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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 29

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

Instancen = 29
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.481866395NUE_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 = 29, 4.481866395
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
    4.481831263→4.481866395
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": "0.832705384",
      "y": "1.104478131"
    },
    {
      "turn": "0.000000000",
      "x": "1.203110441",
      "y": "1.078252551"
    },
    {
      "turn": "0.000000000",
      "x": "1.009136137",
      "y": "0.767685248"
    },
    {
      "turn": "0.000000000",
      "x": "1.053874462",
      "y": "1.436138329"
    },
    {
      "turn": "0.000000000",
      "x": "0.611173801",
      "y": "0.797211842"
    },
    {
      "turn": "0.000000000",
      "x": "1.391951963",
      "y": "0.720669279"
    },
    {
      "turn": "0.000000000",
      "x": "0.683733009",
      "y": "1.450986793"
    },
    {
      "turn": "0.000000000",
      "x": "0.444577205",
      "y": "1.140750936"
    },
    {
      "turn": "0.000000000",
      "x": "1.421011650",
      "y": "1.376289034"
    },
    {
      "turn": "0.000000000",
      "x": "0.800980660",
      "y": "0.462869444"
    },
    {
      "turn": "0.000000000",
      "x": "1.586613504",
      "y": "1.013056080"
    },
    {
      "turn": "0.000000000",
      "x": "1.201702977",
      "y": "0.433116574"
    },
    {
      "turn": "0.000000000",
      "x": "0.815477964",
      "y": "1.783220514"
    },
    {
      "turn": "0.000000000",
      "x": "0.205684188",
      "y": "0.872394473"
    },
    {
      "turn": "0.000000000",
      "x": "1.310480525",
      "y": "1.744529319"
    },
    {
      "turn": "0.000000000",
      "x": "0.430254097",
      "y": "0.438817709"
    },
    {
      "turn": "0.000000000",
      "x": "1.785786312",
      "y": "0.661594315"
    },
    {
      "turn": "0.000000000",
      "x": "1.612667069",
      "y": "0.395500425"
    },
    {
      "turn": "0.000000000",
      "x": "0.292431774",
      "y": "1.475887850"
    },
    {
      "turn": "0.000000000",
      "x": "1.788245120",
      "y": "1.346606273"
    },
    {
      "turn": "0.000000000",
      "x": "0.965803893",
      "y": "0.129998306"
    },
    {
      "turn": "0.000000000",
      "x": "0.487144502",
      "y": "1.792718791"
    },
    {
      "turn": "0.000000000",
      "x": "0.044188368",
      "y": "1.179239534"
    },
    {
      "turn": "0.000000000",
      "x": "1.650428371",
      "y": "1.707170933"
    },
    {
      "turn": "0.000000000",
      "x": "0.574580377",
      "y": "0.138804965"
    },
    {
      "turn": "0.000000000",
      "x": "1.958216094",
      "y": "1.001563474"
    },
    {
      "turn": "0.000000000",
      "x": "1.354855910",
      "y": "0.111282901"
    },
    {
      "turn": "0.000000000",
      "x": "1.069074857",
      "y": "1.862176496"
    },
    {
      "turn": "0.000000000",
      "x": "0.225701471",
      "y": "0.603304543"
    }
  ],
  "radius": "0.223121332"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.832705384",
      "y": "1.104478131"
    },
    {
      "turn": "0.000000000",
      "x": "1.203110441",
      "y": "1.078252551"
    },
    {
      "turn": "0.000000000",
      "x": "1.009136137",
      "y": "0.767685248"
    },
    {
      "turn": "0.000000000",
      "x": "1.053874462",
      "y": "1.436138329"
    },
    {
      "turn": "0.000000000",
      "x": "0.611173801",
      "y": "0.797211842"
    },
    {
      "turn": "0.000000000",
      "x": "1.391951963",
      "y": "0.720669279"
    },
    {
      "turn": "0.000000000",
      "x": "0.683733009",
      "y": "1.450986793"
    },
    {
      "turn": "0.000000000",
      "x": "0.444577205",
      "y": "1.140750936"
    },
    {
      "turn": "0.000000000",
      "x": "1.421011650",
      "y": "1.376289034"
    },
    {
      "turn": "0.000000000",
      "x": "0.800980660",
      "y": "0.462869444"
    },
    {
      "turn": "0.000000000",
      "x": "1.586613504",
      "y": "1.013056080"
    },
    {
      "turn": "0.000000000",
      "x": "1.201702977",
      "y": "0.433116574"
    },
    {
      "turn": "0.000000000",
      "x": "0.815477964",
      "y": "1.783220514"
    },
    {
      "turn": "0.000000000",
      "x": "0.205684188",
      "y": "0.872394473"
    },
    {
      "turn": "0.000000000",
      "x": "1.310480525",
      "y": "1.744529319"
    },
    {
      "turn": "0.000000000",
      "x": "0.430254097",
      "y": "0.438817709"
    },
    {
      "turn": "0.000000000",
      "x": "1.785786312",
      "y": "0.661594315"
    },
    {
      "turn": "0.000000000",
      "x": "1.612667069",
      "y": "0.395500425"
    },
    {
      "turn": "0.000000000",
      "x": "0.292431774",
      "y": "1.475887850"
    },
    {
      "turn": "0.000000000",
      "x": "1.788245120",
      "y": "1.346606273"
    },
    {
      "turn": "0.000000000",
      "x": "0.965803893",
      "y": "0.129998306"
    },
    {
      "turn": "0.000000000",
      "x": "0.487144502",
      "y": "1.792718791"
    },
    {
      "turn": "0.000000000",
      "x": "0.044188368",
      "y": "1.179239534"
    },
    {
      "turn": "0.000000000",
      "x": "1.650428371",
      "y": "1.707170933"
    },
    {
      "turn": "0.000000000",
      "x": "0.574580377",
      "y": "0.138804965"
    },
    {
      "turn": "0.000000000",
      "x": "1.958216094",
      "y": "1.001563474"
    },
    {
      "turn": "0.000000000",
      "x": "1.354855910",
      "y": "0.111282901"
    },
    {
      "turn": "0.000000000",
      "x": "1.069074857",
      "y": "1.862176496"
    },
    {
      "turn": "0.000000000",
      "x": "0.225701471",
      "y": "0.603304543"
    }
  ],
  "radius": "0.223121332"
}

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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