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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 31

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

Instancen = 31
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.644035384NUE_13Record details ↓

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.
Open the full editor ↗

Layout shown: site record

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
The record-holding arrangement for Equal discs covering a disc n = 31, 4.644035384
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13
Solution methodHuman
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)
  1. NUE_13Human
    4.644028137→4.644035384
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

{
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      "x": "1.156330556",
      "y": "1.093562192"
    },
    {
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      "y": "0.940592813"
    },
    {
      "turn": "0.000000000",
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      "y": "0.737362828"
    },
    {
      "turn": "0.000000000",
      "x": "0.878133389",
      "y": "1.290253319"
    },
    {
      "turn": "0.000000000",
      "x": "1.443310549",
      "y": "0.882919618"
    },
    {
      "turn": "0.000000000",
      "x": "1.250881582",
      "y": "1.446018208"
    },
    {
      "turn": "0.000000000",
      "x": "1.519666249",
      "y": "1.242074695"
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    {
      "turn": "0.000000000",
      "x": "0.710436891",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
      "x": "0.421652911",
      "y": "0.791575627"
    },
    {
      "turn": "0.000000000",
      "x": "1.349318015",
      "y": "0.525441133"
    },
    {
      "turn": "0.000000000",
      "x": "0.982853446",
      "y": "0.376537165"
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    {
      "turn": "0.000000000",
      "x": "0.972445830",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
      "x": "1.805194364",
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    },
    {
      "turn": "0.000000000",
      "x": "1.602824581",
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    },
    {
      "turn": "0.000000000",
      "x": "1.707276899",
      "y": "0.607430231"
    },
    {
      "turn": "0.000000000",
      "x": "1.348091321",
      "y": "1.793672489"
    },
    {
      "turn": "0.000000000",
      "x": "0.371511283",
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    {
      "turn": "0.000000000",
      "x": "0.136476702",
      "y": "1.033714750"
    },
    {
      "turn": "0.000000000",
      "x": "1.868414013",
      "y": "1.371189314"
    },
    {
      "turn": "0.000000000",
      "x": "0.613877190",
      "y": "0.228272107"
    },
    {
      "turn": "0.000000000",
      "x": "0.208847325",
      "y": "1.327260444"
    },
    {
      "turn": "0.000000000",
      "x": "1.286417410",
      "y": "0.175754618"
    },
    {
      "turn": "0.000000000",
      "x": "0.670663649",
      "y": "1.810141268"
    },
    {
      "turn": "0.000000000",
      "x": "0.087127624",
      "y": "0.658694051"
    },
    {
      "turn": "0.000000000",
      "x": "1.648530421",
      "y": "0.285630261"
    },
    {
      "turn": "0.000000000",
      "x": "1.015520986",
      "y": "1.949297044"
    },
    {
      "turn": "0.000000000",
      "x": "0.934192994",
      "y": "0.037697133"
    },
    {
      "turn": "0.000000000",
      "x": "0.321656604",
      "y": "1.672997515"
    },
    {
      "turn": "0.000000000",
      "x": "1.851503285",
      "y": "0.817069643"
    }
  ],
  "radius": "0.215329970"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 31,
  "sides": 5
}

The current leader's answer

{
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      "x": "1.156330556",
      "y": "1.093562192"
    },
    {
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      "x": "1.250881582",
      "y": "1.446018208"
    },
    {
      "turn": "0.000000000",
      "x": "1.519666249",
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    {
      "turn": "0.000000000",
      "x": "0.710436891",
      "y": "0.586668154"
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
      "turn": "0.000000000",
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    }
  ],
  "radius": "0.215329970"
}

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. · Verifier v1.0.0

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