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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 31

Pack n equal circles into the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1) so as to maximise their common radius.

Instancem = 5 · n = 31
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.138616362209, 0.138616362211]Eckard SpechtPackomania ↗
Site record0.138616362w33dRecord details ↓

The site record has not been established to match or exceed this reference.

Formal definition

  • ContainerThe editor and certificate normalise the container to the exact regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1), while public scores follow the literature convention. Boundary contact is allowed.
  • 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.
  • ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
  • ObjectiveMaximise the circle radius r (with container circumradius fixed at 1).
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. Pentagon and hexagon rows n = 41–100 and octagon rows n = 41–48 cite Packomania's value and author without importing its coordinates; they start from an ordinary site grid baseline.

Source
The record-holding arrangement for Equal circles in regular polygons m = 5 · n = 31, 0.138616362
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderw33d
Solution methodAI · Anthropic: Claude Fable 5.1
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. w33dAI · Anthropic: Claude Fable 5.1
    0.138616361→0.138616362
ANSWER FORMAT

How to write your answer

the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1)

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": "0.329599368",
      "y": "1.487074570"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "1.116064769"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "0.790158155"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "0.512925431"
    },
    {
      "turn": "0",
      "x": "0.535623433",
      "y": "1.301569669"
    },
    {
      "turn": "0",
      "x": "0.553885353",
      "y": "0.953111462"
    },
    {
      "turn": "0",
      "x": "0.569689950",
      "y": "0.651541793"
    },
    {
      "turn": "0",
      "x": "0.593263357",
      "y": "1.572744193"
    },
    {
      "turn": "0",
      "x": "0.682450657",
      "y": "0.398277097"
    },
    {
      "turn": "0",
      "x": "0.759909418",
      "y": "1.138616362"
    },
    {
      "turn": "0",
      "x": "0.793975935",
      "y": "0.814495100"
    },
    {
      "turn": "0",
      "x": "0.817549342",
      "y": "1.409790886"
    },
    {
      "turn": "0",
      "x": "0.903218965",
      "y": "1.673454875"
    },
    {
      "turn": "0",
      "x": "0.906736642",
      "y": "0.561230404"
    },
    {
      "turn": "0",
      "x": "0.992406265",
      "y": "0.297566415"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.057639924",
      "y": "1.271174524"
    },
    {
      "turn": "0",
      "x": "1.112760707",
      "y": "0.746735304"
    },
    {
      "turn": "0",
      "x": "1.143309547",
      "y": "1.534838513"
    },
    {
      "turn": "0",
      "x": "1.198430330",
      "y": "0.483071315"
    },
    {
      "turn": "0",
      "x": "1.256070254",
      "y": "1.788103209"
    },
    {
      "turn": "0",
      "x": "1.256070255",
      "y": "0.211896791"
    },
    {
      "turn": "0",
      "x": "1.275714015",
      "y": "1.028978710"
    },
    {
      "turn": "0",
      "x": "1.333353939",
      "y": "1.300153234"
    },
    {
      "turn": "0",
      "x": "1.388474722",
      "y": "0.775714014"
    },
    {
      "turn": "0",
      "x": "1.419023562",
      "y": "1.563817224"
    },
    {
      "turn": "0",
      "x": "1.474144345",
      "y": "0.512050025"
    },
    {
      "turn": "0",
      "x": "1.552946739",
      "y": "1.028978711"
    },
    {
      "turn": "0",
      "x": "1.610586663",
      "y": "1.300153235"
    },
    {
      "turn": "0",
      "x": "1.665707446",
      "y": "0.775714014"
    },
    {
      "turn": "0",
      "x": "1.828660753",
      "y": "0.999999999"
    }
  ],
  "radius": "0.138616362"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 31,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "1.487074570"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "1.116064769"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "0.790158155"
    },
    {
      "turn": "0",
      "x": "0.329599368",
      "y": "0.512925431"
    },
    {
      "turn": "0",
      "x": "0.535623433",
      "y": "1.301569669"
    },
    {
      "turn": "0",
      "x": "0.553885353",
      "y": "0.953111462"
    },
    {
      "turn": "0",
      "x": "0.569689950",
      "y": "0.651541793"
    },
    {
      "turn": "0",
      "x": "0.593263357",
      "y": "1.572744193"
    },
    {
      "turn": "0",
      "x": "0.682450657",
      "y": "0.398277097"
    },
    {
      "turn": "0",
      "x": "0.759909418",
      "y": "1.138616362"
    },
    {
      "turn": "0",
      "x": "0.793975935",
      "y": "0.814495100"
    },
    {
      "turn": "0",
      "x": "0.817549342",
      "y": "1.409790886"
    },
    {
      "turn": "0",
      "x": "0.903218965",
      "y": "1.673454875"
    },
    {
      "turn": "0",
      "x": "0.906736642",
      "y": "0.561230404"
    },
    {
      "turn": "0",
      "x": "0.992406265",
      "y": "0.297566415"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.057639924",
      "y": "1.271174524"
    },
    {
      "turn": "0",
      "x": "1.112760707",
      "y": "0.746735304"
    },
    {
      "turn": "0",
      "x": "1.143309547",
      "y": "1.534838513"
    },
    {
      "turn": "0",
      "x": "1.198430330",
      "y": "0.483071315"
    },
    {
      "turn": "0",
      "x": "1.256070254",
      "y": "1.788103209"
    },
    {
      "turn": "0",
      "x": "1.256070255",
      "y": "0.211896791"
    },
    {
      "turn": "0",
      "x": "1.275714015",
      "y": "1.028978710"
    },
    {
      "turn": "0",
      "x": "1.333353939",
      "y": "1.300153234"
    },
    {
      "turn": "0",
      "x": "1.388474722",
      "y": "0.775714014"
    },
    {
      "turn": "0",
      "x": "1.419023562",
      "y": "1.563817224"
    },
    {
      "turn": "0",
      "x": "1.474144345",
      "y": "0.512050025"
    },
    {
      "turn": "0",
      "x": "1.552946739",
      "y": "1.028978711"
    },
    {
      "turn": "0",
      "x": "1.610586663",
      "y": "1.300153235"
    },
    {
      "turn": "0",
      "x": "1.665707446",
      "y": "0.775714014"
    },
    {
      "turn": "0",
      "x": "1.828660753",
      "y": "0.999999999"
    }
  ],
  "radius": "0.138616362"
}

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

Discussion

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本题m=6，n=3的证明NUE_13#310 replies