P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 11

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

Instancen = 11

ObjectiveMaximize covered disc radius

Best known, unproven2.631Source-precision interval: 2.631–2.632 (values inside match)exhibited site certificate 2.631690766Compiled by Erich Friedman; n=11 Found by Hans Melissen in 1997. Source expression: r = 2.631+.

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

2.631690766

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": "0.850572669",
      "y": "1.000592828"
    },
    {
      "turn": "0.000000000",
      "x": "1.272975309",
      "y": "0.998916998"
    },
    {
      "turn": "0.000000000",
      "x": "1.092080797",
      "y": "0.305205089"
    },
    {
      "turn": "0.000000000",
      "x": "1.097590821",
      "y": "1.694042401"
    },
    {
      "turn": "0.000000000",
      "x": "0.646887855",
      "y": "1.675017348"
    },
    {
      "turn": "0.000000000",
      "x": "0.234116102",
      "y": "0.704666383"
    },
    {
      "turn": "0.000000000",
      "x": "0.641542964",
      "y": "0.327805715"
    },
    {
      "turn": "0.000000000",
      "x": "1.613431954",
      "y": "0.398953016"
    },
    {
      "turn": "0.000000000",
      "x": "0.236483575",
      "y": "1.301401318"
    },
    {
      "turn": "0.000000000",
      "x": "1.618181691",
      "y": "1.596160725"
    },
    {
      "turn": "0.000000000",
      "x": "1.892514697",
      "y": "0.996459079"
    }
  ],
  "radius": "0.379983854"
}

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

Instance parameters

{
  "n": 11,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.850572669",
      "y": "1.000592828"
    },
    {
      "turn": "0.000000000",
      "x": "1.272975309",
      "y": "0.998916998"
    },
    {
      "turn": "0.000000000",
      "x": "1.092080797",
      "y": "0.305205089"
    },
    {
      "turn": "0.000000000",
      "x": "1.097590821",
      "y": "1.694042401"
    },
    {
      "turn": "0.000000000",
      "x": "0.646887855",
      "y": "1.675017348"
    },
    {
      "turn": "0.000000000",
      "x": "0.234116102",
      "y": "0.704666383"
    },
    {
      "turn": "0.000000000",
      "x": "0.641542964",
      "y": "0.327805715"
    },
    {
      "turn": "0.000000000",
      "x": "1.613431954",
      "y": "0.398953016"
    },
    {
      "turn": "0.000000000",
      "x": "0.236483575",
      "y": "1.301401318"
    },
    {
      "turn": "0.000000000",
      "x": "1.618181691",
      "y": "1.596160725"
    },
    {
      "turn": "0.000000000",
      "x": "1.892514697",
      "y": "0.996459079"
    }
  ],
  "radius": "0.379983854"
}

DISCUSSION

Discussion

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