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P87 · Packing and covering · Classic · Medium

Equal discs covering a disc · n = 27

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

Instancen = 27
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record4.324476658NUE_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 = 27, 4.324476658
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
    4.324460163→4.324476658
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": "0.923565387",
      "y": "1.212400076"
    },
    {
      "turn": "0.000000000",
      "x": "0.853547846",
      "y": "0.826590446"
    },
    {
      "turn": "0.000000000",
      "x": "1.223025014",
      "y": "0.959759062"
    },
    {
      "turn": "0.000000000",
      "x": "0.540253412",
      "y": "1.081622935"
    },
    {
      "turn": "0.000000000",
      "x": "1.302290293",
      "y": "1.356107713"
    },
    {
      "turn": "0.000000000",
      "x": "1.157974481",
      "y": "0.560771169"
    },
    {
      "turn": "0.000000000",
      "x": "0.614394653",
      "y": "1.444705084"
    },
    {
      "turn": "0.000000000",
      "x": "1.013683011",
      "y": "1.588214186"
    },
    {
      "turn": "0.000000000",
      "x": "0.481842566",
      "y": "0.715623966"
    },
    {
      "turn": "0.000000000",
      "x": "1.579846061",
      "y": "1.111137673"
    },
    {
      "turn": "0.000000000",
      "x": "0.806355632",
      "y": "0.443308364"
    },
    {
      "turn": "0.000000000",
      "x": "1.503656841",
      "y": "0.694277818"
    },
    {
      "turn": "0.000000000",
      "x": "0.210839248",
      "y": "1.302182688"
    },
    {
      "turn": "0.000000000",
      "x": "1.415574742",
      "y": "1.735959480"
    },
    {
      "turn": "0.000000000",
      "x": "0.154909408",
      "y": "0.993008316"
    },
    {
      "turn": "0.000000000",
      "x": "1.656122398",
      "y": "1.533132309"
    },
    {
      "turn": "0.000000000",
      "x": "0.704599492",
      "y": "1.820618616"
    },
    {
      "turn": "0.000000000",
      "x": "1.133800735",
      "y": "0.166385643"
    },
    {
      "turn": "0.000000000",
      "x": "1.428318444",
      "y": "0.272197989"
    },
    {
      "turn": "0.000000000",
      "x": "0.321432629",
      "y": "1.678854203"
    },
    {
      "turn": "0.000000000",
      "x": "1.090115790",
      "y": "1.955207676"
    },
    {
      "turn": "0.000000000",
      "x": "0.435487778",
      "y": "0.331557434"
    },
    {
      "turn": "0.000000000",
      "x": "1.860353575",
      "y": "0.844631408"
    },
    {
      "turn": "0.000000000",
      "x": "0.126351866",
      "y": "0.601190127"
    },
    {
      "turn": "0.000000000",
      "x": "1.926995009",
      "y": "1.248757316"
    },
    {
      "turn": "0.000000000",
      "x": "0.753236306",
      "y": "0.072705172"
    },
    {
      "turn": "0.000000000",
      "x": "1.781211274",
      "y": "0.442924239"
    }
  ],
  "radius": "0.231241854"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.923565387",
      "y": "1.212400076"
    },
    {
      "turn": "0.000000000",
      "x": "0.853547846",
      "y": "0.826590446"
    },
    {
      "turn": "0.000000000",
      "x": "1.223025014",
      "y": "0.959759062"
    },
    {
      "turn": "0.000000000",
      "x": "0.540253412",
      "y": "1.081622935"
    },
    {
      "turn": "0.000000000",
      "x": "1.302290293",
      "y": "1.356107713"
    },
    {
      "turn": "0.000000000",
      "x": "1.157974481",
      "y": "0.560771169"
    },
    {
      "turn": "0.000000000",
      "x": "0.614394653",
      "y": "1.444705084"
    },
    {
      "turn": "0.000000000",
      "x": "1.013683011",
      "y": "1.588214186"
    },
    {
      "turn": "0.000000000",
      "x": "0.481842566",
      "y": "0.715623966"
    },
    {
      "turn": "0.000000000",
      "x": "1.579846061",
      "y": "1.111137673"
    },
    {
      "turn": "0.000000000",
      "x": "0.806355632",
      "y": "0.443308364"
    },
    {
      "turn": "0.000000000",
      "x": "1.503656841",
      "y": "0.694277818"
    },
    {
      "turn": "0.000000000",
      "x": "0.210839248",
      "y": "1.302182688"
    },
    {
      "turn": "0.000000000",
      "x": "1.415574742",
      "y": "1.735959480"
    },
    {
      "turn": "0.000000000",
      "x": "0.154909408",
      "y": "0.993008316"
    },
    {
      "turn": "0.000000000",
      "x": "1.656122398",
      "y": "1.533132309"
    },
    {
      "turn": "0.000000000",
      "x": "0.704599492",
      "y": "1.820618616"
    },
    {
      "turn": "0.000000000",
      "x": "1.133800735",
      "y": "0.166385643"
    },
    {
      "turn": "0.000000000",
      "x": "1.428318444",
      "y": "0.272197989"
    },
    {
      "turn": "0.000000000",
      "x": "0.321432629",
      "y": "1.678854203"
    },
    {
      "turn": "0.000000000",
      "x": "1.090115790",
      "y": "1.955207676"
    },
    {
      "turn": "0.000000000",
      "x": "0.435487778",
      "y": "0.331557434"
    },
    {
      "turn": "0.000000000",
      "x": "1.860353575",
      "y": "0.844631408"
    },
    {
      "turn": "0.000000000",
      "x": "0.126351866",
      "y": "0.601190127"
    },
    {
      "turn": "0.000000000",
      "x": "1.926995009",
      "y": "1.248757316"
    },
    {
      "turn": "0.000000000",
      "x": "0.753236306",
      "y": "0.072705172"
    },
    {
      "turn": "0.000000000",
      "x": "1.781211274",
      "y": "0.442924239"
    }
  ],
  "radius": "0.231241854"
}

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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