": "0.672272028"
    },
    {
      "turn": "0.000000000",
      "x": "1.831180227",
      "y": "1.179322265"
    },
    {
      "turn": "0.000000000",
      "x": "1.831662393",
      "y": "0.822927410"
    },
    {
      "turn": "0.000000000",
      "x": "1.404570789",
      "y": "0.274010055"
    },
    {
      "turn": "0.000000000",
      "x": "1.402604934",
      "y": "1.727081946"
    },
    {
      "turn": "0.000000000",
      "x": "0.738264145",
      "y": "1.772632137"
    },
    {
      "turn": "0.000000000",
      "x": "0.390053518",
      "y": "1.663604263"
    },
    {
      "turn": "0.000000000",
      "x": "0.391851305",
      "y": "0.334747799"
    },
    {
      "turn": "0.000000000",
      "x": "0.740355660",
      "y": "0.226662495"
    },
    {
      "turn": "0.000000000",
      "x": "0.111861225",
      "y": "0.998798471"
    },
    {
      "turn": "0.000000000",
      "x": "1.695084496",
      "y": "0.478222915"
    },
    {
      "turn": "0.000000000",
      "x": "1.071735063",
      "y": "1.877277575"
    },
    {
      "turn": "0.000000000",
      "x": "1.074108500",
      "y": "0.122919743"
    },
    {
      "turn": "0.000000000",
      "x": "1.693670110",
      "y": "1.523655925"
    }
  ],
  "radius": "0.245301651"
}

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

DISCUSSION

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