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

Equal circles in regular polygons · m = 8 · n = 38

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 = 38
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.136097981332, 0.136097981334]Paolo AmorePackomania ↗
Site record0.136097981w33dRecord 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 = 38, 0.136097981
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
    0.13609798→0.136097981
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": "0.573571170",
      "y": "0.323943998"
    },
    {
      "turn": "0",
      "x": "0.825047451",
      "y": "0.219779113"
    },
    {
      "turn": "0",
      "x": "0.358328125",
      "y": "0.490560544"
    },
    {
      "turn": "0",
      "x": "1.095233058",
      "y": "0.186758218"
    },
    {
      "turn": "0",
      "x": "0.857830114",
      "y": "0.489993730"
    },
    {
      "turn": "0",
      "x": "0.254163240",
      "y": "0.742036823"
    },
    {
      "turn": "0",
      "x": "0.608201941",
      "y": "0.598512932"
    },
    {
      "turn": "0",
      "x": "1.349181390",
      "y": "0.291947061"
    },
    {
      "turn": "0",
      "x": "1.132295322",
      "y": "0.456419173"
    },
    {
      "turn": "0",
      "x": "0.149998354",
      "y": "0.993513103"
    },
    {
      "turn": "0",
      "x": "0.504037056",
      "y": "0.849989211"
    },
    {
      "turn": "0",
      "x": "1.602941895",
      "y": "0.397058104"
    },
    {
      "turn": "0",
      "x": "1.023532518",
      "y": "0.705941305"
    },
    {
      "turn": "0",
      "x": "0.773904345",
      "y": "0.814460508"
    },
    {
      "turn": "0",
      "x": "1.386126997",
      "y": "0.561624024"
    },
    {
      "turn": "0",
      "x": "0.399872170",
      "y": "1.101465490"
    },
    {
      "turn": "0",
      "x": "1.275008796",
      "y": "0.810106191"
    },
    {
      "turn": "0",
      "x": "0.669739459",
      "y": "1.065936787"
    },
    {
      "turn": "0",
      "x": "1.703388588",
      "y": "0.650042499"
    },
    {
      "turn": "0",
      "x": "0.994315784",
      "y": "0.976564705"
    },
    {
      "turn": "0",
      "x": "0.293040160",
      "y": "1.351820365"
    },
    {
      "turn": "0",
      "x": "1.540887354",
      "y": "0.868409143"
    },
    {
      "turn": "0",
      "x": "0.563269847",
      "y": "1.319162162"
    },
    {
      "turn": "0",
      "x": "0.888813639",
      "y": "1.227482898"
    },
    {
      "turn": "0",
      "x": "1.245792062",
      "y": "1.080729591"
    },
    {
      "turn": "0",
      "x": "1.811249032",
      "y": "0.899956017"
    },
    {
      "turn": "0",
      "x": "1.511670620",
      "y": "1.139032543"
    },
    {
      "turn": "0",
      "x": "0.783860544",
      "y": "1.478631240"
    },
    {
      "turn": "0",
      "x": "1.140289917",
      "y": "1.331647784"
    },
    {
      "turn": "0",
      "x": "0.462549601",
      "y": "1.628934058"
    },
    {
      "turn": "0",
      "x": "1.782032298",
      "y": "1.170579417"
    },
    {
      "turn": "0",
      "x": "1.406168477",
      "y": "1.389950736"
    },
    {
      "turn": "0",
      "x": "1.035336822",
      "y": "1.582796126"
    },
    {
      "turn": "0",
      "x": "0.748523721",
      "y": "1.748523720"
    },
    {
      "turn": "0",
      "x": "1.677603467",
      "y": "1.422520333"
    },
    {
      "turn": "0",
      "x": "1.251402156",
      "y": "1.748344952"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.852688606"
    },
    {
      "turn": "0",
      "x": "1.502952432",
      "y": "1.644358888"
    }
  ],
  "radius": "0.136097981"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 38,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.573571170",
      "y": "0.323943998"
    },
    {
      "turn": "0",
      "x": "0.825047451",
      "y": "0.219779113"
    },
    {
      "turn": "0",
      "x": "0.358328125",
      "y": "0.490560544"
    },
    {
      "turn": "0",
      "x": "1.095233058",
      "y": "0.186758218"
    },
    {
      "turn": "0",
      "x": "0.857830114",
      "y": "0.489993730"
    },
    {
      "turn": "0",
      "x": "0.254163240",
      "y": "0.742036823"
    },
    {
      "turn": "0",
      "x": "0.608201941",
      "y": "0.598512932"
    },
    {
      "turn": "0",
      "x": "1.349181390",
      "y": "0.291947061"
    },
    {
      "turn": "0",
      "x": "1.132295322",
      "y": "0.456419173"
    },
    {
      "turn": "0",
      "x": "0.149998354",
      "y": "0.993513103"
    },
    {
      "turn": "0",
      "x": "0.504037056",
      "y": "0.849989211"
    },
    {
      "turn": "0",
      "x": "1.602941895",
      "y": "0.397058104"
    },
    {
      "turn": "0",
      "x": "1.023532518",
      "y": "0.705941305"
    },
    {
      "turn": "0",
      "x": "0.773904345",
      "y": "0.814460508"
    },
    {
      "turn": "0",
      "x": "1.386126997",
      "y": "0.561624024"
    },
    {
      "turn": "0",
      "x": "0.399872170",
      "y": "1.101465490"
    },
    {
      "turn": "0",
      "x": "1.275008796",
      "y": "0.810106191"
    },
    {
      "turn": "0",
      "x": "0.669739459",
      "y": "1.065936787"
    },
    {
      "turn": "0",
      "x": "1.703388588",
      "y": "0.650042499"
    },
    {
      "turn": "0",
      "x": "0.994315784",
      "y": "0.976564705"
    },
    {
      "turn": "0",
      "x": "0.293040160",
      "y": "1.351820365"
    },
    {
      "turn": "0",
      "x": "1.540887354",
      "y": "0.868409143"
    },
    {
      "turn": "0",
      "x": "0.563269847",
      "y": "1.319162162"
    },
    {
      "turn": "0",
      "x": "0.888813639",
      "y": "1.227482898"
    },
    {
      "turn": "0",
      "x": "1.245792062",
      "y": "1.080729591"
    },
    {
      "turn": "0",
      "x": "1.811249032",
      "y": "0.899956017"
    },
    {
      "turn": "0",
      "x": "1.511670620",
      "y": "1.139032543"
    },
    {
      "turn": "0",
      "x": "0.783860544",
      "y": "1.478631240"
    },
    {
      "turn": "0",
      "x": "1.140289917",
      "y": "1.331647784"
    },
    {
      "turn": "0",
      "x": "0.462549601",
      "y": "1.628934058"
    },
    {
      "turn": "0",
      "x": "1.782032298",
      "y": "1.170579417"
    },
    {
      "turn": "0",
      "x": "1.406168477",
      "y": "1.389950736"
    },
    {
      "turn": "0",
      "x": "1.035336822",
      "y": "1.582796126"
    },
    {
      "turn": "0",
      "x": "0.748523721",
      "y": "1.748523720"
    },
    {
      "turn": "0",
      "x": "1.677603467",
      "y": "1.422520333"
    },
    {
      "turn": "0",
      "x": "1.251402156",
      "y": "1.748344952"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.852688606"
    },
    {
      "turn": "0",
      "x": "1.502952432",
      "y": "1.644358888"
    }
  ],
  "radius": "0.136097981"
}

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