P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 15

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

Instancen = 15

ObjectiveMaximize covered disc radius

Best known, unproven3.143Source-precision interval: 3.143–3.144 (values inside match)exhibited site certificate 3.14324129Compiled by Erich Friedman; n=15 Found by Jeremy Tan in 2018. Source expression: r = 3.143+.

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

Current leader

3.14324129

covered disc radius

Matches the best known

Answer sourceJeremy Tan

Solution methodPublished reference construction

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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": "0.984338489",
      "y": "1.362179841"
    },
    {
      "turn": "0.000000000",
      "x": "0.946468039",
      "y": "0.641455833"
    },
    {
      "turn": "0.000000000",
      "x": "1.309147234",
      "y": "0.983755862"
    },
    {
      "turn": "0.000000000",
      "x": "0.679859154",
      "y": "1.016821698"
    },
    {
      "turn": "0.000000000",
      "x": "0.329516437",
      "y": "0.610209655"
    },
    {
      "turn": "0.000000000",
      "x": "0.374058969",
      "y": "1.457910907"
    },
    {
      "turn": "0.000000000",
      "x": "1.471239390",
      "y": "0.428026684"
    },
    {
      "turn": "0.000000000",
      "x": "1.528587537",
      "y": "1.519437471"
    },
    {
      "turn": "0.000000000",
      "x": "0.686244978",
      "y": "1.803478024"
    },
    {
      "turn": "0.000000000",
      "x": "1.231259260",
      "y": "1.874881126"
    },
    {
      "turn": "0.000000000",
      "x": "0.603767835",
      "y": "0.233828280"
    },
    {
      "turn": "0.000000000",
      "x": "1.138297777",
      "y": "0.105700358"
    },
    {
      "turn": "0.000000000",
      "x": "1.799199051",
      "y": "0.699645918"
    },
    {
      "turn": "0.000000000",
      "x": "1.826275195",
      "y": "1.214943584"
    },
    {
      "turn": "0.000000000",
      "x": "0.081916112",
      "y": "1.048240750"
    }
  ],
  "radius": "0.318142932"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.984338489",
      "y": "1.362179841"
    },
    {
      "turn": "0.000000000",
      "x": "0.946468039",
      "y": "0.641455833"
    },
    {
      "turn": "0.000000000",
      "x": "1.309147234",
      "y": "0.983755862"
    },
    {
      "turn": "0.000000000",
      "x": "0.679859154",
      "y": "1.016821698"
    },
    {
      "turn": "0.000000000",
      "x": "0.329516437",
      "y": "0.610209655"
    },
    {
      "turn": "0.000000000",
      "x": "0.374058969",
      "y": "1.457910907"
    },
    {
      "turn": "0.000000000",
      "x": "1.471239390",
      "y": "0.428026684"
    },
    {
      "turn": "0.000000000",
      "x": "1.528587537",
      "y": "1.519437471"
    },
    {
      "turn": "0.000000000",
      "x": "0.686244978",
      "y": "1.803478024"
    },
    {
      "turn": "0.000000000",
      "x": "1.231259260",
      "y": "1.874881126"
    },
    {
      "turn": "0.000000000",
      "x": "0.603767835",
      "y": "0.233828280"
    },
    {
      "turn": "0.000000000",
      "x": "1.138297777",
      "y": "0.105700358"
    },
    {
      "turn": "0.000000000",
      "x": "1.799199051",
      "y": "0.699645918"
    },
    {
      "turn": "0.000000000",
      "x": "1.826275195",
      "y": "1.214943584"
    },
    {
      "turn": "0.000000000",
      "x": "0.081916112",
      "y": "1.048240750"
    }
  ],
  "radius": "0.318142932"
}

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