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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 17

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 = 17
ObjectiveMaximize circle radius
Best known, unproven0.194813312769Source-precision interval: 0.194813312768–0.19481331277 (values inside match)exhibited site certificate 0.194813312Packomania (compiled by Eckard Specht), m=6, n=17. Paolo Amore, Circle packing in regular polygons , Phys. Fluids 35 , 027130 (2023). 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 = 17, 0.194813312
Current leader

0.194813312

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

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": "0.663571921",
      "y": "0.328787909"
    },
    {
      "turn": "0",
      "x": "1.363659581",
      "y": "0.328787909"
    },
    {
      "turn": "0",
      "x": "1.013615751",
      "y": "0.499896717"
    },
    {
      "turn": "0",
      "x": "0.431696800",
      "y": "0.641905835"
    },
    {
      "turn": "0",
      "x": "1.576097300",
      "y": "0.655405611"
    },
    {
      "turn": "0",
      "x": "1.005625554",
      "y": "0.889441406"
    },
    {
      "turn": "0",
      "x": "0.236883487",
      "y": "0.979332390"
    },
    {
      "turn": "0",
      "x": "0.626510113",
      "y": "0.979332390"
    },
    {
      "turn": "0",
      "x": "1.381283987",
      "y": "0.992832167"
    },
    {
      "turn": "0",
      "x": "1.770910612",
      "y": "0.992832167"
    },
    {
      "turn": "0",
      "x": "0.413523855",
      "y": "1.326617702"
    },
    {
      "turn": "0",
      "x": "0.803150481",
      "y": "1.326617702"
    },
    {
      "turn": "0",
      "x": "1.192711169",
      "y": "1.333785535"
    },
    {
      "turn": "0",
      "x": "1.582337794",
      "y": "1.333785535"
    },
    {
      "turn": "0",
      "x": "0.608337168",
      "y": "1.664044258"
    },
    {
      "turn": "0",
      "x": "0.997897856",
      "y": "1.671212091"
    },
    {
      "turn": "0",
      "x": "1.387524481",
      "y": "1.671212091"
    }
  ],
  "radius": "0.194813312"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.663571921",
      "y": "0.328787909"
    },
    {
      "turn": "0",
      "x": "1.363659581",
      "y": "0.328787909"
    },
    {
      "turn": "0",
      "x": "1.013615751",
      "y": "0.499896717"
    },
    {
      "turn": "0",
      "x": "0.431696800",
      "y": "0.641905835"
    },
    {
      "turn": "0",
      "x": "1.576097300",
      "y": "0.655405611"
    },
    {
      "turn": "0",
      "x": "1.005625554",
      "y": "0.889441406"
    },
    {
      "turn": "0",
      "x": "0.236883487",
      "y": "0.979332390"
    },
    {
      "turn": "0",
      "x": "0.626510113",
      "y": "0.979332390"
    },
    {
      "turn": "0",
      "x": "1.381283987",
      "y": "0.992832167"
    },
    {
      "turn": "0",
      "x": "1.770910612",
      "y": "0.992832167"
    },
    {
      "turn": "0",
      "x": "0.413523855",
      "y": "1.326617702"
    },
    {
      "turn": "0",
      "x": "0.803150481",
      "y": "1.326617702"
    },
    {
      "turn": "0",
      "x": "1.192711169",
      "y": "1.333785535"
    },
    {
      "turn": "0",
      "x": "1.582337794",
      "y": "1.333785535"
    },
    {
      "turn": "0",
      "x": "0.608337168",
      "y": "1.664044258"
    },
    {
      "turn": "0",
      "x": "0.997897856",
      "y": "1.671212091"
    },
    {
      "turn": "0",
      "x": "1.387524481",
      "y": "1.671212091"
    }
  ],
  "radius": "0.194813312"
}

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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