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

Equal circles in regular polygons · m = 8 · 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 = 8 · n = 23
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.172561559661, 0.172561559663]Eckard SpechtPackomania ↗
Site recordNo player record yet——

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: starter layout

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 current arrangement for Equal circles in regular polygons m = 8 · n = 23
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
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.

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.599898627",
      "y": "0.352506701"
    },
    {
      "turn": "0",
      "x": "0.918750813",
      "y": "0.220433801"
    },
    {
      "turn": "0",
      "x": "0.352506701",
      "y": "0.599898627"
    },
    {
      "turn": "0",
      "x": "1.256181695",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.686569707",
      "y": "0.686569707"
    },
    {
      "turn": "0",
      "x": "1.005421893",
      "y": "0.554496807"
    },
    {
      "turn": "0",
      "x": "0.220433801",
      "y": "0.918750813"
    },
    {
      "turn": "0",
      "x": "1.575033881",
      "y": "0.424966119"
    },
    {
      "turn": "0",
      "x": "1.330994983",
      "y": "0.669005017"
    },
    {
      "turn": "0",
      "x": "0.554496807",
      "y": "1.005421893"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.256181695"
    },
    {
      "turn": "0",
      "x": "1.011552730",
      "y": "1.023691546"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.743818305"
    },
    {
      "turn": "0",
      "x": "1.445503193",
      "y": "0.994578107"
    },
    {
      "turn": "0",
      "x": "0.669005017",
      "y": "1.330994983"
    },
    {
      "turn": "0",
      "x": "0.424966119",
      "y": "1.575033881"
    },
    {
      "turn": "0",
      "x": "1.779566199",
      "y": "1.081249187"
    },
    {
      "turn": "0",
      "x": "0.994578107",
      "y": "1.445503193"
    },
    {
      "turn": "0",
      "x": "1.313430293",
      "y": "1.313430293"
    },
    {
      "turn": "0",
      "x": "0.743818305",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.647493299",
      "y": "1.400101373"
    },
    {
      "turn": "0",
      "x": "1.081249187",
      "y": "1.779566199"
    },
    {
      "turn": "0",
      "x": "1.400101373",
      "y": "1.647493299"
    }
  ],
  "radius": "0.172561559"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 8
}

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.599898627",
      "y": "0.352506701"
    },
    {
      "turn": "0",
      "x": "0.918750813",
      "y": "0.220433801"
    },
    {
      "turn": "0",
      "x": "0.352506701",
      "y": "0.599898627"
    },
    {
      "turn": "0",
      "x": "1.256181695",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.686569707",
      "y": "0.686569707"
    },
    {
      "turn": "0",
      "x": "1.005421893",
      "y": "0.554496807"
    },
    {
      "turn": "0",
      "x": "0.220433801",
      "y": "0.918750813"
    },
    {
      "turn": "0",
      "x": "1.575033881",
      "y": "0.424966119"
    },
    {
      "turn": "0",
      "x": "1.330994983",
      "y": "0.669005017"
    },
    {
      "turn": "0",
      "x": "0.554496807",
      "y": "1.005421893"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.256181695"
    },
    {
      "turn": "0",
      "x": "1.011552730",
      "y": "1.023691546"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.743818305"
    },
    {
      "turn": "0",
      "x": "1.445503193",
      "y": "0.994578107"
    },
    {
      "turn": "0",
      "x": "0.669005017",
      "y": "1.330994983"
    },
    {
      "turn": "0",
      "x": "0.424966119",
      "y": "1.575033881"
    },
    {
      "turn": "0",
      "x": "1.779566199",
      "y": "1.081249187"
    },
    {
      "turn": "0",
      "x": "0.994578107",
      "y": "1.445503193"
    },
    {
      "turn": "0",
      "x": "1.313430293",
      "y": "1.313430293"
    },
    {
      "turn": "0",
      "x": "0.743818305",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.647493299",
      "y": "1.400101373"
    },
    {
      "turn": "0",
      "x": "1.081249187",
      "y": "1.779566199"
    },
    {
      "turn": "0",
      "x": "1.400101373",
      "y": "1.647493299"
    }
  ],
  "radius": "0.172561559"
}

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