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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 32

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 = 32
ObjectiveMaximize circle radius
Best known, unproven0.142897923362Source-precision interval: 0.142897923361–0.142897923363 (values inside match)exhibited site certificate 0.142897922Packomania (compiled by Eckard Specht), m=6, n=32. 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 = 32, 0.142897922
Current leader

0.142897922

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": "0.622971630",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "0.908767477",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "1.194563324",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "1.467125991",
      "y": "0.362830989"
    },
    {
      "turn": "0",
      "x": "0.450800156",
      "y": "0.504987073"
    },
    {
      "turn": "0",
      "x": "0.765732223",
      "y": "0.524458220"
    },
    {
      "turn": "0",
      "x": "1.051528070",
      "y": "0.524458220"
    },
    {
      "turn": "0",
      "x": "1.324133006",
      "y": "0.610282544"
    },
    {
      "turn": "0",
      "x": "1.610023914",
      "y": "0.610337452"
    },
    {
      "turn": "0",
      "x": "0.307902232",
      "y": "0.752493536"
    },
    {
      "turn": "0",
      "x": "0.593698021",
      "y": "0.752676316"
    },
    {
      "turn": "0",
      "x": "0.877357790",
      "y": "0.787553258"
    },
    {
      "turn": "0",
      "x": "1.157786748",
      "y": "0.842679163"
    },
    {
      "turn": "0",
      "x": "1.467030930",
      "y": "0.857789008"
    },
    {
      "turn": "0",
      "x": "1.752921838",
      "y": "0.857843916"
    },
    {
      "turn": "0",
      "x": "0.165004309",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.456624477",
      "y": "1.003455321"
    },
    {
      "turn": "0",
      "x": "0.900496194",
      "y": "1.072410907"
    },
    {
      "turn": "0",
      "x": "1.180925152",
      "y": "1.127536813"
    },
    {
      "turn": "0",
      "x": "1.466270134",
      "y": "1.143583842"
    },
    {
      "turn": "0",
      "x": "1.752065975",
      "y": "1.143638481"
    },
    {
      "turn": "0",
      "x": "0.651430037",
      "y": "1.212572656"
    },
    {
      "turn": "0",
      "x": "0.307902232",
      "y": "1.247506464"
    },
    {
      "turn": "0",
      "x": "1.038027228",
      "y": "1.375043276"
    },
    {
      "turn": "0",
      "x": "1.323372210",
      "y": "1.391090306"
    },
    {
      "turn": "0",
      "x": "1.609168052",
      "y": "1.391144945"
    },
    {
      "turn": "0",
      "x": "0.483690854",
      "y": "1.472845378"
    },
    {
      "turn": "0",
      "x": "0.769486701",
      "y": "1.472845378"
    },
    {
      "turn": "0",
      "x": "1.180474287",
      "y": "1.638596769"
    },
    {
      "turn": "0",
      "x": "1.466270129",
      "y": "1.638651408"
    },
    {
      "turn": "0",
      "x": "0.621669604",
      "y": "1.723127480"
    },
    {
      "turn": "0",
      "x": "0.907465450",
      "y": "1.723127480"
    }
  ],
  "radius": "0.142897922"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.622971630",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "0.908767477",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "1.194563324",
      "y": "0.276872520"
    },
    {
      "turn": "0",
      "x": "1.467125991",
      "y": "0.362830989"
    },
    {
      "turn": "0",
      "x": "0.450800156",
      "y": "0.504987073"
    },
    {
      "turn": "0",
      "x": "0.765732223",
      "y": "0.524458220"
    },
    {
      "turn": "0",
      "x": "1.051528070",
      "y": "0.524458220"
    },
    {
      "turn": "0",
      "x": "1.324133006",
      "y": "0.610282544"
    },
    {
      "turn": "0",
      "x": "1.610023914",
      "y": "0.610337452"
    },
    {
      "turn": "0",
      "x": "0.307902232",
      "y": "0.752493536"
    },
    {
      "turn": "0",
      "x": "0.593698021",
      "y": "0.752676316"
    },
    {
      "turn": "0",
      "x": "0.877357790",
      "y": "0.787553258"
    },
    {
      "turn": "0",
      "x": "1.157786748",
      "y": "0.842679163"
    },
    {
      "turn": "0",
      "x": "1.467030930",
      "y": "0.857789008"
    },
    {
      "turn": "0",
      "x": "1.752921838",
      "y": "0.857843916"
    },
    {
      "turn": "0",
      "x": "0.165004309",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.456624477",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.466270134",
      "y": "1.143583842"
    },
    {
      "turn": "0",
      "x": "1.752065975",
      "y": "1.143638481"
    },
    {
      "turn": "0",
      "x": "0.651430037",
      "y": "1.212572656"
    },
    {
      "turn": "0",
      "x": "0.307902232",
      "y": "1.247506464"
    },
    {
      "turn": "0",
      "x": "1.038027228",
      "y": "1.375043276"
    },
    {
      "turn": "0",
      "x": "1.323372210",
      "y": "1.391090306"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "0.483690854",
      "y": "1.472845378"
    },
    {
      "turn": "0",
      "x": "0.769486701",
      "y": "1.472845378"
    },
    {
      "turn": "0",
      "x": "1.180474287",
      "y": "1.638596769"
    },
    {
      "turn": "0",
      "x": "1.466270129",
      "y": "1.638651408"
    },
    {
      "turn": "0",
      "x": "0.621669604",
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    },
    {
      "turn": "0",
      "x": "0.907465450",
      "y": "1.723127480"
    }
  ],
  "radius": "0.142897922"
}

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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