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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 32

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 = 8 · n = 32
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.147297380424, 0.147297380426]Stefan ZorcicPackomania ↗
Site record0.14729738zzzcyRecord 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 = 8 · n = 32, 0.14729738
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra
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. zzzcyAI · OpenAI: GPT-6 Astra
    0.147297379→0.14729738
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": "0.000000000",
      "x": "0.756432053",
      "y": "0.973003714"
    },
    {
      "turn": "0.000000000",
      "x": "1.262628489",
      "y": "0.893860195"
    },
    {
      "turn": "0.000000000",
      "x": "1.122013966",
      "y": "1.152730102"
    },
    {
      "turn": "0.000000000",
      "x": "0.987059160",
      "y": "0.789708296"
    },
    {
      "turn": "0.000000000",
      "x": "0.577967316",
      "y": "0.656247874"
    },
    {
      "turn": "0.000000000",
      "x": "0.845323961",
      "y": "1.253867208"
    },
    {
      "turn": "0.000000000",
      "x": "0.554122686",
      "y": "1.209281438"
    },
    {
      "turn": "0.000000000",
      "x": "1.411652743",
      "y": "0.639738483"
    },
    {
      "turn": "0.000000000",
      "x": "1.442762684",
      "y": "1.197854942"
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    {
      "turn": "0.000000000",
      "x": "1.555499218",
      "y": "0.925684872"
    },
    {
      "turn": "0.000000000",
      "x": "0.971415441",
      "y": "1.520113356"
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    {
      "turn": "0.000000000",
      "x": "0.844924895",
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    {
      "turn": "0.000000000",
      "x": "1.248105446",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
      "x": "0.398629385",
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    },
    {
      "turn": "0.000000000",
      "x": "0.667746169",
      "y": "1.488925075"
    },
    {
      "turn": "0.000000000",
      "x": "0.173156317",
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    {
      "turn": "0.000000000",
      "x": "1.612775173",
      "y": "1.543912172"
    },
    {
      "turn": "0.000000000",
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    {
      "turn": "0.000000000",
      "x": "1.727829929",
      "y": "1.272170072"
    },
    {
      "turn": "0.000000000",
      "x": "1.704523470",
      "y": "0.671563159"
    },
    {
      "turn": "0.000000000",
      "x": "0.938219357",
      "y": "0.185137578"
    },
    {
      "turn": "0.000000000",
      "x": "1.084058299",
      "y": "1.805748376"
    },
    {
      "turn": "0.000000000",
      "x": "1.840566463",
      "y": "1.000000002"
    },
    {
      "turn": "0.000000000",
      "x": "0.793837650",
      "y": "1.755171223"
    },
    {
      "turn": "0.000000000",
      "x": "0.665586967",
      "y": "0.297951949"
    },
    {
      "turn": "0.000000000",
      "x": "0.374784757",
      "y": "1.519903840"
    },
    {
      "turn": "0.000000000",
      "x": "0.285892851",
      "y": "0.694700206"
    },
    {
      "turn": "0.000000000",
      "x": "1.356228369",
      "y": "1.693011843"
    },
    {
      "turn": "0.000000000",
      "x": "1.313479016",
      "y": "0.289280796"
    }
  ],
  "radius": "0.147297380"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.756432053",
      "y": "0.973003714"
    },
    {
      "turn": "0.000000000",
      "x": "1.262628489",
      "y": "0.893860195"
    },
    {
      "turn": "0.000000000",
      "x": "1.122013966",
      "y": "1.152730102"
    },
    {
      "turn": "0.000000000",
      "x": "0.987059160",
      "y": "0.789708296"
    },
    {
      "turn": "0.000000000",
      "x": "0.577967316",
      "y": "0.656247874"
    },
    {
      "turn": "0.000000000",
      "x": "0.845323961",
      "y": "1.253867208"
    },
    {
      "turn": "0.000000000",
      "x": "0.554122686",
      "y": "1.209281438"
    },
    {
      "turn": "0.000000000",
      "x": "1.411652743",
      "y": "0.639738483"
    },
    {
      "turn": "0.000000000",
      "x": "1.442762684",
      "y": "1.197854942"
    },
    {
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      "x": "1.555499218",
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    },
    {
      "turn": "0.000000000",
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      "y": "1.520113356"
    },
    {
      "turn": "0.000000000",
      "x": "0.844924895",
      "y": "0.531669687"
    },
    {
      "turn": "0.000000000",
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
      "turn": "0.000000000",
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    },
    {
      "turn": "0.000000000",
      "x": "1.313479016",
      "y": "0.289280796"
    }
  ],
  "radius": "0.147297380"
}

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