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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 36

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 = 36
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.128215138244, 0.128215138246]Paolo AmorePackomania ↗
Site record0.128215138w33dRecord 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 = 36, 0.128215138
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
    0.128215137→0.128215138
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": "0.319198144",
      "y": "1.414318579"
    },
    {
      "turn": "0",
      "x": "0.319198144",
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    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.881154335"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.505368499"
    },
    {
      "turn": "0",
      "x": "0.493705273",
      "y": "0.693261417"
    },
    {
      "turn": "0",
      "x": "0.528404113",
      "y": "1.562606640"
    },
    {
      "turn": "0",
      "x": "0.535094101",
      "y": "1.275951595"
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    {
      "turn": "0",
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      "y": "1.019796717"
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    {
      "turn": "0",
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    {
      "turn": "0",
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      "y": "0.620788912"
    },
    {
      "turn": "0",
      "x": "0.759594194",
      "y": "0.876444862"
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    {
      "turn": "0",
      "x": "0.766959488",
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    {
      "turn": "0",
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    {
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    {
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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      "y": "1.464714272"
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    {
      "turn": "0",
      "x": "1.016163483",
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    {
      "turn": "0",
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    {
      "turn": "0",
      "x": "1.067387189",
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    {
      "turn": "0",
      "x": "1.129670491",
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    },
    {
      "turn": "0",
      "x": "1.236592575",
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    {
      "turn": "0",
      "x": "1.255721368",
      "y": "1.388627499"
    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    {
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    },
    {
      "turn": "0",
      "x": "1.388559163",
      "y": "0.754373293"
    },
    {
      "turn": "0",
      "x": "1.406447302",
      "y": "1.181171047"
    },
    {
      "turn": "0",
      "x": "1.410769104",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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      "y": "0.961829746"
    },
    {
      "turn": "0",
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      "y": "1.385417678"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.712220973",
      "y": "1.177961227"
    },
    {
      "turn": "0",
      "x": "1.794310623",
      "y": "0.935025482"
    }
  ],
  "radius": "0.128215138"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 36,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "1.414318579"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "1.137584611"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.881154335"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.505368499"
    },
    {
      "turn": "0",
      "x": "0.493705273",
      "y": "0.693261417"
    },
    {
      "turn": "0",
      "x": "0.528404113",
      "y": "1.562606640"
    },
    {
      "turn": "0",
      "x": "0.535094101",
      "y": "1.275951595"
    },
    {
      "turn": "0",
      "x": "0.546975445",
      "y": "1.019796717"
    },
    {
      "turn": "0",
      "x": "0.565174205",
      "y": "0.432895991"
    },
    {
      "turn": "0",
      "x": "0.739681334",
      "y": "0.620788912"
    },
    {
      "turn": "0",
      "x": "0.759594194",
      "y": "0.876444862"
    },
    {
      "turn": "0",
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    },
    {
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      "y": "1.641847954"
    },
    {
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      "x": "0.797956380",
      "y": "1.130137248"
    },
    {
      "turn": "0",
      "x": "0.806771289",
      "y": "0.346946380"
    },
    {
      "turn": "0",
      "x": "0.981278417",
      "y": "0.534839299"
    },
    {
      "turn": "0",
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    },
    {
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    },
    {
      "turn": "0",
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    },
    {
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      "y": "0.267705067"
    },
    {
      "turn": "0",
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    {
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      "y": "0.745795799"
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    {
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      "y": "0.510940193"
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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",
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    },
    {
      "turn": "0",
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    {
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    },
    {
      "turn": "0",
      "x": "1.794310623",
      "y": "0.935025482"
    }
  ],
  "radius": "0.128215138"
}

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