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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 26

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 = 26
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.163606136258, 0.16360613626]Paolo AmorePackomania ↗
Site record0.163606136w33dRecord 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 = 26, 0.163606136
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
    0.163606135→0.163606136
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": "0.997247695",
      "y": "0.178226048"
    },
    {
      "turn": "0",
      "x": "0.527592545",
      "y": "0.372927963"
    },
    {
      "turn": "0",
      "x": "0.327033380",
      "y": "0.637995015"
    },
    {
      "turn": "0",
      "x": "0.841497851",
      "y": "0.465993067"
    },
    {
      "turn": "0",
      "x": "1.224893317",
      "y": "0.413269325"
    },
    {
      "turn": "0",
      "x": "0.642303642",
      "y": "0.725588017"
    },
    {
      "turn": "0",
      "x": "0.201814665",
      "y": "0.940299737"
    },
    {
      "turn": "0",
      "x": "1.552018945",
      "y": "0.405739740"
    },
    {
      "turn": "0",
      "x": "0.966716568",
      "y": "0.768297788"
    },
    {
      "turn": "0",
      "x": "0.517084926",
      "y": "1.027892738"
    },
    {
      "turn": "0",
      "x": "1.292074422",
      "y": "0.733510750"
    },
    {
      "turn": "0",
      "x": "0.283592032",
      "y": "1.257128293"
    },
    {
      "turn": "0",
      "x": "1.668477173",
      "y": "0.711526189"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "1.070602509"
    },
    {
      "turn": "0",
      "x": "1.166855706",
      "y": "1.035815471"
    },
    {
      "turn": "0",
      "x": "0.642303642",
      "y": "1.330197459"
    },
    {
      "turn": "0",
      "x": "1.495869935",
      "y": "0.989509305"
    },
    {
      "turn": "0",
      "x": "0.408810748",
      "y": "1.559433014"
    },
    {
      "turn": "0",
      "x": "1.822913994",
      "y": "1.000000001"
    },
    {
      "turn": "0",
      "x": "0.970127676",
      "y": "1.373205624"
    },
    {
      "turn": "0",
      "x": "1.370651219",
      "y": "1.291814026"
    },
    {
      "turn": "0",
      "x": "0.724081009",
      "y": "1.647026014"
    },
    {
      "turn": "0",
      "x": "1.697695278",
      "y": "1.302304722"
    },
    {
      "turn": "0",
      "x": "1.223837030",
      "y": "1.584240880"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.822913994"
    },
    {
      "turn": "0",
      "x": "1.550881089",
      "y": "1.594731576"
    }
  ],
  "radius": "0.163606136"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.997247695",
      "y": "0.178226048"
    },
    {
      "turn": "0",
      "x": "0.527592545",
      "y": "0.372927963"
    },
    {
      "turn": "0",
      "x": "0.327033380",
      "y": "0.637995015"
    },
    {
      "turn": "0",
      "x": "0.841497851",
      "y": "0.465993067"
    },
    {
      "turn": "0",
      "x": "1.224893317",
      "y": "0.413269325"
    },
    {
      "turn": "0",
      "x": "0.642303642",
      "y": "0.725588017"
    },
    {
      "turn": "0",
      "x": "0.201814665",
      "y": "0.940299737"
    },
    {
      "turn": "0",
      "x": "1.552018945",
      "y": "0.405739740"
    },
    {
      "turn": "0",
      "x": "0.966716568",
      "y": "0.768297788"
    },
    {
      "turn": "0",
      "x": "0.517084926",
      "y": "1.027892738"
    },
    {
      "turn": "0",
      "x": "1.292074422",
      "y": "0.733510750"
    },
    {
      "turn": "0",
      "x": "0.283592032",
      "y": "1.257128293"
    },
    {
      "turn": "0",
      "x": "1.668477173",
      "y": "0.711526189"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "1.070602509"
    },
    {
      "turn": "0",
      "x": "1.166855706",
      "y": "1.035815471"
    },
    {
      "turn": "0",
      "x": "0.642303642",
      "y": "1.330197459"
    },
    {
      "turn": "0",
      "x": "1.495869935",
      "y": "0.989509305"
    },
    {
      "turn": "0",
      "x": "0.408810748",
      "y": "1.559433014"
    },
    {
      "turn": "0",
      "x": "1.822913994",
      "y": "1.000000001"
    },
    {
      "turn": "0",
      "x": "0.970127676",
      "y": "1.373205624"
    },
    {
      "turn": "0",
      "x": "1.370651219",
      "y": "1.291814026"
    },
    {
      "turn": "0",
      "x": "0.724081009",
      "y": "1.647026014"
    },
    {
      "turn": "0",
      "x": "1.697695278",
      "y": "1.302304722"
    },
    {
      "turn": "0",
      "x": "1.223837030",
      "y": "1.584240880"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.822913994"
    },
    {
      "turn": "0",
      "x": "1.550881089",
      "y": "1.594731576"
    }
  ],
  "radius": "0.163606136"
}

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