P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 16

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

Instancen = 16

ObjectiveMaximize covered disc radius

Best known, unproven3.244Source-precision interval: 3.244–3.245 (values inside match)exhibited site certificate 3.244437866Compiled by Erich Friedman; n=16 Found by Jeremy Tan in 2018. Source expression: r = 3.244+.

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

3.244437866

covered disc radius

Matches the best known

Answer sourceJeremy Tan

Solution methodPublished reference construction

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Record holder's solver noteNo solver note yet (expand)＋

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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": "1.000793180",
      "y": "0.694366509"
    },
    {
      "turn": "0.000000000",
      "x": "0.999205989",
      "y": "1.305633823"
    },
    {
      "turn": "0.000000000",
      "x": "0.651945240",
      "y": "0.999096513"
    },
    {
      "turn": "0.000000000",
      "x": "1.348053929",
      "y": "1.000903976"
    },
    {
      "turn": "0.000000000",
      "x": "1.518002010",
      "y": "1.531504552"
    },
    {
      "turn": "0.000000000",
      "x": "0.479244964",
      "y": "1.528808293"
    },
    {
      "turn": "0.000000000",
      "x": "1.520754621",
      "y": "0.471192589"
    },
    {
      "turn": "0.000000000",
      "x": "0.481998949",
      "y": "0.468494263"
    },
    {
      "turn": "0.000000000",
      "x": "0.172529239",
      "y": "0.741992772"
    },
    {
      "turn": "0.000000000",
      "x": "1.828799033",
      "y": "0.746293866"
    },
    {
      "turn": "0.000000000",
      "x": "1.827470290",
      "y": "1.258005775"
    },
    {
      "turn": "0.000000000",
      "x": "0.757651506",
      "y": "1.806300777"
    },
    {
      "turn": "0.000000000",
      "x": "0.761842290",
      "y": "0.192451237"
    },
    {
      "turn": "0.000000000",
      "x": "0.171200614",
      "y": "1.253705239"
    },
    {
      "turn": "0.000000000",
      "x": "1.242350086",
      "y": "0.193699886"
    },
    {
      "turn": "0.000000000",
      "x": "1.238159572",
      "y": "1.807548442"
    }
  ],
  "radius": "0.308219803"
}

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

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.000793180",
      "y": "0.694366509"
    },
    {
      "turn": "0.000000000",
      "x": "0.999205989",
      "y": "1.305633823"
    },
    {
      "turn": "0.000000000",
      "x": "0.651945240",
      "y": "0.999096513"
    },
    {
      "turn": "0.000000000",
      "x": "1.348053929",
      "y": "1.000903976"
    },
    {
      "turn": "0.000000000",
      "x": "1.518002010",
      "y": "1.531504552"
    },
    {
      "turn": "0.000000000",
      "x": "0.479244964",
      "y": "1.528808293"
    },
    {
      "turn": "0.000000000",
      "x": "1.520754621",
      "y": "0.471192589"
    },
    {
      "turn": "0.000000000",
      "x": "0.481998949",
      "y": "0.468494263"
    },
    {
      "turn": "0.000000000",
      "x": "0.172529239",
      "y": "0.741992772"
    },
    {
      "turn": "0.000000000",
      "x": "1.828799033",
      "y": "0.746293866"
    },
    {
      "turn": "0.000000000",
      "x": "1.827470290",
      "y": "1.258005775"
    },
    {
      "turn": "0.000000000",
      "x": "0.757651506",
      "y": "1.806300777"
    },
    {
      "turn": "0.000000000",
      "x": "0.761842290",
      "y": "0.192451237"
    },
    {
      "turn": "0.000000000",
      "x": "0.171200614",
      "y": "1.253705239"
    },
    {
      "turn": "0.000000000",
      "x": "1.242350086",
      "y": "0.193699886"
    },
    {
      "turn": "0.000000000",
      "x": "1.238159572",
      "y": "1.807548442"
    }
  ],
  "radius": "0.308219803"
}

DISCUSSION

Discussion

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