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

Equal circles in regular polygons · m = 8 · n = 13

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 = 13
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.22353723521, 0.223537235212]Eckard SpechtPackomania ↗
Site record0.223537235w33dRecord 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 = 13, 0.223537235
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
    0.223537234→0.223537235
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": "0.699726357",
      "y": "0.366332375"
    },
    {
      "turn": "0",
      "x": "1.182358486",
      "y": "0.319809651"
    },
    {
      "turn": "0",
      "x": "0.376493576",
      "y": "0.675195049"
    },
    {
      "turn": "0",
      "x": "0.902555835",
      "y": "0.764748972"
    },
    {
      "turn": "0",
      "x": "1.569937041",
      "y": "0.550882476"
    },
    {
      "turn": "0",
      "x": "0.289021587",
      "y": "1.113628890"
    },
    {
      "turn": "0",
      "x": "0.731467841",
      "y": "1.177791925"
    },
    {
      "turn": "0",
      "x": "1.315598787",
      "y": "0.935836966"
    },
    {
      "turn": "0",
      "x": "0.463981211",
      "y": "1.536018787"
    },
    {
      "turn": "0",
      "x": "1.758045040",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.144510794",
      "y": "1.348879919"
    },
    {
      "turn": "0",
      "x": "0.877024164",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.586957047",
      "y": "1.413042953"
    }
  ],
  "radius": "0.223537235"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.699726357",
      "y": "0.366332375"
    },
    {
      "turn": "0",
      "x": "1.182358486",
      "y": "0.319809651"
    },
    {
      "turn": "0",
      "x": "0.376493576",
      "y": "0.675195049"
    },
    {
      "turn": "0",
      "x": "0.902555835",
      "y": "0.764748972"
    },
    {
      "turn": "0",
      "x": "1.569937041",
      "y": "0.550882476"
    },
    {
      "turn": "0",
      "x": "0.289021587",
      "y": "1.113628890"
    },
    {
      "turn": "0",
      "x": "0.731467841",
      "y": "1.177791925"
    },
    {
      "turn": "0",
      "x": "1.315598787",
      "y": "0.935836966"
    },
    {
      "turn": "0",
      "x": "0.463981211",
      "y": "1.536018787"
    },
    {
      "turn": "0",
      "x": "1.758045040",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.144510794",
      "y": "1.348879919"
    },
    {
      "turn": "0",
      "x": "0.877024164",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.586957047",
      "y": "1.413042953"
    }
  ],
  "radius": "0.223537235"
}

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