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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · 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 = 6 · n = 31
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.144498618554, 0.144498618556]Eckard SpechtPackomania ↗
Site record0.144498618w33dRecord 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 = 6 · n = 31, 0.144498618
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
    0.144498617→0.144498618
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": "0.642108169",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "0.931105406",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "1.220102643",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "1.484255699",
      "y": "0.395701905"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "0.499442102"
    },
    {
      "turn": "0",
      "x": "0.786606787",
      "y": "0.528752164"
    },
    {
      "turn": "0",
      "x": "1.075604024",
      "y": "0.528752164"
    },
    {
      "turn": "0",
      "x": "1.339757081",
      "y": "0.645980854"
    },
    {
      "turn": "0",
      "x": "1.628754318",
      "y": "0.645980854"
    },
    {
      "turn": "0",
      "x": "0.311351251",
      "y": "0.749721051"
    },
    {
      "turn": "0",
      "x": "0.600348488",
      "y": "0.749721051"
    },
    {
      "turn": "0",
      "x": "0.931105406",
      "y": "0.779031113"
    },
    {
      "turn": "0",
      "x": "1.195258462",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "1.484255699",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "1.773252937",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "0.166852633",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.744847107",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.150232234",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "1.439229471",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "1.728226708",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "0.311351251",
      "y": "1.250278949"
    },
    {
      "turn": "0",
      "x": "0.600348488",
      "y": "1.250278949"
    },
    {
      "turn": "0",
      "x": "0.885455103",
      "y": "1.297540150"
    },
    {
      "turn": "0",
      "x": "1.583728089",
      "y": "1.432006862"
    },
    {
      "turn": "0",
      "x": "1.262895384",
      "y": "1.452626101"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "1.500557898"
    },
    {
      "turn": "0",
      "x": "0.897508465",
      "y": "1.586285919"
    },
    {
      "turn": "0",
      "x": "1.439229471",
      "y": "1.682285811"
    },
    {
      "turn": "0",
      "x": "0.642108169",
      "y": "1.721526785"
    },
    {
      "turn": "0",
      "x": "1.152908761",
      "y": "1.721526785"
    }
  ],
  "radius": "0.144498618"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 31,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.642108169",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "0.931105406",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "1.220102643",
      "y": "0.278473215"
    },
    {
      "turn": "0",
      "x": "1.484255699",
      "y": "0.395701905"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "0.499442102"
    },
    {
      "turn": "0",
      "x": "0.786606787",
      "y": "0.528752164"
    },
    {
      "turn": "0",
      "x": "1.075604024",
      "y": "0.528752164"
    },
    {
      "turn": "0",
      "x": "1.339757081",
      "y": "0.645980854"
    },
    {
      "turn": "0",
      "x": "1.628754318",
      "y": "0.645980854"
    },
    {
      "turn": "0",
      "x": "0.311351251",
      "y": "0.749721051"
    },
    {
      "turn": "0",
      "x": "0.600348488",
      "y": "0.749721051"
    },
    {
      "turn": "0",
      "x": "0.931105406",
      "y": "0.779031113"
    },
    {
      "turn": "0",
      "x": "1.195258462",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "1.484255699",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "1.773252937",
      "y": "0.896259803"
    },
    {
      "turn": "0",
      "x": "0.166852633",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.744847107",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.150232234",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "1.439229471",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "1.728226708",
      "y": "1.181727913"
    },
    {
      "turn": "0",
      "x": "0.311351251",
      "y": "1.250278949"
    },
    {
      "turn": "0",
      "x": "0.600348488",
      "y": "1.250278949"
    },
    {
      "turn": "0",
      "x": "0.885455103",
      "y": "1.297540150"
    },
    {
      "turn": "0",
      "x": "1.583728089",
      "y": "1.432006862"
    },
    {
      "turn": "0",
      "x": "1.262895384",
      "y": "1.452626101"
    },
    {
      "turn": "0",
      "x": "0.455849870",
      "y": "1.500557898"
    },
    {
      "turn": "0",
      "x": "0.897508465",
      "y": "1.586285919"
    },
    {
      "turn": "0",
      "x": "1.439229471",
      "y": "1.682285811"
    },
    {
      "turn": "0",
      "x": "0.642108169",
      "y": "1.721526785"
    },
    {
      "turn": "0",
      "x": "1.152908761",
      "y": "1.721526785"
    }
  ],
  "radius": "0.144498618"
}

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