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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 11

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 = 11
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.224333593339, 0.224333593341]Eckard SpechtPackomania ↗
Site record0.224333593w33dRecord 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 = 11, 0.224333593
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
    0.224333592→0.224333593
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": "0.415316599",
      "y": "1.338667972"
    },
    {
      "turn": "0",
      "x": "0.415316599",
      "y": "0.685175206"
    },
    {
      "turn": "0",
      "x": "0.722788623",
      "y": "1.011921589"
    },
    {
      "turn": "0",
      "x": "0.797948384",
      "y": "0.450878041"
    },
    {
      "turn": "0",
      "x": "0.809539789",
      "y": "1.552888235"
    },
    {
      "turn": "0",
      "x": "1.105420408",
      "y": "0.777624424"
    },
    {
      "turn": "0",
      "x": "1.117011813",
      "y": "1.226141852"
    },
    {
      "turn": "0",
      "x": "1.245464170",
      "y": "0.344851074"
    },
    {
      "turn": "0",
      "x": "1.246246469",
      "y": "1.655793627"
    },
    {
      "turn": "0",
      "x": "1.553979233",
      "y": "0.767764185"
    },
    {
      "turn": "0",
      "x": "1.565570638",
      "y": "1.216281613"
    }
  ],
  "radius": "0.224333593"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.415316599",
      "y": "1.338667972"
    },
    {
      "turn": "0",
      "x": "0.415316599",
      "y": "0.685175206"
    },
    {
      "turn": "0",
      "x": "0.722788623",
      "y": "1.011921589"
    },
    {
      "turn": "0",
      "x": "0.797948384",
      "y": "0.450878041"
    },
    {
      "turn": "0",
      "x": "0.809539789",
      "y": "1.552888235"
    },
    {
      "turn": "0",
      "x": "1.105420408",
      "y": "0.777624424"
    },
    {
      "turn": "0",
      "x": "1.117011813",
      "y": "1.226141852"
    },
    {
      "turn": "0",
      "x": "1.245464170",
      "y": "0.344851074"
    },
    {
      "turn": "0",
      "x": "1.246246469",
      "y": "1.655793627"
    },
    {
      "turn": "0",
      "x": "1.553979233",
      "y": "0.767764185"
    },
    {
      "turn": "0",
      "x": "1.565570638",
      "y": "1.216281613"
    }
  ],
  "radius": "0.224333593"
}

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