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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 40

Pack n equal circles into the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1), maximising their radius.

Instancem = 6 · n = 40
ObjectiveMaximize circle radius
Best known, unproven0.129711778826Source-precision interval: 0.129711778825–0.129711778827 (values inside match)exhibited site certificate 0.129711778Packomania (compiled by Eckard Specht), m=6, n=40. Eckard Specht, program chx , 2020. Coordinates imported from the public archive, normalized, decimalized and exactly verified by MinMax Arena. The web-table target and grid certificate are separate.

Formal definition

  • ContainerThe editor and certificate normalize to the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCircle radius r, larger is better; the container circumradius is 1.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equal circles in regular polygons m = 6 · n = 40, 0.129711778
Current leader

0.129711778

circle radius

Answer sourceEckard Specht
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
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      "x": "0.790742637",
      "y": "0.263686375"
    },
    {
      "turn": "0",
      "x": "1.050166195",
      "y": "0.263686375"
    },
    {
      "turn": "0",
      "x": "1.309589752",
      "y": "0.263686375"
    },
    {
      "turn": "0",
      "x": "0.538913597",
      "y": "0.325997826"
    },
    {
      "turn": "0",
      "x": "1.515909051",
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    },
    {
      "turn": "0",
      "x": "0.920454416",
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    },
    {
      "turn": "0",
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    {
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    {
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    {
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      "y": "0.713021158"
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
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      "y": "1.579046561"
    },
    {
      "turn": "0",
      "x": "0.538913597",
      "y": "1.674002174"
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    {
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      "y": "1.736313625"
    },
    {
      "turn": "0",
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      "y": "1.736313625"
    },
    {
      "turn": "0",
      "x": "1.309589752",
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  ],
  "radius": "0.129711778"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 40,
  "sides": 6
}

The current leader's answer

{
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      "y": "0.263686375"
    },
    {
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    {
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    },
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    {
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  ],
  "radius": "0.129711778"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

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