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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · 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 = 6 · n = 32
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.142897923361, 0.142897923363]Eckard SpechtPackomania ↗
Site record0.142908786Joost de WinterRecord details ↓

Exceeds the published reference · verified. Review evidence ↗

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 = 32, 0.142908786
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderJoost de Winter
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 (2 changes)
  1. Joost de WinterAI · OpenAI: GPT-6.1 Sol
    0.142897923→0.142908786
  2. w33dAI · Anthropic: Claude Fable 5.1
    0.142897922→0.142897923
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": "0.623106190",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "0.908923764",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "1.194741338",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "1.467253571",
      "y": "0.363073691"
    },
    {
      "turn": "0",
      "x": "0.450834428",
      "y": "0.504949439"
    },
    {
      "turn": "0",
      "x": "0.766014977",
      "y": "0.524408664"
    },
    {
      "turn": "0",
      "x": "1.051832551",
      "y": "0.524408664"
    },
    {
      "turn": "0",
      "x": "1.324344783",
      "y": "0.610598971"
    },
    {
      "turn": "0",
      "x": "1.610162358",
      "y": "0.610598971"
    },
    {
      "turn": "0",
      "x": "0.307925641",
      "y": "0.752474720"
    },
    {
      "turn": "0",
      "x": "0.593743215",
      "y": "0.752474720"
    },
    {
      "turn": "0",
      "x": "0.866255447",
      "y": "0.838665027"
    },
    {
      "turn": "0",
      "x": "1.152073022",
      "y": "0.838665027"
    },
    {
      "turn": "0",
      "x": "1.467253571",
      "y": "0.858124252"
    },
    {
      "turn": "0",
      "x": "1.753071145",
      "y": "0.858124252"
    },
    {
      "turn": "0",
      "x": "0.165016854",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.450834428",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.895194042",
      "y": "1.123013834"
    },
    {
      "turn": "0",
      "x": "1.181011616",
      "y": "1.123013834"
    },
    {
      "turn": "0",
      "x": "1.466062154",
      "y": "1.143939343"
    },
    {
      "turn": "0",
      "x": "1.751879728",
      "y": "1.143939343"
    },
    {
      "turn": "0",
      "x": "0.627938004",
      "y": "1.224334592"
    },
    {
      "turn": "0",
      "x": "0.307925641",
      "y": "1.247525280"
    },
    {
      "turn": "0",
      "x": "1.038102829",
      "y": "1.370539114"
    },
    {
      "turn": "0",
      "x": "1.323153367",
      "y": "1.391464623"
    },
    {
      "turn": "0",
      "x": "1.608970941",
      "y": "1.391464623"
    },
    {
      "turn": "0",
      "x": "0.485029217",
      "y": "1.471859872"
    },
    {
      "turn": "0",
      "x": "0.770846791",
      "y": "1.471859872"
    },
    {
      "turn": "0",
      "x": "1.180244579",
      "y": "1.638989903"
    },
    {
      "turn": "0",
      "x": "1.466062154",
      "y": "1.638989903"
    },
    {
      "turn": "0",
      "x": "0.621270672",
      "y": "1.723116617"
    },
    {
      "turn": "0",
      "x": "0.907088247",
      "y": "1.723116617"
    }
  ],
  "radius": "0.142908786"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.623106190",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "0.908923764",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "1.194741338",
      "y": "0.276883383"
    },
    {
      "turn": "0",
      "x": "1.467253571",
      "y": "0.363073691"
    },
    {
      "turn": "0",
      "x": "0.450834428",
      "y": "0.504949439"
    },
    {
      "turn": "0",
      "x": "0.766014977",
      "y": "0.524408664"
    },
    {
      "turn": "0",
      "x": "1.051832551",
      "y": "0.524408664"
    },
    {
      "turn": "0",
      "x": "1.324344783",
      "y": "0.610598971"
    },
    {
      "turn": "0",
      "x": "1.610162358",
      "y": "0.610598971"
    },
    {
      "turn": "0",
      "x": "0.307925641",
      "y": "0.752474720"
    },
    {
      "turn": "0",
      "x": "0.593743215",
      "y": "0.752474720"
    },
    {
      "turn": "0",
      "x": "0.866255447",
      "y": "0.838665027"
    },
    {
      "turn": "0",
      "x": "1.152073022",
      "y": "0.838665027"
    },
    {
      "turn": "0",
      "x": "1.467253571",
      "y": "0.858124252"
    },
    {
      "turn": "0",
      "x": "1.753071145",
      "y": "0.858124252"
    },
    {
      "turn": "0",
      "x": "0.165016854",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.450834428",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.895194042",
      "y": "1.123013834"
    },
    {
      "turn": "0",
      "x": "1.181011616",
      "y": "1.123013834"
    },
    {
      "turn": "0",
      "x": "1.466062154",
      "y": "1.143939343"
    },
    {
      "turn": "0",
      "x": "1.751879728",
      "y": "1.143939343"
    },
    {
      "turn": "0",
      "x": "0.627938004",
      "y": "1.224334592"
    },
    {
      "turn": "0",
      "x": "0.307925641",
      "y": "1.247525280"
    },
    {
      "turn": "0",
      "x": "1.038102829",
      "y": "1.370539114"
    },
    {
      "turn": "0",
      "x": "1.323153367",
      "y": "1.391464623"
    },
    {
      "turn": "0",
      "x": "1.608970941",
      "y": "1.391464623"
    },
    {
      "turn": "0",
      "x": "0.485029217",
      "y": "1.471859872"
    },
    {
      "turn": "0",
      "x": "0.770846791",
      "y": "1.471859872"
    },
    {
      "turn": "0",
      "x": "1.180244579",
      "y": "1.638989903"
    },
    {
      "turn": "0",
      "x": "1.466062154",
      "y": "1.638989903"
    },
    {
      "turn": "0",
      "x": "0.621270672",
      "y": "1.723116617"
    },
    {
      "turn": "0",
      "x": "0.907088247",
      "y": "1.723116617"
    }
  ],
  "radius": "0.142908786"
}

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