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P87 · Packing and covering · Classic

Equal discs covering a disc · n = 16

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

Instancen = 16
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum[3.244437877904185050606331665494, 3.244437877904185050606331665495]VonEquinoxDiskCoveringSolve ↗
Site record3.244437876空投Record 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 = 16, 3.244437876
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder空投
Solution methodAI · OpenAI: GPT-6.1 Sol
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. 空投AI · OpenAI: GPT-6.1 Sol
    3.244437866→3.244437876
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": "0.000000000",
      "x": "1.000793219",
      "y": "0.694366522"
    },
    {
      "turn": "0.000000000",
      "x": "0.999206010",
      "y": "1.305633840"
    },
    {
      "turn": "0.000000000",
      "x": "0.651945288",
      "y": "0.999096470"
    },
    {
      "turn": "0.000000000",
      "x": "1.348053943",
      "y": "1.000903963"
    },
    {
      "turn": "0.000000000",
      "x": "1.518001977",
      "y": "1.531504591"
    },
    {
      "turn": "0.000000000",
      "x": "0.479244981",
      "y": "1.528808250"
    },
    {
      "turn": "0.000000000",
      "x": "1.520754641",
      "y": "0.471192555"
    },
    {
      "turn": "0.000000000",
      "x": "0.481998905",
      "y": "0.468494295"
    },
    {
      "turn": "0.000000000",
      "x": "0.172529284",
      "y": "0.741992842"
    },
    {
      "turn": "0.000000000",
      "x": "1.828799021",
      "y": "0.746293908"
    },
    {
      "turn": "0.000000000",
      "x": "1.827470317",
      "y": "1.258005827"
    },
    {
      "turn": "0.000000000",
      "x": "0.757651435",
      "y": "1.806300824"
    },
    {
      "turn": "0.000000000",
      "x": "0.761842241",
      "y": "0.192451255"
    },
    {
      "turn": "0.000000000",
      "x": "0.171200616",
      "y": "1.253705277"
    },
    {
      "turn": "0.000000000",
      "x": "1.242350026",
      "y": "0.193699881"
    },
    {
      "turn": "0.000000000",
      "x": "1.238159515",
      "y": "1.807548413"
    }
  ],
  "radius": "0.308219802"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.000793219",
      "y": "0.694366522"
    },
    {
      "turn": "0.000000000",
      "x": "0.999206010",
      "y": "1.305633840"
    },
    {
      "turn": "0.000000000",
      "x": "0.651945288",
      "y": "0.999096470"
    },
    {
      "turn": "0.000000000",
      "x": "1.348053943",
      "y": "1.000903963"
    },
    {
      "turn": "0.000000000",
      "x": "1.518001977",
      "y": "1.531504591"
    },
    {
      "turn": "0.000000000",
      "x": "0.479244981",
      "y": "1.528808250"
    },
    {
      "turn": "0.000000000",
      "x": "1.520754641",
      "y": "0.471192555"
    },
    {
      "turn": "0.000000000",
      "x": "0.481998905",
      "y": "0.468494295"
    },
    {
      "turn": "0.000000000",
      "x": "0.172529284",
      "y": "0.741992842"
    },
    {
      "turn": "0.000000000",
      "x": "1.828799021",
      "y": "0.746293908"
    },
    {
      "turn": "0.000000000",
      "x": "1.827470317",
      "y": "1.258005827"
    },
    {
      "turn": "0.000000000",
      "x": "0.757651435",
      "y": "1.806300824"
    },
    {
      "turn": "0.000000000",
      "x": "0.761842241",
      "y": "0.192451255"
    },
    {
      "turn": "0.000000000",
      "x": "0.171200616",
      "y": "1.253705277"
    },
    {
      "turn": "0.000000000",
      "x": "1.242350026",
      "y": "0.193699881"
    },
    {
      "turn": "0.000000000",
      "x": "1.238159515",
      "y": "1.807548413"
    }
  ],
  "radius": "0.308219802"
}

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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