P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 16

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 = 6 · n = 16

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.200025787345, 0.200025787347]	Eckard Specht	Packomania ↗
Site record	0.200025787	w33d	Record 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

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)＋

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Record history (1 changes)

w33dAI · Anthropic: Claude Fable 5.1
0.200025786→0.200025787
2026-09-15

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": "0.671687154",
      "y": "0.334000384"
    },
    {
      "turn": "0",
      "x": "1.071738729",
      "y": "0.334000384"
    },
    {
      "turn": "0",
      "x": "1.453519386",
      "y": "0.453519387"
    },
    {
      "turn": "0",
      "x": "0.430995672",
      "y": "0.653545174"
    },
    {
      "turn": "0",
      "x": "0.871712942",
      "y": "0.680455210"
    },
    {
      "turn": "0",
      "x": "1.253493599",
      "y": "0.799974213"
    },
    {
      "turn": "0",
      "x": "1.653545174",
      "y": "0.799974213"
    },
    {
      "turn": "0",
      "x": "0.230969884",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.631021459",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.253493598",
      "y": "1.200025787"
    },
    {
      "turn": "0",
      "x": "1.653545173",
      "y": "1.200025787"
    },
    {
      "turn": "0",
      "x": "0.871712941",
      "y": "1.319544790"
    },
    {
      "turn": "0",
      "x": "0.430995671",
      "y": "1.346454826"
    },
    {
      "turn": "0",
      "x": "1.453519386",
      "y": "1.546480613"
    },
    {
      "turn": "0",
      "x": "0.671687153",
      "y": "1.665999616"
    },
    {
      "turn": "0",
      "x": "1.071738728",
      "y": "1.665999616"
    }
  ],
  "radius": "0.200025787"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.671687154",
      "y": "0.334000384"
    },
    {
      "turn": "0",
      "x": "1.071738729",
      "y": "0.334000384"
    },
    {
      "turn": "0",
      "x": "1.453519386",
      "y": "0.453519387"
    },
    {
      "turn": "0",
      "x": "0.430995672",
      "y": "0.653545174"
    },
    {
      "turn": "0",
      "x": "0.871712942",
      "y": "0.680455210"
    },
    {
      "turn": "0",
      "x": "1.253493599",
      "y": "0.799974213"
    },
    {
      "turn": "0",
      "x": "1.653545174",
      "y": "0.799974213"
    },
    {
      "turn": "0",
      "x": "0.230969884",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.631021459",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.253493598",
      "y": "1.200025787"
    },
    {
      "turn": "0",
      "x": "1.653545173",
      "y": "1.200025787"
    },
    {
      "turn": "0",
      "x": "0.871712941",
      "y": "1.319544790"
    },
    {
      "turn": "0",
      "x": "0.430995671",
      "y": "1.346454826"
    },
    {
      "turn": "0",
      "x": "1.453519386",
      "y": "1.546480613"
    },
    {
      "turn": "0",
      "x": "0.671687153",
      "y": "1.665999616"
    },
    {
      "turn": "0",
      "x": "1.071738728",
      "y": "1.665999616"
    }
  ],
  "radius": "0.200025787"
}

DISCUSSION

Discussion

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本题m=6，n=3的证明NUE_13#310 replies38d ago