MinMax Arena
中/EN
P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 36

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 = 36
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.139934447381, 0.139934447383]Eckard SpechtPackomania ↗
Site record0.139934447w33dRecord 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 = 36, 0.139934447
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
    0.139934446→0.139934447
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": "0.485703659",
      "y": "0.364492475"
    },
    {
      "turn": "0",
      "x": "0.744268802",
      "y": "0.257391285"
    },
    {
      "turn": "0",
      "x": "1.003986078",
      "y": "0.153115042"
    },
    {
      "turn": "0",
      "x": "0.320980483",
      "y": "0.590750897"
    },
    {
      "turn": "0",
      "x": "0.856283748",
      "y": "0.513865943"
    },
    {
      "turn": "0",
      "x": "0.598895338",
      "y": "0.623765008"
    },
    {
      "turn": "0",
      "x": "1.116001025",
      "y": "0.409589699"
    },
    {
      "turn": "0",
      "x": "1.376562523",
      "y": "0.307441259"
    },
    {
      "turn": "0",
      "x": "0.213879294",
      "y": "0.849316040"
    },
    {
      "turn": "0",
      "x": "0.491926270",
      "y": "0.885323101"
    },
    {
      "turn": "0",
      "x": "1.619340477",
      "y": "0.446672949"
    },
    {
      "turn": "0",
      "x": "0.821721712",
      "y": "0.793101965"
    },
    {
      "turn": "0",
      "x": "1.080794796",
      "y": "0.687235372"
    },
    {
      "turn": "0",
      "x": "1.341356294",
      "y": "0.585086932"
    },
    {
      "turn": "0",
      "x": "0.204914621",
      "y": "1.129041321"
    },
    {
      "turn": "0",
      "x": "0.482389198",
      "y": "1.165571543"
    },
    {
      "turn": "0",
      "x": "0.742157700",
      "y": "1.061422978"
    },
    {
      "turn": "0",
      "x": "1.584134247",
      "y": "0.724318622"
    },
    {
      "turn": "0",
      "x": "1.042941414",
      "y": "0.964532540"
    },
    {
      "turn": "0",
      "x": "1.303502912",
      "y": "0.862384100"
    },
    {
      "turn": "0",
      "x": "0.312015809",
      "y": "1.387606465"
    },
    {
      "turn": "0",
      "x": "1.546280865",
      "y": "1.001615790"
    },
    {
      "turn": "0",
      "x": "1.805353948",
      "y": "0.895749197"
    },
    {
      "turn": "0",
      "x": "0.642171457",
      "y": "1.397434802"
    },
    {
      "turn": "0",
      "x": "0.902369456",
      "y": "1.290898013"
    },
    {
      "turn": "0",
      "x": "1.171252613",
      "y": "1.213254945"
    },
    {
      "turn": "0",
      "x": "0.460279390",
      "y": "1.624976448"
    },
    {
      "turn": "0",
      "x": "1.446638677",
      "y": "1.263145966"
    },
    {
      "turn": "0",
      "x": "1.776414088",
      "y": "1.174117806"
    },
    {
      "turn": "0",
      "x": "0.737889415",
      "y": "1.660462674"
    },
    {
      "turn": "0",
      "x": "0.996855597",
      "y": "1.554334852"
    },
    {
      "turn": "0",
      "x": "1.265738754",
      "y": "1.476691783"
    },
    {
      "turn": "0",
      "x": "1.669312899",
      "y": "1.432682950"
    },
    {
      "turn": "0",
      "x": "0.959281896",
      "y": "1.831670055"
    },
    {
      "turn": "0",
      "x": "1.228165053",
      "y": "1.754026986"
    },
    {
      "turn": "0",
      "x": "1.488412976",
      "y": "1.646228767"
    }
  ],
  "radius": "0.139934447"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 36,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.485703659",
      "y": "0.364492475"
    },
    {
      "turn": "0",
      "x": "0.744268802",
      "y": "0.257391285"
    },
    {
      "turn": "0",
      "x": "1.003986078",
      "y": "0.153115042"
    },
    {
      "turn": "0",
      "x": "0.320980483",
      "y": "0.590750897"
    },
    {
      "turn": "0",
      "x": "0.856283748",
      "y": "0.513865943"
    },
    {
      "turn": "0",
      "x": "0.598895338",
      "y": "0.623765008"
    },
    {
      "turn": "0",
      "x": "1.116001025",
      "y": "0.409589699"
    },
    {
      "turn": "0",
      "x": "1.376562523",
      "y": "0.307441259"
    },
    {
      "turn": "0",
      "x": "0.213879294",
      "y": "0.849316040"
    },
    {
      "turn": "0",
      "x": "0.491926270",
      "y": "0.885323101"
    },
    {
      "turn": "0",
      "x": "1.619340477",
      "y": "0.446672949"
    },
    {
      "turn": "0",
      "x": "0.821721712",
      "y": "0.793101965"
    },
    {
      "turn": "0",
      "x": "1.080794796",
      "y": "0.687235372"
    },
    {
      "turn": "0",
      "x": "1.341356294",
      "y": "0.585086932"
    },
    {
      "turn": "0",
      "x": "0.204914621",
      "y": "1.129041321"
    },
    {
      "turn": "0",
      "x": "0.482389198",
      "y": "1.165571543"
    },
    {
      "turn": "0",
      "x": "0.742157700",
      "y": "1.061422978"
    },
    {
      "turn": "0",
      "x": "1.584134247",
      "y": "0.724318622"
    },
    {
      "turn": "0",
      "x": "1.042941414",
      "y": "0.964532540"
    },
    {
      "turn": "0",
      "x": "1.303502912",
      "y": "0.862384100"
    },
    {
      "turn": "0",
      "x": "0.312015809",
      "y": "1.387606465"
    },
    {
      "turn": "0",
      "x": "1.546280865",
      "y": "1.001615790"
    },
    {
      "turn": "0",
      "x": "1.805353948",
      "y": "0.895749197"
    },
    {
      "turn": "0",
      "x": "0.642171457",
      "y": "1.397434802"
    },
    {
      "turn": "0",
      "x": "0.902369456",
      "y": "1.290898013"
    },
    {
      "turn": "0",
      "x": "1.171252613",
      "y": "1.213254945"
    },
    {
      "turn": "0",
      "x": "0.460279390",
      "y": "1.624976448"
    },
    {
      "turn": "0",
      "x": "1.446638677",
      "y": "1.263145966"
    },
    {
      "turn": "0",
      "x": "1.776414088",
      "y": "1.174117806"
    },
    {
      "turn": "0",
      "x": "0.737889415",
      "y": "1.660462674"
    },
    {
      "turn": "0",
      "x": "0.996855597",
      "y": "1.554334852"
    },
    {
      "turn": "0",
      "x": "1.265738754",
      "y": "1.476691783"
    },
    {
      "turn": "0",
      "x": "1.669312899",
      "y": "1.432682950"
    },
    {
      "turn": "0",
      "x": "0.959281896",
      "y": "1.831670055"
    },
    {
      "turn": "0",
      "x": "1.228165053",
      "y": "1.754026986"
    },
    {
      "turn": "0",
      "x": "1.488412976",
      "y": "1.646228767"
    }
  ],
  "radius": "0.139934447"
}

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

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

本题m=6，n=3的证明NUE_13#310 replies