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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 19

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 = 19
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.191508243903, 0.191508243905]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).
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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 = 8 · n = 19
LAYOUT & HISTORY

Starter layout

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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": "0.451626612",
      "y": "0.434430724"
    },
    {
      "turn": "0",
      "x": "0.805487706",
      "y": "0.287856660"
    },
    {
      "turn": "0",
      "x": "1.188496087",
      "y": "0.285364665"
    },
    {
      "turn": "0",
      "x": "0.296497877",
      "y": "0.784625963"
    },
    {
      "turn": "0",
      "x": "0.809096629",
      "y": "0.670856145"
    },
    {
      "turn": "0",
      "x": "1.542357181",
      "y": "0.431938729"
    },
    {
      "turn": "0",
      "x": "1.196477159",
      "y": "0.668297991"
    },
    {
      "turn": "0",
      "x": "0.613224342",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.296497877",
      "y": "1.215374037"
    },
    {
      "turn": "0",
      "x": "1.004968915",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.701686585",
      "y": "0.780242865"
    },
    {
      "turn": "0",
      "x": "1.387985403",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.809096629",
      "y": "1.329143855"
    },
    {
      "turn": "0",
      "x": "0.451626612",
      "y": "1.565569276"
    },
    {
      "turn": "0",
      "x": "1.196477159",
      "y": "1.331702009"
    },
    {
      "turn": "0",
      "x": "1.701686585",
      "y": "1.219757135"
    },
    {
      "turn": "0",
      "x": "0.805487706",
      "y": "1.712143340"
    },
    {
      "turn": "0",
      "x": "1.188496087",
      "y": "1.714635335"
    },
    {
      "turn": "0",
      "x": "1.542357181",
      "y": "1.568061271"
    }
  ],
  "radius": "0.191508243"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 8
}

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.451626612",
      "y": "0.434430724"
    },
    {
      "turn": "0",
      "x": "0.805487706",
      "y": "0.287856660"
    },
    {
      "turn": "0",
      "x": "1.188496087",
      "y": "0.285364665"
    },
    {
      "turn": "0",
      "x": "0.296497877",
      "y": "0.784625963"
    },
    {
      "turn": "0",
      "x": "0.809096629",
      "y": "0.670856145"
    },
    {
      "turn": "0",
      "x": "1.542357181",
      "y": "0.431938729"
    },
    {
      "turn": "0",
      "x": "1.196477159",
      "y": "0.668297991"
    },
    {
      "turn": "0",
      "x": "0.613224342",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.296497877",
      "y": "1.215374037"
    },
    {
      "turn": "0",
      "x": "1.004968915",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.701686585",
      "y": "0.780242865"
    },
    {
      "turn": "0",
      "x": "1.387985403",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.809096629",
      "y": "1.329143855"
    },
    {
      "turn": "0",
      "x": "0.451626612",
      "y": "1.565569276"
    },
    {
      "turn": "0",
      "x": "1.196477159",
      "y": "1.331702009"
    },
    {
      "turn": "0",
      "x": "1.701686585",
      "y": "1.219757135"
    },
    {
      "turn": "0",
      "x": "0.805487706",
      "y": "1.712143340"
    },
    {
      "turn": "0",
      "x": "1.188496087",
      "y": "1.714635335"
    },
    {
      "turn": "0",
      "x": "1.542357181",
      "y": "1.568061271"
    }
  ],
  "radius": "0.191508243"
}

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