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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 25

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 = 25
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.166450891365, 0.166450891367]Paolo AmorePackomania ↗
Site record0.166450891w33dRecord details ↓

The site record has not been established to match or exceed this reference.

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: site record

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 record-holding arrangement for Equal circles in regular polygons m = 8 · n = 25, 0.166450891
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderw33d
Solution methodAI · Anthropic: Claude Fable 5.1
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (1 changes)
  1. w33dAI · Anthropic: Claude Fable 5.1
    0.16645089→0.166450891
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.

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.420289216",
      "y": "0.420289216"
    },
    {
      "turn": "0",
      "x": "0.955377095",
      "y": "0.198648559"
    },
    {
      "turn": "0",
      "x": "0.750671001",
      "y": "0.461172866"
    },
    {
      "turn": "0",
      "x": "1.256476965",
      "y": "0.340643156"
    },
    {
      "turn": "0",
      "x": "0.464256960",
      "y": "0.750274722"
    },
    {
      "turn": "0",
      "x": "0.199832080",
      "y": "0.952519822"
    },
    {
      "turn": "0",
      "x": "1.051770870",
      "y": "0.603167463"
    },
    {
      "turn": "0",
      "x": "1.579710784",
      "y": "0.420289216"
    },
    {
      "turn": "0",
      "x": "0.791599982",
      "y": "0.810856427"
    },
    {
      "turn": "0",
      "x": "1.375004690",
      "y": "0.682813523"
    },
    {
      "turn": "0",
      "x": "0.557318098",
      "y": "1.069904542"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.272149641"
    },
    {
      "turn": "0",
      "x": "1.146508336",
      "y": "0.924914769"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.727850359"
    },
    {
      "turn": "0",
      "x": "0.884661120",
      "y": "1.130486246"
    },
    {
      "turn": "0",
      "x": "1.470422342",
      "y": "1.001747793"
    },
    {
      "turn": "0",
      "x": "0.708369572",
      "y": "1.412877618"
    },
    {
      "turn": "0",
      "x": "0.420289216",
      "y": "1.579710784"
    },
    {
      "turn": "0",
      "x": "1.800167919",
      "y": "1.047480178"
    },
    {
      "turn": "0",
      "x": "1.343026345",
      "y": "1.309308936"
    },
    {
      "turn": "0",
      "x": "1.039594454",
      "y": "1.446249377"
    },
    {
      "turn": "0",
      "x": "1.672771923",
      "y": "1.355041322"
    },
    {
      "turn": "0",
      "x": "0.794104282",
      "y": "1.734550055"
    },
    {
      "turn": "0",
      "x": "1.471815301",
      "y": "1.624289014"
    },
    {
      "turn": "0",
      "x": "1.125329164",
      "y": "1.767921814"
    }
  ],
  "radius": "0.166450891"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.420289216",
      "y": "0.420289216"
    },
    {
      "turn": "0",
      "x": "0.955377095",
      "y": "0.198648559"
    },
    {
      "turn": "0",
      "x": "0.750671001",
      "y": "0.461172866"
    },
    {
      "turn": "0",
      "x": "1.256476965",
      "y": "0.340643156"
    },
    {
      "turn": "0",
      "x": "0.464256960",
      "y": "0.750274722"
    },
    {
      "turn": "0",
      "x": "0.199832080",
      "y": "0.952519822"
    },
    {
      "turn": "0",
      "x": "1.051770870",
      "y": "0.603167463"
    },
    {
      "turn": "0",
      "x": "1.579710784",
      "y": "0.420289216"
    },
    {
      "turn": "0",
      "x": "0.791599982",
      "y": "0.810856427"
    },
    {
      "turn": "0",
      "x": "1.375004690",
      "y": "0.682813523"
    },
    {
      "turn": "0",
      "x": "0.557318098",
      "y": "1.069904542"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.272149641"
    },
    {
      "turn": "0",
      "x": "1.146508336",
      "y": "0.924914769"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.727850359"
    },
    {
      "turn": "0",
      "x": "0.884661120",
      "y": "1.130486246"
    },
    {
      "turn": "0",
      "x": "1.470422342",
      "y": "1.001747793"
    },
    {
      "turn": "0",
      "x": "0.708369572",
      "y": "1.412877618"
    },
    {
      "turn": "0",
      "x": "0.420289216",
      "y": "1.579710784"
    },
    {
      "turn": "0",
      "x": "1.800167919",
      "y": "1.047480178"
    },
    {
      "turn": "0",
      "x": "1.343026345",
      "y": "1.309308936"
    },
    {
      "turn": "0",
      "x": "1.039594454",
      "y": "1.446249377"
    },
    {
      "turn": "0",
      "x": "1.672771923",
      "y": "1.355041322"
    },
    {
      "turn": "0",
      "x": "0.794104282",
      "y": "1.734550055"
    },
    {
      "turn": "0",
      "x": "1.471815301",
      "y": "1.624289014"
    },
    {
      "turn": "0",
      "x": "1.125329164",
      "y": "1.767921814"
    }
  ],
  "radius": "0.166450891"
}

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