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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 30

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 = 30
ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[0.152319579408, 0.15231957941]Eckard SpechtPackomania ↗
Site record0.152319579w33dRecord 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 = 30, 0.152319579
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
    0.152319578→0.152319579
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": "0.738763877",
      "y": "0.273077070"
    },
    {
      "turn": "0",
      "x": "1.027626655",
      "y": "0.176312860"
    },
    {
      "turn": "0",
      "x": "0.495645041",
      "y": "0.478168596"
    },
    {
      "turn": "0",
      "x": "0.281449884",
      "y": "0.718550116"
    },
    {
      "turn": "0",
      "x": "1.309076539",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.805687506",
      "y": "0.570274386"
    },
    {
      "turn": "0",
      "x": "1.106170498",
      "y": "0.520124721"
    },
    {
      "turn": "0",
      "x": "0.580689196",
      "y": "0.775653979"
    },
    {
      "turn": "0",
      "x": "0.164869525",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.590526422",
      "y": "0.409473578"
    },
    {
      "turn": "0",
      "x": "1.387620381",
      "y": "0.636705080"
    },
    {
      "turn": "0",
      "x": "0.934400975",
      "y": "0.846386429"
    },
    {
      "turn": "0",
      "x": "0.464108837",
      "y": "1.057103863"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.690923461"
    },
    {
      "turn": "0",
      "x": "1.234294772",
      "y": "0.899946936"
    },
    {
      "turn": "0",
      "x": "0.765705228",
      "y": "1.100053064"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.309076539"
    },
    {
      "turn": "0",
      "x": "1.535891163",
      "y": "0.942896137"
    },
    {
      "turn": "0",
      "x": "1.065599025",
      "y": "1.153613571"
    },
    {
      "turn": "0",
      "x": "0.612379617",
      "y": "1.363294919"
    },
    {
      "turn": "0",
      "x": "0.409473578",
      "y": "1.590526422"
    },
    {
      "turn": "0",
      "x": "1.835130475",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.419310804",
      "y": "1.224346021"
    },
    {
      "turn": "0",
      "x": "0.893829502",
      "y": "1.479875279"
    },
    {
      "turn": "0",
      "x": "1.194312494",
      "y": "1.429725614"
    },
    {
      "turn": "0",
      "x": "0.690923461",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.718550116",
      "y": "1.281449884"
    },
    {
      "turn": "0",
      "x": "1.511866093",
      "y": "1.523568988"
    },
    {
      "turn": "0",
      "x": "0.972373345",
      "y": "1.823687140"
    },
    {
      "turn": "0",
      "x": "1.261236123",
      "y": "1.726922930"
    }
  ],
  "radius": "0.152319579"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.738763877",
      "y": "0.273077070"
    },
    {
      "turn": "0",
      "x": "1.027626655",
      "y": "0.176312860"
    },
    {
      "turn": "0",
      "x": "0.495645041",
      "y": "0.478168596"
    },
    {
      "turn": "0",
      "x": "0.281449884",
      "y": "0.718550116"
    },
    {
      "turn": "0",
      "x": "1.309076539",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.805687506",
      "y": "0.570274386"
    },
    {
      "turn": "0",
      "x": "1.106170498",
      "y": "0.520124721"
    },
    {
      "turn": "0",
      "x": "0.580689196",
      "y": "0.775653979"
    },
    {
      "turn": "0",
      "x": "0.164869525",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.590526422",
      "y": "0.409473578"
    },
    {
      "turn": "0",
      "x": "1.387620381",
      "y": "0.636705080"
    },
    {
      "turn": "0",
      "x": "0.934400975",
      "y": "0.846386429"
    },
    {
      "turn": "0",
      "x": "0.464108837",
      "y": "1.057103863"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.690923461"
    },
    {
      "turn": "0",
      "x": "1.234294772",
      "y": "0.899946936"
    },
    {
      "turn": "0",
      "x": "0.765705228",
      "y": "1.100053064"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.309076539"
    },
    {
      "turn": "0",
      "x": "1.535891163",
      "y": "0.942896137"
    },
    {
      "turn": "0",
      "x": "1.065599025",
      "y": "1.153613571"
    },
    {
      "turn": "0",
      "x": "0.612379617",
      "y": "1.363294919"
    },
    {
      "turn": "0",
      "x": "0.409473578",
      "y": "1.590526422"
    },
    {
      "turn": "0",
      "x": "1.835130475",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.419310804",
      "y": "1.224346021"
    },
    {
      "turn": "0",
      "x": "0.893829502",
      "y": "1.479875279"
    },
    {
      "turn": "0",
      "x": "1.194312494",
      "y": "1.429725614"
    },
    {
      "turn": "0",
      "x": "0.690923461",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.718550116",
      "y": "1.281449884"
    },
    {
      "turn": "0",
      "x": "1.511866093",
      "y": "1.523568988"
    },
    {
      "turn": "0",
      "x": "0.972373345",
      "y": "1.823687140"
    },
    {
      "turn": "0",
      "x": "1.261236123",
      "y": "1.726922930"
    }
  ],
  "radius": "0.152319579"
}

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