P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 12

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

Instancen = 12

ObjectiveMaximize covered disc radius

Best known, unproven2.769Source-precision interval: 2.769–2.77 (values inside match)exhibited site certificate 2.76929324Compiled by Erich Friedman; n=12 Found by Hans Melissen in 1997. Source expression: r = 2.769+.

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

2.76929324

covered disc radius

Matches the best known

Answer sourceHans Melissen

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": "0.862295984",
      "y": "1.241776507"
    },
    {
      "turn": "0.000000000",
      "x": "1.278236113",
      "y": "0.998366661"
    },
    {
      "turn": "0.000000000",
      "x": "0.859466950",
      "y": "0.759855763"
    },
    {
      "turn": "0.000000000",
      "x": "0.230844292",
      "y": "1.004514577"
    },
    {
      "turn": "0.000000000",
      "x": "1.388487487",
      "y": "1.663851189"
    },
    {
      "turn": "0.000000000",
      "x": "1.380668491",
      "y": "0.331634250"
    },
    {
      "turn": "0.000000000",
      "x": "0.856020690",
      "y": "0.134545810"
    },
    {
      "turn": "0.000000000",
      "x": "0.322484935",
      "y": "1.557416403"
    },
    {
      "turn": "0.000000000",
      "x": "0.315988081",
      "y": "0.450574864"
    },
    {
      "turn": "0.000000000",
      "x": "1.821494071",
      "y": "1.308037152"
    },
    {
      "turn": "0.000000000",
      "x": "0.866189760",
      "y": "1.867083925"
    },
    {
      "turn": "0.000000000",
      "x": "1.817821566",
      "y": "0.682341482"
    }
  ],
  "radius": "0.361102965"
}

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

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.862295984",
      "y": "1.241776507"
    },
    {
      "turn": "0.000000000",
      "x": "1.278236113",
      "y": "0.998366661"
    },
    {
      "turn": "0.000000000",
      "x": "0.859466950",
      "y": "0.759855763"
    },
    {
      "turn": "0.000000000",
      "x": "0.230844292",
      "y": "1.004514577"
    },
    {
      "turn": "0.000000000",
      "x": "1.388487487",
      "y": "1.663851189"
    },
    {
      "turn": "0.000000000",
      "x": "1.380668491",
      "y": "0.331634250"
    },
    {
      "turn": "0.000000000",
      "x": "0.856020690",
      "y": "0.134545810"
    },
    {
      "turn": "0.000000000",
      "x": "0.322484935",
      "y": "1.557416403"
    },
    {
      "turn": "0.000000000",
      "x": "0.315988081",
      "y": "0.450574864"
    },
    {
      "turn": "0.000000000",
      "x": "1.821494071",
      "y": "1.308037152"
    },
    {
      "turn": "0.000000000",
      "x": "0.866189760",
      "y": "1.867083925"
    },
    {
      "turn": "0.000000000",
      "x": "1.817821566",
      "y": "0.682341482"
    }
  ],
  "radius": "0.361102965"
}

DISCUSSION

Discussion

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