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P87 · Packing and covering · Classic · Hard

Equal discs covering a disc · n = 25

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

Instancen = 25
ObjectiveMaximize covered disc radius

Record comparison

covered disc radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[4.181, 4.182)Jeremy TanErich Friedman ↗
Site record4.181540545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 = 25, 4.181540545
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
    4.181540528→4.181540545
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": "1.254916246",
      "y": "1.360943015"
    },
    {
      "turn": "0.000000000",
      "x": "1.435478254",
      "y": "1.074972244"
    },
    {
      "turn": "0.000000000",
      "x": "0.564521760",
      "y": "0.925027765"
    },
    {
      "turn": "0.000000000",
      "x": "1.360943019",
      "y": "0.745083744"
    },
    {
      "turn": "0.000000000",
      "x": "0.925027757",
      "y": "1.435478240"
    },
    {
      "turn": "0.000000000",
      "x": "1.074972237",
      "y": "0.564521746"
    },
    {
      "turn": "0.000000000",
      "x": "0.745083747",
      "y": "0.639056993"
    },
    {
      "turn": "0.000000000",
      "x": "0.639056997",
      "y": "1.254916243"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000012",
      "y": "0.999999993"
    },
    {
      "turn": "0.000000000",
      "x": "0.309605494",
      "y": "0.564084767"
    },
    {
      "turn": "0.000000000",
      "x": "1.435915240",
      "y": "0.309605494"
    },
    {
      "turn": "0.000000000",
      "x": "0.564084751",
      "y": "1.690394495"
    },
    {
      "turn": "0.000000000",
      "x": "1.690394495",
      "y": "1.435915256"
    },
    {
      "turn": "0.000000000",
      "x": "0.820055984",
      "y": "0.203578743"
    },
    {
      "turn": "0.000000000",
      "x": "0.203578744",
      "y": "1.179944006"
    },
    {
      "turn": "0.000000000",
      "x": "1.796421263",
      "y": "0.820056001"
    },
    {
      "turn": "0.000000000",
      "x": "1.179943989",
      "y": "1.796421262"
    },
    {
      "turn": "0.000000000",
      "x": "0.850055509",
      "y": "1.870956492"
    },
    {
      "turn": "0.000000000",
      "x": "1.149944470",
      "y": "0.129043493"
    },
    {
      "turn": "0.000000000",
      "x": "0.278113989",
      "y": "1.509832490"
    },
    {
      "turn": "0.000000000",
      "x": "1.721886023",
      "y": "0.490167498"
    },
    {
      "turn": "0.000000000",
      "x": "1.509832484",
      "y": "1.721886027"
    },
    {
      "turn": "0.000000000",
      "x": "1.870956504",
      "y": "1.149944491"
    },
    {
      "turn": "0.000000000",
      "x": "0.129043505",
      "y": "0.850055530"
    },
    {
      "turn": "0.000000000",
      "x": "0.490167492",
      "y": "0.278113992"
    }
  ],
  "radius": "0.239146312"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "1.254916246",
      "y": "1.360943015"
    },
    {
      "turn": "0.000000000",
      "x": "1.435478254",
      "y": "1.074972244"
    },
    {
      "turn": "0.000000000",
      "x": "0.564521760",
      "y": "0.925027765"
    },
    {
      "turn": "0.000000000",
      "x": "1.360943019",
      "y": "0.745083744"
    },
    {
      "turn": "0.000000000",
      "x": "0.925027757",
      "y": "1.435478240"
    },
    {
      "turn": "0.000000000",
      "x": "1.074972237",
      "y": "0.564521746"
    },
    {
      "turn": "0.000000000",
      "x": "0.745083747",
      "y": "0.639056993"
    },
    {
      "turn": "0.000000000",
      "x": "0.639056997",
      "y": "1.254916243"
    },
    {
      "turn": "0.000000000",
      "x": "1.000000012",
      "y": "0.999999993"
    },
    {
      "turn": "0.000000000",
      "x": "0.309605494",
      "y": "0.564084767"
    },
    {
      "turn": "0.000000000",
      "x": "1.435915240",
      "y": "0.309605494"
    },
    {
      "turn": "0.000000000",
      "x": "0.564084751",
      "y": "1.690394495"
    },
    {
      "turn": "0.000000000",
      "x": "1.690394495",
      "y": "1.435915256"
    },
    {
      "turn": "0.000000000",
      "x": "0.820055984",
      "y": "0.203578743"
    },
    {
      "turn": "0.000000000",
      "x": "0.203578744",
      "y": "1.179944006"
    },
    {
      "turn": "0.000000000",
      "x": "1.796421263",
      "y": "0.820056001"
    },
    {
      "turn": "0.000000000",
      "x": "1.179943989",
      "y": "1.796421262"
    },
    {
      "turn": "0.000000000",
      "x": "0.850055509",
      "y": "1.870956492"
    },
    {
      "turn": "0.000000000",
      "x": "1.149944470",
      "y": "0.129043493"
    },
    {
      "turn": "0.000000000",
      "x": "0.278113989",
      "y": "1.509832490"
    },
    {
      "turn": "0.000000000",
      "x": "1.721886023",
      "y": "0.490167498"
    },
    {
      "turn": "0.000000000",
      "x": "1.509832484",
      "y": "1.721886027"
    },
    {
      "turn": "0.000000000",
      "x": "1.870956504",
      "y": "1.149944491"
    },
    {
      "turn": "0.000000000",
      "x": "0.129043505",
      "y": "0.850055530"
    },
    {
      "turn": "0.000000000",
      "x": "0.490167492",
      "y": "0.278113992"
    }
  ],
  "radius": "0.239146312"
}

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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