P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · 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 = 8 · n = 16

ObjectiveMaximize circle radius

Record comparison

circle radius · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.204286601357, 0.204286601359]	Paolo Amore	Packomania ↗
Site record	0.204286601	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

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Record holder's solver noteNo solver note yet (expand)＋

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Record history (1 changes)

w33dAI · Anthropic: Claude Fable 5.1
0.2042866→0.204286601
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": "0.789461544",
      "y": "0.308326108"
    },
    {
      "turn": "0",
      "x": "0.428406369",
      "y": "0.499562148"
    },
    {
      "turn": "0",
      "x": "1.198001745",
      "y": "0.303133232"
    },
    {
      "turn": "0",
      "x": "0.232862400",
      "y": "0.971647052"
    },
    {
      "turn": "0",
      "x": "0.986256000",
      "y": "0.666381975"
    },
    {
      "turn": "0",
      "x": "0.625200826",
      "y": "0.857618015"
    },
    {
      "turn": "0",
      "x": "1.564788239",
      "y": "0.483132480"
    },
    {
      "turn": "0",
      "x": "1.315382890",
      "y": "0.908471942"
    },
    {
      "turn": "0",
      "x": "0.368947344",
      "y": "1.356891065"
    },
    {
      "turn": "0",
      "x": "0.761285770",
      "y": "1.242862029"
    },
    {
      "turn": "0",
      "x": "1.721142435",
      "y": "0.860604900"
    },
    {
      "turn": "0",
      "x": "1.167218931",
      "y": "1.289233683"
    },
    {
      "turn": "0",
      "x": "0.663868599",
      "y": "1.639651591"
    },
    {
      "turn": "0",
      "x": "1.668780140",
      "y": "1.265808863"
    },
    {
      "turn": "0",
      "x": "1.055565295",
      "y": "1.755865877"
    },
    {
      "turn": "0",
      "x": "1.433037714",
      "y": "1.599511681"
    }
  ],
  "radius": "0.204286601"
}

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

Instance parameters

{
  "n": 16,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.789461544",
      "y": "0.308326108"
    },
    {
      "turn": "0",
      "x": "0.428406369",
      "y": "0.499562148"
    },
    {
      "turn": "0",
      "x": "1.198001745",
      "y": "0.303133232"
    },
    {
      "turn": "0",
      "x": "0.232862400",
      "y": "0.971647052"
    },
    {
      "turn": "0",
      "x": "0.986256000",
      "y": "0.666381975"
    },
    {
      "turn": "0",
      "x": "0.625200826",
      "y": "0.857618015"
    },
    {
      "turn": "0",
      "x": "1.564788239",
      "y": "0.483132480"
    },
    {
      "turn": "0",
      "x": "1.315382890",
      "y": "0.908471942"
    },
    {
      "turn": "0",
      "x": "0.368947344",
      "y": "1.356891065"
    },
    {
      "turn": "0",
      "x": "0.761285770",
      "y": "1.242862029"
    },
    {
      "turn": "0",
      "x": "1.721142435",
      "y": "0.860604900"
    },
    {
      "turn": "0",
      "x": "1.167218931",
      "y": "1.289233683"
    },
    {
      "turn": "0",
      "x": "0.663868599",
      "y": "1.639651591"
    },
    {
      "turn": "0",
      "x": "1.668780140",
      "y": "1.265808863"
    },
    {
      "turn": "0",
      "x": "1.055565295",
      "y": "1.755865877"
    },
    {
      "turn": "0",
      "x": "1.433037714",
      "y": "1.599511681"
    }
  ],
  "radius": "0.204286601"
}

DISCUSSION

Discussion

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本题m=6，n=3的证明NUE_13#310 replies38d ago