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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 32

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 = 32
ObjectiveMaximize circle radius
Best known, unproven0.147282442434Source-precision interval: 0.147282442433–0.147282442435 (values inside match)exhibited site certificate 0.147282442Packomania (compiled by Eckard Specht), m=8, n=32. 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 = 32, 0.147282442
Current leader

0.147282442

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": [
    {
      "turn": "0",
      "x": "0.649291608",
      "y": "0.304685539"
    },
    {
      "turn": "0",
      "x": "0.921434076",
      "y": "0.191960438"
    },
    {
      "turn": "0",
      "x": "0.386641040",
      "y": "0.453290653"
    },
    {
      "turn": "0",
      "x": "1.210857253",
      "y": "0.246757301"
    },
    {
      "turn": "0",
      "x": "0.272142468",
      "y": "0.727857532"
    },
    {
      "turn": "0",
      "x": "0.755526223",
      "y": "0.579426650"
    },
    {
      "turn": "0",
      "x": "1.033490761",
      "y": "0.481937230"
    },
    {
      "turn": "0",
      "x": "0.558202065",
      "y": "0.798131127"
    },
    {
      "turn": "0",
      "x": "0.159417367",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.330811589",
      "y": "0.515791552"
    },
    {
      "turn": "0",
      "x": "1.622229504",
      "y": "0.472848914"
    },
    {
      "turn": "0",
      "x": "0.875480559",
      "y": "0.848460902"
    },
    {
      "turn": "0",
      "x": "1.153445097",
      "y": "0.750971482"
    },
    {
      "turn": "0",
      "x": "0.445476964",
      "y": "1.070273595"
    },
    {
      "turn": "0",
      "x": "1.444684495",
      "y": "0.795108592"
    },
    {
      "turn": "0",
      "x": "0.737521806",
      "y": "1.108722028"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.322239212"
    },
    {
      "turn": "0",
      "x": "1.734954605",
      "y": "0.744991382"
    },
    {
      "turn": "0",
      "x": "1.249334111",
      "y": "1.029492160"
    },
    {
      "turn": "0",
      "x": "0.584938060",
      "y": "1.360687645"
    },
    {
      "turn": "0",
      "x": "1.014905872",
      "y": "1.207850990"
    },
    {
      "turn": "0",
      "x": "1.540573509",
      "y": "1.073629270"
    },
    {
      "turn": "0",
      "x": "0.405618320",
      "y": "1.594381680"
    },
    {
      "turn": "0",
      "x": "1.830843619",
      "y": "1.023512060"
    },
    {
      "turn": "0",
      "x": "0.857080528",
      "y": "1.473412746"
    },
    {
      "turn": "0",
      "x": "1.417036569",
      "y": "1.341037378"
    },
    {
      "turn": "0",
      "x": "1.151608884",
      "y": "1.468773886"
    },
    {
      "turn": "0",
      "x": "0.677760788",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.710387253",
      "y": "1.314319453"
    },
    {
      "turn": "0",
      "x": "0.973600743",
      "y": "1.824958386"
    },
    {
      "turn": "0",
      "x": "1.597662152",
      "y": "1.586461921"
    },
    {
      "turn": "0",
      "x": "1.327896953",
      "y": "1.704763268"
    }
  ],
  "radius": "0.147282442"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.649291608",
      "y": "0.304685539"
    },
    {
      "turn": "0",
      "x": "0.921434076",
      "y": "0.191960438"
    },
    {
      "turn": "0",
      "x": "0.386641040",
      "y": "0.453290653"
    },
    {
      "turn": "0",
      "x": "1.210857253",
      "y": "0.246757301"
    },
    {
      "turn": "0",
      "x": "0.272142468",
      "y": "0.727857532"
    },
    {
      "turn": "0",
      "x": "0.755526223",
      "y": "0.579426650"
    },
    {
      "turn": "0",
      "x": "1.033490761",
      "y": "0.481937230"
    },
    {
      "turn": "0",
      "x": "0.558202065",
      "y": "0.798131127"
    },
    {
      "turn": "0",
      "x": "0.159417367",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.330811589",
      "y": "0.515791552"
    },
    {
      "turn": "0",
      "x": "1.622229504",
      "y": "0.472848914"
    },
    {
      "turn": "0",
      "x": "0.875480559",
      "y": "0.848460902"
    },
    {
      "turn": "0",
      "x": "1.153445097",
      "y": "0.750971482"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    {
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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      "y": "1.029492160"
    },
    {
      "turn": "0",
      "x": "0.584938060",
      "y": "1.360687645"
    },
    {
      "turn": "0",
      "x": "1.014905872",
      "y": "1.207850990"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "0.857080528",
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    {
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    {
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    {
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    {
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      "y": "1.314319453"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.597662152",
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    },
    {
      "turn": "0",
      "x": "1.327896953",
      "y": "1.704763268"
    }
  ],
  "radius": "0.147282442"
}

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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