P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 17

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

Instancen = 17

ObjectiveMaximize covered disc radius

Best known, unproven3.349Source-precision interval: 3.349–3.35 (values inside match)exhibited site certificate 3.349729253Compiled by Erich Friedman; n=17 Found by Jeremy Tan in 2018. Source expression: r = 3.349+.

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

3.349729253

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": "0.737111109",
      "y": "1.187633634"
    },
    {
      "turn": "0.000000000",
      "x": "0.734942002",
      "y": "0.815443203"
    },
    {
      "turn": "0.000000000",
      "x": "1.267976840",
      "y": "0.998438241"
    },
    {
      "turn": "0.000000000",
      "x": "1.114946894",
      "y": "1.403379291"
    },
    {
      "turn": "0.000000000",
      "x": "1.110237504",
      "y": "0.595308332"
    },
    {
      "turn": "0.000000000",
      "x": "0.608821804",
      "y": "1.679283883"
    },
    {
      "turn": "0.000000000",
      "x": "0.600930977",
      "y": "0.325321640"
    },
    {
      "turn": "0.000000000",
      "x": "1.673718719",
      "y": "1.344736246"
    },
    {
      "turn": "0.000000000",
      "x": "1.669654885",
      "y": "0.647434643"
    },
    {
      "turn": "0.000000000",
      "x": "1.004947314",
      "y": "1.888764566"
    },
    {
      "turn": "0.000000000",
      "x": "0.994587976",
      "y": "0.111238138"
    },
    {
      "turn": "0.000000000",
      "x": "0.209492350",
      "y": "1.004607040"
    },
    {
      "turn": "0.000000000",
      "x": "1.846987813",
      "y": "0.995063803"
    },
    {
      "turn": "0.000000000",
      "x": "0.259215756",
      "y": "1.434442312"
    },
    {
      "turn": "0.000000000",
      "x": "1.503065826",
      "y": "0.245139481"
    },
    {
      "turn": "0.000000000",
      "x": "0.254202431",
      "y": "0.574221416"
    },
    {
      "turn": "0.000000000",
      "x": "1.511829926",
      "y": "1.748945747"
    }
  ],
  "radius": "0.298531590"
}

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

Instance parameters

{
  "n": 17,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.737111109",
      "y": "1.187633634"
    },
    {
      "turn": "0.000000000",
      "x": "0.734942002",
      "y": "0.815443203"
    },
    {
      "turn": "0.000000000",
      "x": "1.267976840",
      "y": "0.998438241"
    },
    {
      "turn": "0.000000000",
      "x": "1.114946894",
      "y": "1.403379291"
    },
    {
      "turn": "0.000000000",
      "x": "1.110237504",
      "y": "0.595308332"
    },
    {
      "turn": "0.000000000",
      "x": "0.608821804",
      "y": "1.679283883"
    },
    {
      "turn": "0.000000000",
      "x": "0.600930977",
      "y": "0.325321640"
    },
    {
      "turn": "0.000000000",
      "x": "1.673718719",
      "y": "1.344736246"
    },
    {
      "turn": "0.000000000",
      "x": "1.669654885",
      "y": "0.647434643"
    },
    {
      "turn": "0.000000000",
      "x": "1.004947314",
      "y": "1.888764566"
    },
    {
      "turn": "0.000000000",
      "x": "0.994587976",
      "y": "0.111238138"
    },
    {
      "turn": "0.000000000",
      "x": "0.209492350",
      "y": "1.004607040"
    },
    {
      "turn": "0.000000000",
      "x": "1.846987813",
      "y": "0.995063803"
    },
    {
      "turn": "0.000000000",
      "x": "0.259215756",
      "y": "1.434442312"
    },
    {
      "turn": "0.000000000",
      "x": "1.503065826",
      "y": "0.245139481"
    },
    {
      "turn": "0.000000000",
      "x": "0.254202431",
      "y": "0.574221416"
    },
    {
      "turn": "0.000000000",
      "x": "1.511829926",
      "y": "1.748945747"
    }
  ],
  "radius": "0.298531590"
}

DISCUSSION

Discussion

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