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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 26

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

Instancen = 26
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.193904837NUE_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 = 26, 4.193904837
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
    4.193888814→4.193904837
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": "0.862897716",
      "y": "0.910270323"
    },
    {
      "turn": "0.000000000",
      "x": "1.115941229",
      "y": "1.183833172"
    },
    {
      "turn": "0.000000000",
      "x": "1.287679772",
      "y": "0.800761592"
    },
    {
      "turn": "0.000000000",
      "x": "0.696819221",
      "y": "1.300469931"
    },
    {
      "turn": "0.000000000",
      "x": "0.960275548",
      "y": "0.544888199"
    },
    {
      "turn": "0.000000000",
      "x": "0.454804813",
      "y": "1.038153398"
    },
    {
      "turn": "0.000000000",
      "x": "0.570290400",
      "y": "0.642072457"
    },
    {
      "turn": "0.000000000",
      "x": "1.532838360",
      "y": "1.073194587"
    },
    {
      "turn": "0.000000000",
      "x": "0.940148801",
      "y": "1.551463876"
    },
    {
      "turn": "0.000000000",
      "x": "1.307250196",
      "y": "0.516360728"
    },
    {
      "turn": "0.000000000",
      "x": "1.374273613",
      "y": "1.464032721"
    },
    {
      "turn": "0.000000000",
      "x": "1.724020012",
      "y": "0.706013813"
    },
    {
      "turn": "0.000000000",
      "x": "0.296808214",
      "y": "1.450699791"
    },
    {
      "turn": "0.000000000",
      "x": "0.690999583",
      "y": "0.234514225"
    },
    {
      "turn": "0.000000000",
      "x": "0.512769388",
      "y": "1.683734599"
    },
    {
      "turn": "0.000000000",
      "x": "0.974049144",
      "y": "0.180169245"
    },
    {
      "turn": "0.000000000",
      "x": "0.171090508",
      "y": "0.732768482"
    },
    {
      "turn": "0.000000000",
      "x": "0.066953717",
      "y": "1.133589795"
    },
    {
      "turn": "0.000000000",
      "x": "0.308136387",
      "y": "0.332381222"
    },
    {
      "turn": "0.000000000",
      "x": "1.792003570",
      "y": "1.348263426"
    },
    {
      "turn": "0.000000000",
      "x": "1.955822228",
      "y": "0.963917398"
    },
    {
      "turn": "0.000000000",
      "x": "1.217732475",
      "y": "1.828886312"
    },
    {
      "turn": "0.000000000",
      "x": "1.699968262",
      "y": "0.371239426"
    },
    {
      "turn": "0.000000000",
      "x": "0.824439570",
      "y": "1.923151203"
    },
    {
      "turn": "0.000000000",
      "x": "1.327202186",
      "y": "0.158580722"
    },
    {
      "turn": "0.000000000",
      "x": "1.628123312",
      "y": "1.739612410"
    }
  ],
  "radius": "0.238441271"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.862897716",
      "y": "0.910270323"
    },
    {
      "turn": "0.000000000",
      "x": "1.115941229",
      "y": "1.183833172"
    },
    {
      "turn": "0.000000000",
      "x": "1.287679772",
      "y": "0.800761592"
    },
    {
      "turn": "0.000000000",
      "x": "0.696819221",
      "y": "1.300469931"
    },
    {
      "turn": "0.000000000",
      "x": "0.960275548",
      "y": "0.544888199"
    },
    {
      "turn": "0.000000000",
      "x": "0.454804813",
      "y": "1.038153398"
    },
    {
      "turn": "0.000000000",
      "x": "0.570290400",
      "y": "0.642072457"
    },
    {
      "turn": "0.000000000",
      "x": "1.532838360",
      "y": "1.073194587"
    },
    {
      "turn": "0.000000000",
      "x": "0.940148801",
      "y": "1.551463876"
    },
    {
      "turn": "0.000000000",
      "x": "1.307250196",
      "y": "0.516360728"
    },
    {
      "turn": "0.000000000",
      "x": "1.374273613",
      "y": "1.464032721"
    },
    {
      "turn": "0.000000000",
      "x": "1.724020012",
      "y": "0.706013813"
    },
    {
      "turn": "0.000000000",
      "x": "0.296808214",
      "y": "1.450699791"
    },
    {
      "turn": "0.000000000",
      "x": "0.690999583",
      "y": "0.234514225"
    },
    {
      "turn": "0.000000000",
      "x": "0.512769388",
      "y": "1.683734599"
    },
    {
      "turn": "0.000000000",
      "x": "0.974049144",
      "y": "0.180169245"
    },
    {
      "turn": "0.000000000",
      "x": "0.171090508",
      "y": "0.732768482"
    },
    {
      "turn": "0.000000000",
      "x": "0.066953717",
      "y": "1.133589795"
    },
    {
      "turn": "0.000000000",
      "x": "0.308136387",
      "y": "0.332381222"
    },
    {
      "turn": "0.000000000",
      "x": "1.792003570",
      "y": "1.348263426"
    },
    {
      "turn": "0.000000000",
      "x": "1.955822228",
      "y": "0.963917398"
    },
    {
      "turn": "0.000000000",
      "x": "1.217732475",
      "y": "1.828886312"
    },
    {
      "turn": "0.000000000",
      "x": "1.699968262",
      "y": "0.371239426"
    },
    {
      "turn": "0.000000000",
      "x": "0.824439570",
      "y": "1.923151203"
    },
    {
      "turn": "0.000000000",
      "x": "1.327202186",
      "y": "0.158580722"
    },
    {
      "turn": "0.000000000",
      "x": "1.628123312",
      "y": "1.739612410"
    }
  ],
  "radius": "0.238441271"
}

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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