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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 15

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 = 6 · n = 15
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.201559210542, 0.201559210544]Paolo AmorePackomania ↗
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).
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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 = 6 · n = 15
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

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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": "0.616370264",
      "y": "0.335533807"
    },
    {
      "turn": "0",
      "x": "1.019488686",
      "y": "0.335533807"
    },
    {
      "turn": "0",
      "x": "1.418148678",
      "y": "0.395322368"
    },
    {
      "turn": "0",
      "x": "0.616370264",
      "y": "0.738652228"
    },
    {
      "turn": "0",
      "x": "1.019488686",
      "y": "0.738652228"
    },
    {
      "turn": "0",
      "x": "1.418148678",
      "y": "0.798440789"
    },
    {
      "turn": "0",
      "x": "0.267259471",
      "y": "0.940211438"
    },
    {
      "turn": "0",
      "x": "1.767259471",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.763921847",
      "y": "1.113796189"
    },
    {
      "turn": "0",
      "x": "1.216589467",
      "y": "1.147551583"
    },
    {
      "turn": "0",
      "x": "0.414811054",
      "y": "1.315355400"
    },
    {
      "turn": "0",
      "x": "1.565700261",
      "y": "1.349110793"
    },
    {
      "turn": "0",
      "x": "0.965481058",
      "y": "1.462906983"
    },
    {
      "turn": "0",
      "x": "0.616370264",
      "y": "1.664466193"
    },
    {
      "turn": "0",
      "x": "1.314591851",
      "y": "1.664466193"
    }
  ],
  "radius": "0.201559210"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "sides": 6
}

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.616370264",
      "y": "0.335533807"
    },
    {
      "turn": "0",
      "x": "1.019488686",
      "y": "0.335533807"
    },
    {
      "turn": "0",
      "x": "1.418148678",
      "y": "0.395322368"
    },
    {
      "turn": "0",
      "x": "0.616370264",
      "y": "0.738652228"
    },
    {
      "turn": "0",
      "x": "1.019488686",
      "y": "0.738652228"
    },
    {
      "turn": "0",
      "x": "1.418148678",
      "y": "0.798440789"
    },
    {
      "turn": "0",
      "x": "0.267259471",
      "y": "0.940211438"
    },
    {
      "turn": "0",
      "x": "1.767259471",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.763921847",
      "y": "1.113796189"
    },
    {
      "turn": "0",
      "x": "1.216589467",
      "y": "1.147551583"
    },
    {
      "turn": "0",
      "x": "0.414811054",
      "y": "1.315355400"
    },
    {
      "turn": "0",
      "x": "1.565700261",
      "y": "1.349110793"
    },
    {
      "turn": "0",
      "x": "0.965481058",
      "y": "1.462906983"
    },
    {
      "turn": "0",
      "x": "0.616370264",
      "y": "1.664466193"
    },
    {
      "turn": "0",
      "x": "1.314591851",
      "y": "1.664466193"
    }
  ],
  "radius": "0.201559210"
}

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