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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 31

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 = 31
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.149878153405, 0.149878153407]Eckard SpechtPackomania ↗
Site record0.149878153w33dRecord 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 = 31, 0.149878153
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
    0.149878152→0.149878153
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": "0.638229777",
      "y": "0.312077077"
    },
    {
      "turn": "0",
      "x": "0.915168493",
      "y": "0.197365305"
    },
    {
      "turn": "0",
      "x": "0.382438813",
      "y": "0.468361519"
    },
    {
      "turn": "0",
      "x": "1.210379377",
      "y": "0.249368935"
    },
    {
      "turn": "0",
      "x": "0.267727041",
      "y": "0.745300235"
    },
    {
      "turn": "0",
      "x": "0.739027206",
      "y": "0.594377842"
    },
    {
      "turn": "0",
      "x": "1.015965923",
      "y": "0.479666069"
    },
    {
      "turn": "0",
      "x": "0.554978021",
      "y": "0.830978225"
    },
    {
      "turn": "0",
      "x": "1.321803704",
      "y": "0.527646607"
    },
    {
      "turn": "0",
      "x": "0.174768693",
      "y": "1.030278461"
    },
    {
      "turn": "0",
      "x": "1.613474896",
      "y": "0.458496341"
    },
    {
      "turn": "0",
      "x": "0.852161260",
      "y": "0.871964818"
    },
    {
      "turn": "0",
      "x": "1.129099976",
      "y": "0.757253046"
    },
    {
      "turn": "0",
      "x": "0.531079321",
      "y": "1.129780326"
    },
    {
      "turn": "0",
      "x": "1.436515477",
      "y": "0.804585324"
    },
    {
      "turn": "0",
      "x": "0.289480466",
      "y": "1.307217178"
    },
    {
      "turn": "0",
      "x": "1.728186669",
      "y": "0.735435059"
    },
    {
      "turn": "0",
      "x": "0.830927068",
      "y": "1.170968083"
    },
    {
      "turn": "0",
      "x": "1.243811749",
      "y": "1.034191762"
    },
    {
      "turn": "0",
      "x": "0.645791094",
      "y": "1.406719042"
    },
    {
      "turn": "0",
      "x": "1.539008621",
      "y": "1.086274867"
    },
    {
      "turn": "0",
      "x": "0.404192238",
      "y": "1.584155894"
    },
    {
      "turn": "0",
      "x": "1.102242490",
      "y": "1.298411346"
    },
    {
      "turn": "0",
      "x": "1.830679813",
      "y": "1.017124602"
    },
    {
      "turn": "0",
      "x": "0.932760390",
      "y": "1.545655479"
    },
    {
      "turn": "0",
      "x": "1.396851248",
      "y": "1.353724299"
    },
    {
      "turn": "0",
      "x": "0.678656487",
      "y": "1.704668214"
    },
    {
      "turn": "0",
      "x": "1.696354677",
      "y": "1.341414166"
    },
    {
      "turn": "0",
      "x": "1.228088349",
      "y": "1.624504064"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.837773056"
    },
    {
      "turn": "0",
      "x": "1.557263851",
      "y": "1.606946811"
    }
  ],
  "radius": "0.149878153"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 31,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.638229777",
      "y": "0.312077077"
    },
    {
      "turn": "0",
      "x": "0.915168493",
      "y": "0.197365305"
    },
    {
      "turn": "0",
      "x": "0.382438813",
      "y": "0.468361519"
    },
    {
      "turn": "0",
      "x": "1.210379377",
      "y": "0.249368935"
    },
    {
      "turn": "0",
      "x": "0.267727041",
      "y": "0.745300235"
    },
    {
      "turn": "0",
      "x": "0.739027206",
      "y": "0.594377842"
    },
    {
      "turn": "0",
      "x": "1.015965923",
      "y": "0.479666069"
    },
    {
      "turn": "0",
      "x": "0.554978021",
      "y": "0.830978225"
    },
    {
      "turn": "0",
      "x": "1.321803704",
      "y": "0.527646607"
    },
    {
      "turn": "0",
      "x": "0.174768693",
      "y": "1.030278461"
    },
    {
      "turn": "0",
      "x": "1.613474896",
      "y": "0.458496341"
    },
    {
      "turn": "0",
      "x": "0.852161260",
      "y": "0.871964818"
    },
    {
      "turn": "0",
      "x": "1.129099976",
      "y": "0.757253046"
    },
    {
      "turn": "0",
      "x": "0.531079321",
      "y": "1.129780326"
    },
    {
      "turn": "0",
      "x": "1.436515477",
      "y": "0.804585324"
    },
    {
      "turn": "0",
      "x": "0.289480466",
      "y": "1.307217178"
    },
    {
      "turn": "0",
      "x": "1.728186669",
      "y": "0.735435059"
    },
    {
      "turn": "0",
      "x": "0.830927068",
      "y": "1.170968083"
    },
    {
      "turn": "0",
      "x": "1.243811749",
      "y": "1.034191762"
    },
    {
      "turn": "0",
      "x": "0.645791094",
      "y": "1.406719042"
    },
    {
      "turn": "0",
      "x": "1.539008621",
      "y": "1.086274867"
    },
    {
      "turn": "0",
      "x": "0.404192238",
      "y": "1.584155894"
    },
    {
      "turn": "0",
      "x": "1.102242490",
      "y": "1.298411346"
    },
    {
      "turn": "0",
      "x": "1.830679813",
      "y": "1.017124602"
    },
    {
      "turn": "0",
      "x": "0.932760390",
      "y": "1.545655479"
    },
    {
      "turn": "0",
      "x": "1.396851248",
      "y": "1.353724299"
    },
    {
      "turn": "0",
      "x": "0.678656487",
      "y": "1.704668214"
    },
    {
      "turn": "0",
      "x": "1.696354677",
      "y": "1.341414166"
    },
    {
      "turn": "0",
      "x": "1.228088349",
      "y": "1.624504064"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.837773056"
    },
    {
      "turn": "0",
      "x": "1.557263851",
      "y": "1.606946811"
    }
  ],
  "radius": "0.149878153"
}

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