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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 12

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 = 12
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.234100964946, 0.234100964948]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 = 8 · n = 12
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": "0.641616747",
      "y": "0.401836263"
    },
    {
      "turn": "0",
      "x": "1.097142557",
      "y": "0.293626823"
    },
    {
      "turn": "0",
      "x": "0.349572865",
      "y": "0.767791750"
    },
    {
      "turn": "0",
      "x": "1.528969018",
      "y": "0.474565916"
    },
    {
      "turn": "0",
      "x": "1.156357937",
      "y": "0.758069054"
    },
    {
      "turn": "0",
      "x": "0.755828805",
      "y": "1.000534437"
    },
    {
      "turn": "0",
      "x": "0.351139755",
      "y": "1.235991058"
    },
    {
      "turn": "0",
      "x": "1.166074552",
      "y": "1.226170149"
    },
    {
      "turn": "0",
      "x": "1.703232941",
      "y": "1.018527687"
    },
    {
      "turn": "0",
      "x": "0.645756461",
      "y": "1.599878463"
    },
    {
      "turn": "0",
      "x": "1.531883793",
      "y": "1.518397195"
    },
    {
      "turn": "0",
      "x": "1.102167343",
      "y": "1.704291842"
    }
  ],
  "radius": "0.234100964"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 8
}

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.641616747",
      "y": "0.401836263"
    },
    {
      "turn": "0",
      "x": "1.097142557",
      "y": "0.293626823"
    },
    {
      "turn": "0",
      "x": "0.349572865",
      "y": "0.767791750"
    },
    {
      "turn": "0",
      "x": "1.528969018",
      "y": "0.474565916"
    },
    {
      "turn": "0",
      "x": "1.156357937",
      "y": "0.758069054"
    },
    {
      "turn": "0",
      "x": "0.755828805",
      "y": "1.000534437"
    },
    {
      "turn": "0",
      "x": "0.351139755",
      "y": "1.235991058"
    },
    {
      "turn": "0",
      "x": "1.166074552",
      "y": "1.226170149"
    },
    {
      "turn": "0",
      "x": "1.703232941",
      "y": "1.018527687"
    },
    {
      "turn": "0",
      "x": "0.645756461",
      "y": "1.599878463"
    },
    {
      "turn": "0",
      "x": "1.531883793",
      "y": "1.518397195"
    },
    {
      "turn": "0",
      "x": "1.102167343",
      "y": "1.704291842"
    }
  ],
  "radius": "0.234100964"
}

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