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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 21

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 = 21
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.180670260258, 0.18067026026]Eckard SpechtPackomania ↗
Site record0.18067026w33dRecord 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 = 21, 0.18067026
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
    0.180670259→0.18067026
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": "0.678086425",
      "y": "0.328897049"
    },
    {
      "turn": "0",
      "x": "1.035288788",
      "y": "0.211460335"
    },
    {
      "turn": "0",
      "x": "0.385631727",
      "y": "0.541116798"
    },
    {
      "turn": "0",
      "x": "1.419083975",
      "y": "0.369146347"
    },
    {
      "turn": "0",
      "x": "0.247352696",
      "y": "0.874951909"
    },
    {
      "turn": "0",
      "x": "0.843509747",
      "y": "0.650147880"
    },
    {
      "turn": "0",
      "x": "1.204715127",
      "y": "0.660029431"
    },
    {
      "turn": "0",
      "x": "0.605601898",
      "y": "0.922116311"
    },
    {
      "turn": "0",
      "x": "1.656107992",
      "y": "0.641885393"
    },
    {
      "turn": "0",
      "x": "0.292270680",
      "y": "1.233489698"
    },
    {
      "turn": "0",
      "x": "0.950255540",
      "y": "1.055142567"
    },
    {
      "turn": "0",
      "x": "1.441739144",
      "y": "0.932768477"
    },
    {
      "turn": "0",
      "x": "0.650519882",
      "y": "1.280654100"
    },
    {
      "turn": "0",
      "x": "1.799006211",
      "y": "0.986872213"
    },
    {
      "turn": "0",
      "x": "0.430549711",
      "y": "1.567324809"
    },
    {
      "turn": "0",
      "x": "1.312796330",
      "y": "1.270319543"
    },
    {
      "turn": "0",
      "x": "0.983913101",
      "y": "1.419995170"
    },
    {
      "turn": "0",
      "x": "1.670063397",
      "y": "1.324423279"
    },
    {
      "turn": "0",
      "x": "0.763942929",
      "y": "1.706665879"
    },
    {
      "turn": "0",
      "x": "1.122192131",
      "y": "1.753830281"
    },
    {
      "turn": "0",
      "x": "1.456027242",
      "y": "1.615551251"
    }
  ],
  "radius": "0.180670260"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.678086425",
      "y": "0.328897049"
    },
    {
      "turn": "0",
      "x": "1.035288788",
      "y": "0.211460335"
    },
    {
      "turn": "0",
      "x": "0.385631727",
      "y": "0.541116798"
    },
    {
      "turn": "0",
      "x": "1.419083975",
      "y": "0.369146347"
    },
    {
      "turn": "0",
      "x": "0.247352696",
      "y": "0.874951909"
    },
    {
      "turn": "0",
      "x": "0.843509747",
      "y": "0.650147880"
    },
    {
      "turn": "0",
      "x": "1.204715127",
      "y": "0.660029431"
    },
    {
      "turn": "0",
      "x": "0.605601898",
      "y": "0.922116311"
    },
    {
      "turn": "0",
      "x": "1.656107992",
      "y": "0.641885393"
    },
    {
      "turn": "0",
      "x": "0.292270680",
      "y": "1.233489698"
    },
    {
      "turn": "0",
      "x": "0.950255540",
      "y": "1.055142567"
    },
    {
      "turn": "0",
      "x": "1.441739144",
      "y": "0.932768477"
    },
    {
      "turn": "0",
      "x": "0.650519882",
      "y": "1.280654100"
    },
    {
      "turn": "0",
      "x": "1.799006211",
      "y": "0.986872213"
    },
    {
      "turn": "0",
      "x": "0.430549711",
      "y": "1.567324809"
    },
    {
      "turn": "0",
      "x": "1.312796330",
      "y": "1.270319543"
    },
    {
      "turn": "0",
      "x": "0.983913101",
      "y": "1.419995170"
    },
    {
      "turn": "0",
      "x": "1.670063397",
      "y": "1.324423279"
    },
    {
      "turn": "0",
      "x": "0.763942929",
      "y": "1.706665879"
    },
    {
      "turn": "0",
      "x": "1.122192131",
      "y": "1.753830281"
    },
    {
      "turn": "0",
      "x": "1.456027242",
      "y": "1.615551251"
    }
  ],
  "radius": "0.180670260"
}

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