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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 23

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

Instancen = 23
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4, 4.001)Jeremy TanErich Friedman ↗
Site record4.000739704空投Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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.
Open full 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 = 23, 4.000739704
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder空投
Solution methodAI · OpenAI: GPT-6.1 Sol
Challenge this record ↗Submit 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
    4.000739688→4.000739704
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": "0.870780029",
      "y": "0.565617776"
    },
    {
      "turn": "0.000000000",
      "x": "0.889771564",
      "y": "1.439585500"
    },
    {
      "turn": "0.000000000",
      "x": "1.435910456",
      "y": "0.990527557"
    },
    {
      "turn": "0.000000000",
      "x": "1.253933057",
      "y": "0.632240514"
    },
    {
      "turn": "0.000000000",
      "x": "1.269668818",
      "y": "1.356381487"
    },
    {
      "turn": "0.000000000",
      "x": "0.579313145",
      "y": "0.817520464"
    },
    {
      "turn": "0.000000000",
      "x": "0.993879703",
      "y": "1.000132996"
    },
    {
      "turn": "0.000000000",
      "x": "0.587637167",
      "y": "1.200581911"
    },
    {
      "turn": "0.000000000",
      "x": "1.658281669",
      "y": "1.418079221"
    },
    {
      "turn": "0.000000000",
      "x": "1.639498926",
      "y": "0.553719685"
    },
    {
      "turn": "0.000000000",
      "x": "0.491030090",
      "y": "1.645818047"
    },
    {
      "turn": "0.000000000",
      "x": "1.530948112",
      "y": "1.732167732"
    },
    {
      "turn": "0.000000000",
      "x": "0.755818410",
      "y": "0.125151221"
    },
    {
      "turn": "0.000000000",
      "x": "1.498641562",
      "y": "0.245459026"
    },
    {
      "turn": "0.000000000",
      "x": "0.463456213",
      "y": "0.376901224"
    },
    {
      "turn": "0.000000000",
      "x": "1.161420712",
      "y": "1.796682240"
    },
    {
      "turn": "0.000000000",
      "x": "1.126660472",
      "y": "0.197057685"
    },
    {
      "turn": "0.000000000",
      "x": "0.794052323",
      "y": "1.884630203"
    },
    {
      "turn": "0.000000000",
      "x": "1.847009728",
      "y": "1.176120854"
    },
    {
      "turn": "0.000000000",
      "x": "1.838559493",
      "y": "0.787251316"
    },
    {
      "turn": "0.000000000",
      "x": "0.171989328",
      "y": "0.628803914"
    },
    {
      "turn": "0.000000000",
      "x": "0.188895693",
      "y": "1.406814460"
    },
    {
      "turn": "0.000000000",
      "x": "0.173070609",
      "y": "1.017969381"
    }
  ],
  "radius": "0.249953777"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.870780029",
      "y": "0.565617776"
    },
    {
      "turn": "0.000000000",
      "x": "0.889771564",
      "y": "1.439585500"
    },
    {
      "turn": "0.000000000",
      "x": "1.435910456",
      "y": "0.990527557"
    },
    {
      "turn": "0.000000000",
      "x": "1.253933057",
      "y": "0.632240514"
    },
    {
      "turn": "0.000000000",
      "x": "1.269668818",
      "y": "1.356381487"
    },
    {
      "turn": "0.000000000",
      "x": "0.579313145",
      "y": "0.817520464"
    },
    {
      "turn": "0.000000000",
      "x": "0.993879703",
      "y": "1.000132996"
    },
    {
      "turn": "0.000000000",
      "x": "0.587637167",
      "y": "1.200581911"
    },
    {
      "turn": "0.000000000",
      "x": "1.658281669",
      "y": "1.418079221"
    },
    {
      "turn": "0.000000000",
      "x": "1.639498926",
      "y": "0.553719685"
    },
    {
      "turn": "0.000000000",
      "x": "0.491030090",
      "y": "1.645818047"
    },
    {
      "turn": "0.000000000",
      "x": "1.530948112",
      "y": "1.732167732"
    },
    {
      "turn": "0.000000000",
      "x": "0.755818410",
      "y": "0.125151221"
    },
    {
      "turn": "0.000000000",
      "x": "1.498641562",
      "y": "0.245459026"
    },
    {
      "turn": "0.000000000",
      "x": "0.463456213",
      "y": "0.376901224"
    },
    {
      "turn": "0.000000000",
      "x": "1.161420712",
      "y": "1.796682240"
    },
    {
      "turn": "0.000000000",
      "x": "1.126660472",
      "y": "0.197057685"
    },
    {
      "turn": "0.000000000",
      "x": "0.794052323",
      "y": "1.884630203"
    },
    {
      "turn": "0.000000000",
      "x": "1.847009728",
      "y": "1.176120854"
    },
    {
      "turn": "0.000000000",
      "x": "1.838559493",
      "y": "0.787251316"
    },
    {
      "turn": "0.000000000",
      "x": "0.171989328",
      "y": "0.628803914"
    },
    {
      "turn": "0.000000000",
      "x": "0.188895693",
      "y": "1.406814460"
    },
    {
      "turn": "0.000000000",
      "x": "0.173070609",
      "y": "1.017969381"
    }
  ],
  "radius": "0.249953777"
}

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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