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P87 · Packing and covering · Classic

Equal discs covering a disc · n = 12

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

Instancen = 12
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum[2.769293249854162059034535249256, 2.769293249854162059034535249257]VonEquinoxDiskCoveringSolve ↗
Site record2.769293247zzzcyRecord 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 = 12, 2.769293247
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
    2.76929324→2.769293247
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": "0.862295819",
      "y": "1.241776707"
    },
    {
      "turn": "0",
      "x": "1.278236889",
      "y": "0.998366910"
    },
    {
      "turn": "0",
      "x": "0.859467236",
      "y": "0.759856321"
    },
    {
      "turn": "0",
      "x": "0.230844188",
      "y": "1.004514465"
    },
    {
      "turn": "0",
      "x": "1.388487507",
      "y": "1.663851172"
    },
    {
      "turn": "0",
      "x": "1.380668258",
      "y": "0.331634359"
    },
    {
      "turn": "0",
      "x": "0.856020848",
      "y": "0.134546378"
    },
    {
      "turn": "0",
      "x": "0.322484789",
      "y": "1.557416476"
    },
    {
      "turn": "0",
      "x": "0.315988290",
      "y": "0.450574806"
    },
    {
      "turn": "0",
      "x": "1.821494389",
      "y": "1.308037190"
    },
    {
      "turn": "0",
      "x": "0.866189751",
      "y": "1.867084058"
    },
    {
      "turn": "0",
      "x": "1.817822020",
      "y": "0.682341073"
    }
  ],
  "radius": "0.361102964"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.862295819",
      "y": "1.241776707"
    },
    {
      "turn": "0",
      "x": "1.278236889",
      "y": "0.998366910"
    },
    {
      "turn": "0",
      "x": "0.859467236",
      "y": "0.759856321"
    },
    {
      "turn": "0",
      "x": "0.230844188",
      "y": "1.004514465"
    },
    {
      "turn": "0",
      "x": "1.388487507",
      "y": "1.663851172"
    },
    {
      "turn": "0",
      "x": "1.380668258",
      "y": "0.331634359"
    },
    {
      "turn": "0",
      "x": "0.856020848",
      "y": "0.134546378"
    },
    {
      "turn": "0",
      "x": "0.322484789",
      "y": "1.557416476"
    },
    {
      "turn": "0",
      "x": "0.315988290",
      "y": "0.450574806"
    },
    {
      "turn": "0",
      "x": "1.821494389",
      "y": "1.308037190"
    },
    {
      "turn": "0",
      "x": "0.866189751",
      "y": "1.867084058"
    },
    {
      "turn": "0",
      "x": "1.817822020",
      "y": "0.682341073"
    }
  ],
  "radius": "0.361102964"
}

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

DISCUSSION

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