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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 29

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 = 29
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.154929849311, 0.154929849313]Stefan ZorcicPackomania ↗
Site record0.154929849w33dRecord 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 = 29, 0.154929849
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
    0.154929848→0.154929849
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": "1.446916492",
      "y": "0.861648961"
    },
    {
      "turn": "0",
      "x": "0.671545926",
      "y": "0.303744993"
    },
    {
      "turn": "0",
      "x": "0.474903667",
      "y": "1.040444807"
    },
    {
      "turn": "0",
      "x": "1.428526246",
      "y": "1.170962446"
    },
    {
      "turn": "0",
      "x": "1.821137006",
      "y": "0.973311854"
    },
    {
      "turn": "0",
      "x": "0.592180803",
      "y": "1.335603241"
    },
    {
      "turn": "0",
      "x": "1.286273033",
      "y": "0.286273034"
    },
    {
      "turn": "0",
      "x": "0.982788578",
      "y": "1.825175935"
    },
    {
      "turn": "0",
      "x": "1.027549884",
      "y": "0.765251165"
    },
    {
      "turn": "0",
      "x": "1.702333166",
      "y": "0.686219900"
    },
    {
      "turn": "0",
      "x": "0.167694861",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.356325494",
      "y": "0.754171773"
    },
    {
      "turn": "0",
      "x": "1.560116939",
      "y": "1.600297107"
    },
    {
      "turn": "0",
      "x": "0.286273034",
      "y": "1.286273034"
    },
    {
      "turn": "0",
      "x": "1.698492839",
      "y": "1.323051470"
    },
    {
      "turn": "0",
      "x": "0.657236201",
      "y": "0.615271081"
    },
    {
      "turn": "0",
      "x": "0.932898518",
      "y": "0.470201760"
    },
    {
      "turn": "0",
      "x": "0.942343772",
      "y": "1.517967129"
    },
    {
      "turn": "0",
      "x": "1.233399099",
      "y": "1.411666474"
    },
    {
      "turn": "0",
      "x": "1.572546067",
      "y": "0.404851207"
    },
    {
      "turn": "0",
      "x": "0.396770300",
      "y": "0.446962967"
    },
    {
      "turn": "0",
      "x": "1.287187193",
      "y": "0.596131384"
    },
    {
      "turn": "0",
      "x": "0.877585467",
      "y": "1.214949976"
    },
    {
      "turn": "0",
      "x": "0.763234633",
      "y": "0.926962268"
    },
    {
      "turn": "0",
      "x": "1.141900718",
      "y": "1.053238873"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.167694861"
    },
    {
      "turn": "0",
      "x": "0.410753069",
      "y": "1.586794423"
    },
    {
      "turn": "0",
      "x": "0.696515545",
      "y": "1.706597762"
    },
    {
      "turn": "0",
      "x": "1.273843905",
      "y": "1.718875280"
    }
  ],
  "radius": "0.154929849"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.446916492",
      "y": "0.861648961"
    },
    {
      "turn": "0",
      "x": "0.671545926",
      "y": "0.303744993"
    },
    {
      "turn": "0",
      "x": "0.474903667",
      "y": "1.040444807"
    },
    {
      "turn": "0",
      "x": "1.428526246",
      "y": "1.170962446"
    },
    {
      "turn": "0",
      "x": "1.821137006",
      "y": "0.973311854"
    },
    {
      "turn": "0",
      "x": "0.592180803",
      "y": "1.335603241"
    },
    {
      "turn": "0",
      "x": "1.286273033",
      "y": "0.286273034"
    },
    {
      "turn": "0",
      "x": "0.982788578",
      "y": "1.825175935"
    },
    {
      "turn": "0",
      "x": "1.027549884",
      "y": "0.765251165"
    },
    {
      "turn": "0",
      "x": "1.702333166",
      "y": "0.686219900"
    },
    {
      "turn": "0",
      "x": "0.167694861",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.356325494",
      "y": "0.754171773"
    },
    {
      "turn": "0",
      "x": "1.560116939",
      "y": "1.600297107"
    },
    {
      "turn": "0",
      "x": "0.286273034",
      "y": "1.286273034"
    },
    {
      "turn": "0",
      "x": "1.698492839",
      "y": "1.323051470"
    },
    {
      "turn": "0",
      "x": "0.657236201",
      "y": "0.615271081"
    },
    {
      "turn": "0",
      "x": "0.932898518",
      "y": "0.470201760"
    },
    {
      "turn": "0",
      "x": "0.942343772",
      "y": "1.517967129"
    },
    {
      "turn": "0",
      "x": "1.233399099",
      "y": "1.411666474"
    },
    {
      "turn": "0",
      "x": "1.572546067",
      "y": "0.404851207"
    },
    {
      "turn": "0",
      "x": "0.396770300",
      "y": "0.446962967"
    },
    {
      "turn": "0",
      "x": "1.287187193",
      "y": "0.596131384"
    },
    {
      "turn": "0",
      "x": "0.877585467",
      "y": "1.214949976"
    },
    {
      "turn": "0",
      "x": "0.763234633",
      "y": "0.926962268"
    },
    {
      "turn": "0",
      "x": "1.141900718",
      "y": "1.053238873"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.167694861"
    },
    {
      "turn": "0",
      "x": "0.410753069",
      "y": "1.586794423"
    },
    {
      "turn": "0",
      "x": "0.696515545",
      "y": "1.706597762"
    },
    {
      "turn": "0",
      "x": "1.273843905",
      "y": "1.718875280"
    }
  ],
  "radius": "0.154929849"
}

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