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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 34

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 = 8 · n = 34
ObjectiveMaximize circle radius
Best known, unproven0.143477532364Source-precision interval: 0.143477532363–0.143477532365 (values inside match)exhibited site certificate 0.143477531Packomania (compiled by Eckard Specht), m=8, n=34. Eckard Specht, program coc , 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 = 8 · n = 34, 0.143477531
Current leader

0.143477531

circle radius

Answer sourceEckard Specht
Solution methodPublished reference construction
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Record holder's solver noteNo solver note yet (expand)＋

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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

{
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      "y": "0.339228544"
    },
    {
      "turn": "0",
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      "y": "0.168991350"
    },
    {
      "turn": "0",
      "x": "0.346465584",
      "y": "0.538482948"
    },
    {
      "turn": "0",
      "x": "0.830830073",
      "y": "0.421610238"
    },
    {
      "turn": "0",
      "x": "1.240705098",
      "y": "0.255002278"
    },
    {
      "turn": "0",
      "x": "0.622047751",
      "y": "0.618468439"
    },
    {
      "turn": "0",
      "x": "0.359647200",
      "y": "0.825135096"
    },
    {
      "turn": "0",
      "x": "1.122768060",
      "y": "0.518850195"
    },
    {
      "turn": "0",
      "x": "1.505817009",
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    {
      "turn": "0",
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      "y": "0.700850134"
    },
    {
      "turn": "0",
      "x": "0.173230817",
      "y": "1.043291328"
    },
    {
      "turn": "0",
      "x": "0.667013900",
      "y": "0.901878486"
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    {
      "turn": "0",
      "x": "1.391954454",
      "y": "0.628213268"
    },
    {
      "turn": "0",
      "x": "0.455367847",
      "y": "1.095654535"
    },
    {
      "turn": "0",
      "x": "1.164846489",
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    },
    {
      "turn": "0",
      "x": "1.676995126",
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    },
    {
      "turn": "0",
      "x": "0.941889264",
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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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    {
      "turn": "0",
      "x": "1.786808075",
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    },
    {
      "turn": "0",
      "x": "1.007982305",
      "y": "1.263500076"
    },
    {
      "turn": "0",
      "x": "0.652808399",
      "y": "1.456609494"
    },
    {
      "turn": "0",
      "x": "0.402706168",
      "y": "1.597293832"
    },
    {
      "turn": "0",
      "x": "1.499239991",
      "y": "1.184627839"
    },
    {
      "turn": "0",
      "x": "1.783740115",
      "y": "1.147172687"
    },
    {
      "turn": "0",
      "x": "1.275984280",
      "y": "1.373185276"
    },
    {
      "turn": "0",
      "x": "0.945845763",
      "y": "1.543646929"
    },
    {
      "turn": "0",
      "x": "0.730178533",
      "y": "1.732937327"
    },
    {
      "turn": "0",
      "x": "1.673927166",
      "y": "1.412284598"
    },
    {
      "turn": "0",
      "x": "1.210957674",
      "y": "1.653459878"
    },
    {
      "turn": "0",
      "x": "0.995290444",
      "y": "1.842750276"
    },
    {
      "turn": "0",
      "x": "1.497528653",
      "y": "1.638617922"
    }
  ],
  "radius": "0.143477531"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 34,
  "sides": 8
}

The current leader's answer

{
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  ],
  "radius": "0.143477531"
}

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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