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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 21

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 = 21
ObjectiveMaximize circle radius
Best known, unproven0.180670260259Source-precision interval: 0.180670260258–0.18067026026 (values inside match)exhibited site certificate 0.180670259Packomania (compiled by Eckard Specht), m=8, n=21. 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 = 21, 0.180670259
Current leader

0.180670259

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": "0.678086424",
      "y": "0.328897049"
    },
    {
      "turn": "0",
      "x": "1.035288788",
      "y": "0.211460335"
    },
    {
      "turn": "0",
      "x": "0.385631727",
      "y": "0.541116797"
    },
    {
      "turn": "0",
      "x": "1.419083975",
      "y": "0.369146347"
    },
    {
      "turn": "0",
      "x": "0.247352696",
      "y": "0.874951909"
    },
    {
      "turn": "0",
      "x": "0.843509747",
      "y": "0.650147880"
    },
    {
      "turn": "0",
      "x": "1.204715127",
      "y": "0.660029431"
    },
    {
      "turn": "0",
      "x": "0.605601898",
      "y": "0.922116311"
    },
    {
      "turn": "0",
      "x": "1.656107992",
      "y": "0.641885392"
    },
    {
      "turn": "0",
      "x": "0.292270680",
      "y": "1.233489698"
    },
    {
      "turn": "0",
      "x": "0.950255540",
      "y": "1.055142567"
    },
    {
      "turn": "0",
      "x": "1.441739144",
      "y": "0.932768477"
    },
    {
      "turn": "0",
      "x": "0.650519882",
      "y": "1.280654100"
    },
    {
      "turn": "0",
      "x": "1.799006212",
      "y": "0.986872212"
    },
    {
      "turn": "0",
      "x": "0.430549711",
      "y": "1.567324809"
    },
    {
      "turn": "0",
      "x": "1.312796330",
      "y": "1.270319543"
    },
    {
      "turn": "0",
      "x": "0.983913101",
      "y": "1.419995170"
    },
    {
      "turn": "0",
      "x": "1.670063397",
      "y": "1.324423279"
    },
    {
      "turn": "0",
      "x": "0.763942929",
      "y": "1.706665879"
    },
    {
      "turn": "0",
      "x": "1.122192131",
      "y": "1.753830281"
    },
    {
      "turn": "0",
      "x": "1.456027243",
      "y": "1.615551251"
    }
  ],
  "radius": "0.180670259"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.678086424",
      "y": "0.328897049"
    },
    {
      "turn": "0",
      "x": "1.035288788",
      "y": "0.211460335"
    },
    {
      "turn": "0",
      "x": "0.385631727",
      "y": "0.541116797"
    },
    {
      "turn": "0",
      "x": "1.419083975",
      "y": "0.369146347"
    },
    {
      "turn": "0",
      "x": "0.247352696",
      "y": "0.874951909"
    },
    {
      "turn": "0",
      "x": "0.843509747",
      "y": "0.650147880"
    },
    {
      "turn": "0",
      "x": "1.204715127",
      "y": "0.660029431"
    },
    {
      "turn": "0",
      "x": "0.605601898",
      "y": "0.922116311"
    },
    {
      "turn": "0",
      "x": "1.656107992",
      "y": "0.641885392"
    },
    {
      "turn": "0",
      "x": "0.292270680",
      "y": "1.233489698"
    },
    {
      "turn": "0",
      "x": "0.950255540",
      "y": "1.055142567"
    },
    {
      "turn": "0",
      "x": "1.441739144",
      "y": "0.932768477"
    },
    {
      "turn": "0",
      "x": "0.650519882",
      "y": "1.280654100"
    },
    {
      "turn": "0",
      "x": "1.799006212",
      "y": "0.986872212"
    },
    {
      "turn": "0",
      "x": "0.430549711",
      "y": "1.567324809"
    },
    {
      "turn": "0",
      "x": "1.312796330",
      "y": "1.270319543"
    },
    {
      "turn": "0",
      "x": "0.983913101",
      "y": "1.419995170"
    },
    {
      "turn": "0",
      "x": "1.670063397",
      "y": "1.324423279"
    },
    {
      "turn": "0",
      "x": "0.763942929",
      "y": "1.706665879"
    },
    {
      "turn": "0",
      "x": "1.122192131",
      "y": "1.753830281"
    },
    {
      "turn": "0",
      "x": "1.456027243",
      "y": "1.615551251"
    }
  ],
  "radius": "0.180670259"
}

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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