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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 20

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 = 20
ObjectiveMaximize circle radius
Best known, unproven0.176152871147Source-precision interval: 0.176152871146–0.176152871148 (values inside match)exhibited site certificate 0.176152870Packomania (compiled by Eckard Specht), m=6, n=20. 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 = 20, 0.17615287
Current leader

0.17615287

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": "0.898890919",
      "y": "0.310127467"
    },
    {
      "turn": "0",
      "x": "1.251196662",
      "y": "0.310127467"
    },
    {
      "turn": "0",
      "x": "0.555709557",
      "y": "0.389788555"
    },
    {
      "turn": "0",
      "x": "1.525755963",
      "y": "0.530890975"
    },
    {
      "turn": "0",
      "x": "0.887171904",
      "y": "0.662238246"
    },
    {
      "turn": "0",
      "x": "0.379556686",
      "y": "0.694894277"
    },
    {
      "turn": "0",
      "x": "1.233387510",
      "y": "0.727461633"
    },
    {
      "turn": "0",
      "x": "1.701908834",
      "y": "0.835996698"
    },
    {
      "turn": "0",
      "x": "0.649015615",
      "y": "0.921855115"
    },
    {
      "turn": "0",
      "x": "0.203403815",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.992104622",
      "y": "1.013555469"
    },
    {
      "turn": "0",
      "x": "1.361558346",
      "y": "1.055625575"
    },
    {
      "turn": "0",
      "x": "1.688037445",
      "y": "1.188029254"
    },
    {
      "turn": "0",
      "x": "0.472862744",
      "y": "1.226960837"
    },
    {
      "turn": "0",
      "x": "0.805498273",
      "y": "1.343033836"
    },
    {
      "turn": "0",
      "x": "1.157804015",
      "y": "1.343033836"
    },
    {
      "turn": "0",
      "x": "1.511884573",
      "y": "1.493134976"
    },
    {
      "turn": "0",
      "x": "0.534687130",
      "y": "1.573799534"
    },
    {
      "turn": "0",
      "x": "0.867322659",
      "y": "1.689872533"
    },
    {
      "turn": "0",
      "x": "1.219628402",
      "y": "1.689872533"
    }
  ],
  "radius": "0.176152870"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.898890919",
      "y": "0.310127467"
    },
    {
      "turn": "0",
      "x": "1.251196662",
      "y": "0.310127467"
    },
    {
      "turn": "0",
      "x": "0.555709557",
      "y": "0.389788555"
    },
    {
      "turn": "0",
      "x": "1.525755963",
      "y": "0.530890975"
    },
    {
      "turn": "0",
      "x": "0.887171904",
      "y": "0.662238246"
    },
    {
      "turn": "0",
      "x": "0.379556686",
      "y": "0.694894277"
    },
    {
      "turn": "0",
      "x": "1.233387510",
      "y": "0.727461633"
    },
    {
      "turn": "0",
      "x": "1.701908834",
      "y": "0.835996698"
    },
    {
      "turn": "0",
      "x": "0.649015615",
      "y": "0.921855115"
    },
    {
      "turn": "0",
      "x": "0.203403815",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.992104622",
      "y": "1.013555469"
    },
    {
      "turn": "0",
      "x": "1.361558346",
      "y": "1.055625575"
    },
    {
      "turn": "0",
      "x": "1.688037445",
      "y": "1.188029254"
    },
    {
      "turn": "0",
      "x": "0.472862744",
      "y": "1.226960837"
    },
    {
      "turn": "0",
      "x": "0.805498273",
      "y": "1.343033836"
    },
    {
      "turn": "0",
      "x": "1.157804015",
      "y": "1.343033836"
    },
    {
      "turn": "0",
      "x": "1.511884573",
      "y": "1.493134976"
    },
    {
      "turn": "0",
      "x": "0.534687130",
      "y": "1.573799534"
    },
    {
      "turn": "0",
      "x": "0.867322659",
      "y": "1.689872533"
    },
    {
      "turn": "0",
      "x": "1.219628402",
      "y": "1.689872533"
    }
  ],
  "radius": "0.176152870"
}

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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