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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 35

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 = 35
ObjectiveMaximize circle radius
Best known, unproven0.142256369270Source-precision interval: 0.142256369269–0.142256369271 (values inside match)exhibited site certificate 0.142256368Packomania (compiled by Eckard Specht), m=8, n=35. 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 = 35, 0.142256368
Current leader

0.142256368

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

{
  "placements": [
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      "turn": "0",
      "x": "0.437016306",
      "y": "0.387172666"
    },
    {
      "turn": "0",
      "x": "0.911413586",
      "y": "0.190670879"
    },
    {
      "turn": "0",
      "x": "0.721132412",
      "y": "0.402190555"
    },
    {
      "turn": "0",
      "x": "1.192642603",
      "y": "0.233772364"
    },
    {
      "turn": "0",
      "x": "0.303706649",
      "y": "0.638521097"
    },
    {
      "turn": "0",
      "x": "1.011541067",
      "y": "0.457322854"
    },
    {
      "turn": "0",
      "x": "0.587822754",
      "y": "0.653538985"
    },
    {
      "turn": "0",
      "x": "1.551961351",
      "y": "0.382607062"
    },
    {
      "turn": "0",
      "x": "1.292825284",
      "y": "0.500063424"
    },
    {
      "turn": "0",
      "x": "0.866986662",
      "y": "0.708448100"
    },
    {
      "turn": "0",
      "x": "0.411099289",
      "y": "0.901987070"
    },
    {
      "turn": "0",
      "x": "0.184814413",
      "y": "1.074447655"
    },
    {
      "turn": "0",
      "x": "0.693786281",
      "y": "0.934167232"
    },
    {
      "turn": "0",
      "x": "1.149065355",
      "y": "0.745584465"
    },
    {
      "turn": "0",
      "x": "1.692708845",
      "y": "0.629867332"
    },
    {
      "turn": "0",
      "x": "1.433572777",
      "y": "0.747323695"
    },
    {
      "turn": "0",
      "x": "0.975864974",
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    },
    {
      "turn": "0",
      "x": "0.519977600",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.801587156",
      "y": "0.892722828"
    },
    {
      "turn": "0",
      "x": "1.542451089",
      "y": "1.010179191"
    },
    {
      "turn": "0",
      "x": "1.084743285",
      "y": "1.234159092"
    },
    {
      "turn": "0",
      "x": "0.633452259",
      "y": "1.425746763"
    },
    {
      "turn": "0",
      "x": "0.404491264",
      "y": "1.599355020"
    },
    {
      "turn": "0",
      "x": "1.433572777",
      "y": "1.273034686"
    },
    {
      "turn": "0",
      "x": "0.912223486",
      "y": "1.506826270"
    },
    {
      "turn": "0",
      "x": "1.772417276",
      "y": "1.177699491"
    },
    {
      "turn": "0",
      "x": "0.667346760",
      "y": "1.708233332"
    },
    {
      "turn": "0",
      "x": "1.234354397",
      "y": "1.476159120"
    },
    {
      "turn": "0",
      "x": "1.663538965",
      "y": "1.440554987"
    },
    {
      "turn": "0",
      "x": "0.930202255",
      "y": "1.817111643"
    },
    {
      "turn": "0",
      "x": "1.208826000",
      "y": "1.759524254"
    },
    {
      "turn": "0",
      "x": "1.471681496",
      "y": "1.650645943"
    }
  ],
  "radius": "0.142256368"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35,
  "sides": 8
}

The current leader's answer

{
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      "x": "0.437016306",
      "y": "0.387172666"
    },
    {
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      "y": "0.190670879"
    },
    {
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      "y": "0.402190555"
    },
    {
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      "y": "0.233772364"
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    {
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      "y": "0.638521097"
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    {
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      "y": "0.457322854"
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    {
      "turn": "0",
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      "y": "0.653538985"
    },
    {
      "turn": "0",
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    {
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    {
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    {
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    }
  ],
  "radius": "0.142256368"
}

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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