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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 34

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

Instancen = 34
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.834328808NUE_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 = 34, 4.834328808
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
    4.83428597→4.834328808
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": [
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      "turn": "0.000000000",
      "x": "1.020854664",
      "y": "0.839894938"
    },
    {
      "turn": "0.000000000",
      "x": "1.149466170",
      "y": "1.179309576"
    },
    {
      "turn": "0.000000000",
      "x": "0.804541342",
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    {
      "turn": "0.000000000",
      "x": "1.346717055",
      "y": "0.896481416"
    },
    {
      "turn": "0.000000000",
      "x": "0.702327590",
      "y": "0.802387145"
    },
    {
      "turn": "0.000000000",
      "x": "0.935875539",
      "y": "1.469460840"
    },
    {
      "turn": "0.000000000",
      "x": "1.224390915",
      "y": "0.539895925"
    },
    {
      "turn": "0.000000000",
      "x": "0.923521209",
      "y": "0.489950930"
    },
    {
      "turn": "0.000000000",
      "x": "1.484940752",
      "y": "1.252535283"
    },
    {
      "turn": "0.000000000",
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      "y": "1.107522415"
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    {
      "turn": "0.000000000",
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      "y": "1.520740441"
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
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    {
      "turn": "0.000000000",
      "x": "0.381423225",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
      "x": "1.629688599",
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    {
      "turn": "0.000000000",
      "x": "0.218044868",
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    {
      "turn": "0.000000000",
      "x": "1.373337785",
      "y": "0.231879242"
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    {
      "turn": "0.000000000",
      "x": "0.868239670",
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    {
      "turn": "0.000000000",
      "x": "1.790802282",
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    {
      "turn": "0.000000000",
      "x": "0.137191537",
      "y": "1.093167307"
    },
    {
      "turn": "0.000000000",
      "x": "1.397834725",
      "y": "1.865073863"
    },
    {
      "turn": "0.000000000",
      "x": "0.361527099",
      "y": "1.692426115"
    },
    {
      "turn": "0.000000000",
      "x": "1.844113664",
      "y": "0.686279901"
    },
    {
      "turn": "0.000000000",
      "x": "0.302333735",
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    {
      "turn": "0.000000000",
      "x": "0.854549296",
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    {
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    {
      "turn": "0.000000000",
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    },
    {
      "turn": "0.000000000",
      "x": "0.075877593",
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    }
  ],
  "radius": "0.206853948"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 34,
  "sides": 5
}

The current leader's answer

{
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    {
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      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
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  ],
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}

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

DISCUSSION

Discussion

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