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P87 · Packing and covering · Classic

Equal discs covering a disc · n = 15

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

Instancen = 15
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum[3.143241301320540312069843754742, 3.143241301320540312069843754743]VonEquinoxDiskCoveringSolve ↗
Site record3.143241299zzzcyRecord details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

  • ContainerThe editor and certificate normalise the target disc to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-covering-disc scale.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsThe n distinct centres lie inside the target disc. Covering discs may overlap and protrude, and their union must cover every point of the target disc.
  • PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
  • ObjectiveCovered disc radius R = 1/r, where larger is better.
Inspect in editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The record-holding arrangement for Equal discs covering a disc n = 15, 3.143241299
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra
Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions 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. zzzcyAI · OpenAI: GPT-6 Astra
    3.14324129→3.143241299
ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.984338480",
      "y": "1.362179846"
    },
    {
      "turn": "0",
      "x": "0.946468038",
      "y": "0.641455841"
    },
    {
      "turn": "0",
      "x": "1.309147228",
      "y": "0.983755861"
    },
    {
      "turn": "0",
      "x": "0.679859149",
      "y": "1.016821710"
    },
    {
      "turn": "0",
      "x": "0.329516441",
      "y": "0.610209649"
    },
    {
      "turn": "0",
      "x": "0.374058959",
      "y": "1.457910918"
    },
    {
      "turn": "0",
      "x": "1.471239396",
      "y": "0.428026688"
    },
    {
      "turn": "0",
      "x": "1.528587534",
      "y": "1.519437462"
    },
    {
      "turn": "0",
      "x": "0.686244993",
      "y": "1.803478028"
    },
    {
      "turn": "0",
      "x": "1.231259278",
      "y": "1.874881128"
    },
    {
      "turn": "0",
      "x": "0.603767840",
      "y": "0.233828271"
    },
    {
      "turn": "0",
      "x": "1.138297784",
      "y": "0.105700372"
    },
    {
      "turn": "0",
      "x": "1.799199042",
      "y": "0.699645928"
    },
    {
      "turn": "0",
      "x": "1.826275198",
      "y": "1.214943588"
    },
    {
      "turn": "0",
      "x": "0.081916113",
      "y": "1.048240740"
    }
  ],
  "radius": "0.318142931"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.984338480",
      "y": "1.362179846"
    },
    {
      "turn": "0",
      "x": "0.946468038",
      "y": "0.641455841"
    },
    {
      "turn": "0",
      "x": "1.309147228",
      "y": "0.983755861"
    },
    {
      "turn": "0",
      "x": "0.679859149",
      "y": "1.016821710"
    },
    {
      "turn": "0",
      "x": "0.329516441",
      "y": "0.610209649"
    },
    {
      "turn": "0",
      "x": "0.374058959",
      "y": "1.457910918"
    },
    {
      "turn": "0",
      "x": "1.471239396",
      "y": "0.428026688"
    },
    {
      "turn": "0",
      "x": "1.528587534",
      "y": "1.519437462"
    },
    {
      "turn": "0",
      "x": "0.686244993",
      "y": "1.803478028"
    },
    {
      "turn": "0",
      "x": "1.231259278",
      "y": "1.874881128"
    },
    {
      "turn": "0",
      "x": "0.603767840",
      "y": "0.233828271"
    },
    {
      "turn": "0",
      "x": "1.138297784",
      "y": "0.105700372"
    },
    {
      "turn": "0",
      "x": "1.799199042",
      "y": "0.699645928"
    },
    {
      "turn": "0",
      "x": "1.826275198",
      "y": "1.214943588"
    },
    {
      "turn": "0",
      "x": "0.081916113",
      "y": "1.048240740"
    }
  ],
  "radius": "0.318142931"
}

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · Verifier v1.0.0

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