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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 27

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 = 5 · n = 27
ObjectiveMaximize circle radius
Best known, unproven0.148137584795Source-precision interval: 0.148137584794–0.148137584796 (values inside match)exhibited site certificate 0.148137584Packomania (compiled by Eckard Specht), m=5, n=27. Eckard Specht, program cpt , 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.
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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 = 5 · n = 27, 0.148137584
Current leader

0.148137584

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": "0.339120590",
      "y": "1.115080444"
    },
    {
      "turn": "0",
      "x": "0.339120590",
      "y": "0.818805274"
    },
    {
      "turn": "0",
      "x": "0.339282390",
      "y": "1.480039712"
    },
    {
      "turn": "0",
      "x": "0.364739198",
      "y": "0.512621983"
    },
    {
      "turn": "0",
      "x": "0.572650398",
      "y": "1.297406539"
    },
    {
      "turn": "0",
      "x": "0.613784525",
      "y": "1.004000745"
    },
    {
      "turn": "0",
      "x": "0.613784525",
      "y": "0.707725576"
    },
    {
      "turn": "0",
      "x": "0.653716526",
      "y": "1.582375413"
    },
    {
      "turn": "0",
      "x": "0.661043641",
      "y": "0.415243863"
    },
    {
      "turn": "0",
      "x": "0.861037515",
      "y": "1.370722433"
    },
    {
      "turn": "0",
      "x": "0.895558955",
      "y": "0.912446683"
    },
    {
      "turn": "0",
      "x": "0.895558955",
      "y": "0.616171513"
    },
    {
      "turn": "0",
      "x": "0.935490957",
      "y": "1.673929475"
    },
    {
      "turn": "0",
      "x": "0.942818071",
      "y": "0.323689801"
    },
    {
      "turn": "0",
      "x": "1.064790924",
      "y": "1.155632860"
    },
    {
      "turn": "0",
      "x": "1.142811945",
      "y": "1.462276496"
    },
    {
      "turn": "0",
      "x": "1.172485106",
      "y": "0.807133422"
    },
    {
      "turn": "0",
      "x": "1.172485106",
      "y": "0.510858252"
    },
    {
      "turn": "0",
      "x": "1.240980685",
      "y": "1.772086134"
    },
    {
      "turn": "0",
      "x": "1.252032372",
      "y": "0.224322954"
    },
    {
      "turn": "0",
      "x": "1.346565355",
      "y": "1.247186923"
    },
    {
      "turn": "0",
      "x": "1.425324484",
      "y": "0.961571840"
    },
    {
      "turn": "0",
      "x": "1.430961684",
      "y": "1.531187338"
    },
    {
      "turn": "0",
      "x": "1.468599525",
      "y": "0.520616706"
    },
    {
      "turn": "0",
      "x": "1.642745700",
      "y": "1.239691647"
    },
    {
      "turn": "0",
      "x": "1.642745700",
      "y": "0.760308353"
    },
    {
      "turn": "0",
      "x": "1.816891875",
      "y": "1.000000000"
    }
  ],
  "radius": "0.148137584"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.339120590",
      "y": "1.115080444"
    },
    {
      "turn": "0",
      "x": "0.339120590",
      "y": "0.818805274"
    },
    {
      "turn": "0",
      "x": "0.339282390",
      "y": "1.480039712"
    },
    {
      "turn": "0",
      "x": "0.364739198",
      "y": "0.512621983"
    },
    {
      "turn": "0",
      "x": "0.572650398",
      "y": "1.297406539"
    },
    {
      "turn": "0",
      "x": "0.613784525",
      "y": "1.004000745"
    },
    {
      "turn": "0",
      "x": "0.613784525",
      "y": "0.707725576"
    },
    {
      "turn": "0",
      "x": "0.653716526",
      "y": "1.582375413"
    },
    {
      "turn": "0",
      "x": "0.661043641",
      "y": "0.415243863"
    },
    {
      "turn": "0",
      "x": "0.861037515",
      "y": "1.370722433"
    },
    {
      "turn": "0",
      "x": "0.895558955",
      "y": "0.912446683"
    },
    {
      "turn": "0",
      "x": "0.895558955",
      "y": "0.616171513"
    },
    {
      "turn": "0",
      "x": "0.935490957",
      "y": "1.673929475"
    },
    {
      "turn": "0",
      "x": "0.942818071",
      "y": "0.323689801"
    },
    {
      "turn": "0",
      "x": "1.064790924",
      "y": "1.155632860"
    },
    {
      "turn": "0",
      "x": "1.142811945",
      "y": "1.462276496"
    },
    {
      "turn": "0",
      "x": "1.172485106",
      "y": "0.807133422"
    },
    {
      "turn": "0",
      "x": "1.172485106",
      "y": "0.510858252"
    },
    {
      "turn": "0",
      "x": "1.240980685",
      "y": "1.772086134"
    },
    {
      "turn": "0",
      "x": "1.252032372",
      "y": "0.224322954"
    },
    {
      "turn": "0",
      "x": "1.346565355",
      "y": "1.247186923"
    },
    {
      "turn": "0",
      "x": "1.425324484",
      "y": "0.961571840"
    },
    {
      "turn": "0",
      "x": "1.430961684",
      "y": "1.531187338"
    },
    {
      "turn": "0",
      "x": "1.468599525",
      "y": "0.520616706"
    },
    {
      "turn": "0",
      "x": "1.642745700",
      "y": "1.239691647"
    },
    {
      "turn": "0",
      "x": "1.642745700",
      "y": "0.760308353"
    },
    {
      "turn": "0",
      "x": "1.816891875",
      "y": "1.000000000"
    }
  ],
  "radius": "0.148137584"
}

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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