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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 22

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 = 22
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.175500940766, 0.175500940768]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).
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 = 22
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": "0.578883180",
      "y": "0.364393148"
    },
    {
      "turn": "0",
      "x": "0.903166634",
      "y": "0.230070543"
    },
    {
      "turn": "0",
      "x": "0.344873745",
      "y": "0.626007186"
    },
    {
      "turn": "0",
      "x": "1.248500722",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.783320450",
      "y": "0.649713535"
    },
    {
      "turn": "0",
      "x": "0.210551140",
      "y": "0.950290640"
    },
    {
      "turn": "0",
      "x": "1.572784176",
      "y": "0.427215824"
    },
    {
      "turn": "0",
      "x": "1.133414984",
      "y": "0.624491762"
    },
    {
      "turn": "0",
      "x": "0.560308881",
      "y": "0.920763643"
    },
    {
      "turn": "0",
      "x": "0.893422183",
      "y": "1.031388453"
    },
    {
      "turn": "0",
      "x": "1.536251060",
      "y": "0.776311301"
    },
    {
      "turn": "0",
      "x": "0.308932518",
      "y": "1.287223015"
    },
    {
      "turn": "0",
      "x": "1.237055256",
      "y": "0.959843838"
    },
    {
      "turn": "0",
      "x": "0.659879701",
      "y": "1.293419424"
    },
    {
      "turn": "0",
      "x": "1.808931653",
      "y": "0.997326265"
    },
    {
      "turn": "0",
      "x": "0.479039862",
      "y": "1.594250396"
    },
    {
      "turn": "0",
      "x": "0.984163155",
      "y": "1.427742029"
    },
    {
      "turn": "0",
      "x": "1.312144501",
      "y": "1.302719801"
    },
    {
      "turn": "0",
      "x": "1.676501947",
      "y": "1.322387327"
    },
    {
      "turn": "0",
      "x": "0.803323316",
      "y": "1.728573001"
    },
    {
      "turn": "0",
      "x": "1.153877213",
      "y": "1.746301122"
    },
    {
      "turn": "0",
      "x": "1.478160667",
      "y": "1.611978517"
    }
  ],
  "radius": "0.175500940"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 8
}

Starter answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.578883180",
      "y": "0.364393148"
    },
    {
      "turn": "0",
      "x": "0.903166634",
      "y": "0.230070543"
    },
    {
      "turn": "0",
      "x": "0.344873745",
      "y": "0.626007186"
    },
    {
      "turn": "0",
      "x": "1.248500722",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.783320450",
      "y": "0.649713535"
    },
    {
      "turn": "0",
      "x": "0.210551140",
      "y": "0.950290640"
    },
    {
      "turn": "0",
      "x": "1.572784176",
      "y": "0.427215824"
    },
    {
      "turn": "0",
      "x": "1.133414984",
      "y": "0.624491762"
    },
    {
      "turn": "0",
      "x": "0.560308881",
      "y": "0.920763643"
    },
    {
      "turn": "0",
      "x": "0.893422183",
      "y": "1.031388453"
    },
    {
      "turn": "0",
      "x": "1.536251060",
      "y": "0.776311301"
    },
    {
      "turn": "0",
      "x": "0.308932518",
      "y": "1.287223015"
    },
    {
      "turn": "0",
      "x": "1.237055256",
      "y": "0.959843838"
    },
    {
      "turn": "0",
      "x": "0.659879701",
      "y": "1.293419424"
    },
    {
      "turn": "0",
      "x": "1.808931653",
      "y": "0.997326265"
    },
    {
      "turn": "0",
      "x": "0.479039862",
      "y": "1.594250396"
    },
    {
      "turn": "0",
      "x": "0.984163155",
      "y": "1.427742029"
    },
    {
      "turn": "0",
      "x": "1.312144501",
      "y": "1.302719801"
    },
    {
      "turn": "0",
      "x": "1.676501947",
      "y": "1.322387327"
    },
    {
      "turn": "0",
      "x": "0.803323316",
      "y": "1.728573001"
    },
    {
      "turn": "0",
      "x": "1.153877213",
      "y": "1.746301122"
    },
    {
      "turn": "0",
      "x": "1.478160667",
      "y": "1.611978517"
    }
  ],
  "radius": "0.175500940"
}

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