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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · 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 = 8 · n = 20
ObjectiveMaximize circle radius
Best known, unproven0.185530744908Source-precision interval: 0.185530744907–0.185530744909 (values inside match)exhibited site certificate 0.185530744Packomania (compiled by Eckard Specht), m=8, n=20. 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 = 8 · n = 20, 0.185530744
Current leader

0.185530744

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
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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": "0.686545049",
      "y": "0.330654323"
    },
    {
      "turn": "0",
      "x": "1.039879950",
      "y": "0.217335847"
    },
    {
      "turn": "0",
      "x": "0.387228513",
      "y": "0.549962873"
    },
    {
      "turn": "0",
      "x": "1.416814676",
      "y": "0.373467323"
    },
    {
      "turn": "0",
      "x": "0.245229428",
      "y": "0.892778989"
    },
    {
      "turn": "0",
      "x": "0.756397570",
      "y": "0.695081620"
    },
    {
      "turn": "0",
      "x": "1.109732471",
      "y": "0.581763144"
    },
    {
      "turn": "0",
      "x": "0.602278482",
      "y": "1.032622633"
    },
    {
      "turn": "0",
      "x": "1.656866967",
      "y": "0.656418779"
    },
    {
      "turn": "0",
      "x": "0.307895609",
      "y": "1.258510556"
    },
    {
      "turn": "0",
      "x": "1.003621142",
      "y": "1.001377049"
    },
    {
      "turn": "0",
      "x": "1.349784763",
      "y": "0.864714600"
    },
    {
      "turn": "0",
      "x": "0.778038018",
      "y": "1.359418004"
    },
    {
      "turn": "0",
      "x": "1.798866052",
      "y": "0.999234895"
    },
    {
      "turn": "0",
      "x": "0.483655146",
      "y": "1.585305927"
    },
    {
      "turn": "0",
      "x": "1.148967732",
      "y": "1.349529800"
    },
    {
      "turn": "0",
      "x": "1.491783848",
      "y": "1.207530716"
    },
    {
      "turn": "0",
      "x": "0.826471261",
      "y": "1.727305012"
    },
    {
      "turn": "0",
      "x": "1.197400975",
      "y": "1.717416808"
    },
    {
      "turn": "0",
      "x": "1.540217091",
      "y": "1.575417723"
    }
  ],
  "radius": "0.185530744"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.686545049",
      "y": "0.330654323"
    },
    {
      "turn": "0",
      "x": "1.039879950",
      "y": "0.217335847"
    },
    {
      "turn": "0",
      "x": "0.387228513",
      "y": "0.549962873"
    },
    {
      "turn": "0",
      "x": "1.416814676",
      "y": "0.373467323"
    },
    {
      "turn": "0",
      "x": "0.245229428",
      "y": "0.892778989"
    },
    {
      "turn": "0",
      "x": "0.756397570",
      "y": "0.695081620"
    },
    {
      "turn": "0",
      "x": "1.109732471",
      "y": "0.581763144"
    },
    {
      "turn": "0",
      "x": "0.602278482",
      "y": "1.032622633"
    },
    {
      "turn": "0",
      "x": "1.656866967",
      "y": "0.656418779"
    },
    {
      "turn": "0",
      "x": "0.307895609",
      "y": "1.258510556"
    },
    {
      "turn": "0",
      "x": "1.003621142",
      "y": "1.001377049"
    },
    {
      "turn": "0",
      "x": "1.349784763",
      "y": "0.864714600"
    },
    {
      "turn": "0",
      "x": "0.778038018",
      "y": "1.359418004"
    },
    {
      "turn": "0",
      "x": "1.798866052",
      "y": "0.999234895"
    },
    {
      "turn": "0",
      "x": "0.483655146",
      "y": "1.585305927"
    },
    {
      "turn": "0",
      "x": "1.148967732",
      "y": "1.349529800"
    },
    {
      "turn": "0",
      "x": "1.491783848",
      "y": "1.207530716"
    },
    {
      "turn": "0",
      "x": "0.826471261",
      "y": "1.727305012"
    },
    {
      "turn": "0",
      "x": "1.197400975",
      "y": "1.717416808"
    },
    {
      "turn": "0",
      "x": "1.540217091",
      "y": "1.575417723"
    }
  ],
  "radius": "0.185530744"
}

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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