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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 37

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 = 37
ObjectiveMaximize circle radius
Best known, unproven0.138229583693Source-precision interval: 0.138229583692–0.138229583694 (values inside match)exhibited site certificate 0.138229583Packomania (compiled by Eckard Specht), m=8, n=37. 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 = 37, 0.138229583
Current leader

0.138229583

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": "0.561292408",
      "y": "0.331337258"
    },
    {
      "turn": "0",
      "x": "0.816707375",
      "y": "0.225540915"
    },
    {
      "turn": "0",
      "x": "1.090480505",
      "y": "0.187096876"
    },
    {
      "turn": "0",
      "x": "0.331965825",
      "y": "0.560180823"
    },
    {
      "turn": "0",
      "x": "1.345895472",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.605702273",
      "y": "0.604206155"
    },
    {
      "turn": "0",
      "x": "0.861117239",
      "y": "0.498409812"
    },
    {
      "turn": "0",
      "x": "0.225540915",
      "y": "0.816707375"
    },
    {
      "turn": "0",
      "x": "1.136436225",
      "y": "0.473327404"
    },
    {
      "turn": "0",
      "x": "1.601310438",
      "y": "0.398689562"
    },
    {
      "turn": "0",
      "x": "0.498725950",
      "y": "0.859129158"
    },
    {
      "turn": "0",
      "x": "0.840618505",
      "y": "0.774107968"
    },
    {
      "turn": "0",
      "x": "0.187096876",
      "y": "1.090480505"
    },
    {
      "turn": "0",
      "x": "1.392930152",
      "y": "0.595582159"
    },
    {
      "turn": "0",
      "x": "1.115937491",
      "y": "0.749025560"
    },
    {
      "turn": "0",
      "x": "0.724681014",
      "y": "1.025082407"
    },
    {
      "turn": "0",
      "x": "0.470649562",
      "y": "1.134158952"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.654104528"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.345895472"
    },
    {
      "turn": "0",
      "x": "1.529350438",
      "y": "0.865841048"
    },
    {
      "turn": "0",
      "x": "1.275318986",
      "y": "0.974917593"
    },
    {
      "turn": "0",
      "x": "0.884062509",
      "y": "1.250974440"
    },
    {
      "turn": "0",
      "x": "0.607087960",
      "y": "1.404496208"
    },
    {
      "turn": "0",
      "x": "1.812903124",
      "y": "0.909519495"
    },
    {
      "turn": "0",
      "x": "1.159381495",
      "y": "1.225892032"
    },
    {
      "turn": "0",
      "x": "1.501274050",
      "y": "1.140870842"
    },
    {
      "turn": "0",
      "x": "0.398689562",
      "y": "1.601310438"
    },
    {
      "turn": "0",
      "x": "0.863563775",
      "y": "1.526672596"
    },
    {
      "turn": "0",
      "x": "1.774459085",
      "y": "1.183292625"
    },
    {
      "turn": "0",
      "x": "1.138882761",
      "y": "1.501590188"
    },
    {
      "turn": "0",
      "x": "1.394297727",
      "y": "1.395793845"
    },
    {
      "turn": "0",
      "x": "0.654104528",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.668034817",
      "y": "1.439815235"
    },
    {
      "turn": "0",
      "x": "0.909519495",
      "y": "1.812903124"
    },
    {
      "turn": "0",
      "x": "1.183292625",
      "y": "1.774459085"
    },
    {
      "turn": "0",
      "x": "1.438707592",
      "y": "1.668662742"
    }
  ],
  "radius": "0.138229583"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 37,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.561292408",
      "y": "0.331337258"
    },
    {
      "turn": "0",
      "x": "0.816707375",
      "y": "0.225540915"
    },
    {
      "turn": "0",
      "x": "1.090480505",
      "y": "0.187096876"
    },
    {
      "turn": "0",
      "x": "0.331965825",
      "y": "0.560180823"
    },
    {
      "turn": "0",
      "x": "1.345895472",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.605702273",
      "y": "0.604206155"
    },
    {
      "turn": "0",
      "x": "0.861117239",
      "y": "0.498409812"
    },
    {
      "turn": "0",
      "x": "0.225540915",
      "y": "0.816707375"
    },
    {
      "turn": "0",
      "x": "1.136436225",
      "y": "0.473327404"
    },
    {
      "turn": "0",
      "x": "1.601310438",
      "y": "0.398689562"
    },
    {
      "turn": "0",
      "x": "0.498725950",
      "y": "0.859129158"
    },
    {
      "turn": "0",
      "x": "0.840618505",
      "y": "0.774107968"
    },
    {
      "turn": "0",
      "x": "0.187096876",
      "y": "1.090480505"
    },
    {
      "turn": "0",
      "x": "1.392930152",
      "y": "0.595582159"
    },
    {
      "turn": "0",
      "x": "1.115937491",
      "y": "0.749025560"
    },
    {
      "turn": "0",
      "x": "0.724681014",
      "y": "1.025082407"
    },
    {
      "turn": "0",
      "x": "0.470649562",
      "y": "1.134158952"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.654104528"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.345895472"
    },
    {
      "turn": "0",
      "x": "1.529350438",
      "y": "0.865841048"
    },
    {
      "turn": "0",
      "x": "1.275318986",
      "y": "0.974917593"
    },
    {
      "turn": "0",
      "x": "0.884062509",
      "y": "1.250974440"
    },
    {
      "turn": "0",
      "x": "0.607087960",
      "y": "1.404496208"
    },
    {
      "turn": "0",
      "x": "1.812903124",
      "y": "0.909519495"
    },
    {
      "turn": "0",
      "x": "1.159381495",
      "y": "1.225892032"
    },
    {
      "turn": "0",
      "x": "1.501274050",
      "y": "1.140870842"
    },
    {
      "turn": "0",
      "x": "0.398689562",
      "y": "1.601310438"
    },
    {
      "turn": "0",
      "x": "0.863563775",
      "y": "1.526672596"
    },
    {
      "turn": "0",
      "x": "1.774459085",
      "y": "1.183292625"
    },
    {
      "turn": "0",
      "x": "1.138882761",
      "y": "1.501590188"
    },
    {
      "turn": "0",
      "x": "1.394297727",
      "y": "1.395793845"
    },
    {
      "turn": "0",
      "x": "0.654104528",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.668034817",
      "y": "1.439815235"
    },
    {
      "turn": "0",
      "x": "0.909519495",
      "y": "1.812903124"
    },
    {
      "turn": "0",
      "x": "1.183292625",
      "y": "1.774459085"
    },
    {
      "turn": "0",
      "x": "1.438707592",
      "y": "1.668662742"
    }
  ],
  "radius": "0.138229583"
}

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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