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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 23

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 = 23
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.158923367388, 0.15892336739]Eckard SpechtPackomania ↗
Site record0.158923367w33dRecord 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 = 23, 0.158923367
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
    0.158923366→0.158923367
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": "0.349906373",
      "y": "1.472320667"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.041966153"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "0.527679333"
    },
    {
      "turn": "0",
      "x": "0.536731996",
      "y": "0.784822743"
    },
    {
      "turn": "0",
      "x": "0.583840754",
      "y": "1.257143410"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "1.567325155"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "0.432674844"
    },
    {
      "turn": "0",
      "x": "0.770666377",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.840048157",
      "y": "0.689818254"
    },
    {
      "turn": "0",
      "x": "0.899946291",
      "y": "1.290367440"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "1.668749676"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "0.331250323"
    },
    {
      "turn": "0",
      "x": "1.085892279",
      "y": "0.959266998"
    },
    {
      "turn": "0",
      "x": "1.141278324",
      "y": "0.588393733"
    },
    {
      "turn": "0",
      "x": "1.182401875",
      "y": "1.436125425"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "1.743481255"
    },
    {
      "turn": "0",
      "x": "1.263389165",
      "y": "0.256518746"
    },
    {
      "turn": "0",
      "x": "1.348193930",
      "y": "1.138778533"
    },
    {
      "turn": "0",
      "x": "1.372504653",
      "y": "0.821862872"
    },
    {
      "turn": "0",
      "x": "1.450214788",
      "y": "0.513662156"
    },
    {
      "turn": "0",
      "x": "1.499741088",
      "y": "1.418170740"
    },
    {
      "turn": "0",
      "x": "1.686566711",
      "y": "1.161027329"
    },
    {
      "turn": "0",
      "x": "1.689634705",
      "y": "0.843195402"
    }
  ],
  "radius": "0.158923367"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.472320667"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.041966153"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "0.527679333"
    },
    {
      "turn": "0",
      "x": "0.536731996",
      "y": "0.784822743"
    },
    {
      "turn": "0",
      "x": "0.583840754",
      "y": "1.257143410"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "1.567325155"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "0.432674844"
    },
    {
      "turn": "0",
      "x": "0.770666377",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.840048157",
      "y": "0.689818254"
    },
    {
      "turn": "0",
      "x": "0.899946291",
      "y": "1.290367440"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "1.668749676"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "0.331250323"
    },
    {
      "turn": "0",
      "x": "1.085892279",
      "y": "0.959266998"
    },
    {
      "turn": "0",
      "x": "1.141278324",
      "y": "0.588393733"
    },
    {
      "turn": "0",
      "x": "1.182401875",
      "y": "1.436125425"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "1.743481255"
    },
    {
      "turn": "0",
      "x": "1.263389165",
      "y": "0.256518746"
    },
    {
      "turn": "0",
      "x": "1.348193930",
      "y": "1.138778533"
    },
    {
      "turn": "0",
      "x": "1.372504653",
      "y": "0.821862872"
    },
    {
      "turn": "0",
      "x": "1.450214788",
      "y": "0.513662156"
    },
    {
      "turn": "0",
      "x": "1.499741088",
      "y": "1.418170740"
    },
    {
      "turn": "0",
      "x": "1.686566711",
      "y": "1.161027329"
    },
    {
      "turn": "0",
      "x": "1.689634705",
      "y": "0.843195402"
    }
  ],
  "radius": "0.158923367"
}

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