P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · 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 = 8 · n = 27

ObjectiveMaximize circle radius

Best known, unproven0.161395047685Source-precision interval: 0.161395047684–0.161395047686 (values inside match)exhibited site certificate 0.161395047Packomania (compiled by Eckard Specht), m=8, n=27. 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

Current leader

0.161395047

circle radius

Answer sourcePaolo Amore

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": "0.449874147",
      "y": "0.402562330"
    },
    {
      "turn": "0",
      "x": "0.966545494",
      "y": "0.188550051"
    },
    {
      "turn": "0",
      "x": "0.769922565",
      "y": "0.444543937"
    },
    {
      "turn": "0",
      "x": "0.303081062",
      "y": "0.690043175"
    },
    {
      "turn": "0",
      "x": "1.273623534",
      "y": "0.288031320"
    },
    {
      "turn": "0",
      "x": "0.623129480",
      "y": "0.732024781"
    },
    {
      "turn": "0",
      "x": "1.077000606",
      "y": "0.544025206"
    },
    {
      "turn": "0",
      "x": "0.179554640",
      "y": "0.988262337"
    },
    {
      "turn": "0",
      "x": "1.571842696",
      "y": "0.411557741"
    },
    {
      "turn": "0",
      "x": "0.930207521",
      "y": "0.831506050"
    },
    {
      "turn": "0",
      "x": "0.499603059",
      "y": "1.030243944"
    },
    {
      "turn": "0",
      "x": "1.375219768",
      "y": "0.667551627"
    },
    {
      "turn": "0",
      "x": "0.294733431",
      "y": "1.289803863"
    },
    {
      "turn": "0",
      "x": "1.703621050",
      "y": "0.706223503"
    },
    {
      "turn": "0",
      "x": "0.888074959",
      "y": "1.151534631"
    },
    {
      "turn": "0",
      "x": "1.186294121",
      "y": "1.028008210"
    },
    {
      "turn": "0",
      "x": "0.619158561",
      "y": "1.330077110"
    },
    {
      "turn": "0",
      "x": "1.502695327",
      "y": "0.964104199"
    },
    {
      "turn": "0",
      "x": "0.422656401",
      "y": "1.586163710"
    },
    {
      "turn": "0",
      "x": "1.822723908",
      "y": "1.006236761"
    },
    {
      "turn": "0",
      "x": "1.011601380",
      "y": "1.449753794"
    },
    {
      "turn": "0",
      "x": "1.309820543",
      "y": "1.326227372"
    },
    {
      "turn": "0",
      "x": "0.742684983",
      "y": "1.628296273"
    },
    {
      "turn": "0",
      "x": "1.626221749",
      "y": "1.262323361"
    },
    {
      "turn": "0",
      "x": "1.267687981",
      "y": "1.646255954"
    },
    {
      "turn": "0",
      "x": "0.998771583",
      "y": "1.824798432"
    },
    {
      "turn": "0",
      "x": "1.584089187",
      "y": "1.582351943"
    }
  ],
  "radius": "0.161395047"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.449874147",
      "y": "0.402562330"
    },
    {
      "turn": "0",
      "x": "0.966545494",
      "y": "0.188550051"
    },
    {
      "turn": "0",
      "x": "0.769922565",
      "y": "0.444543937"
    },
    {
      "turn": "0",
      "x": "0.303081062",
      "y": "0.690043175"
    },
    {
      "turn": "0",
      "x": "1.273623534",
      "y": "0.288031320"
    },
    {
      "turn": "0",
      "x": "0.623129480",
      "y": "0.732024781"
    },
    {
      "turn": "0",
      "x": "1.077000606",
      "y": "0.544025206"
    },
    {
      "turn": "0",
      "x": "0.179554640",
      "y": "0.988262337"
    },
    {
      "turn": "0",
      "x": "1.571842696",
      "y": "0.411557741"
    },
    {
      "turn": "0",
      "x": "0.930207521",
      "y": "0.831506050"
    },
    {
      "turn": "0",
      "x": "0.499603059",
      "y": "1.030243944"
    },
    {
      "turn": "0",
      "x": "1.375219768",
      "y": "0.667551627"
    },
    {
      "turn": "0",
      "x": "0.294733431",
      "y": "1.289803863"
    },
    {
      "turn": "0",
      "x": "1.703621050",
      "y": "0.706223503"
    },
    {
      "turn": "0",
      "x": "0.888074959",
      "y": "1.151534631"
    },
    {
      "turn": "0",
      "x": "1.186294121",
      "y": "1.028008210"
    },
    {
      "turn": "0",
      "x": "0.619158561",
      "y": "1.330077110"
    },
    {
      "turn": "0",
      "x": "1.502695327",
      "y": "0.964104199"
    },
    {
      "turn": "0",
      "x": "0.422656401",
      "y": "1.586163710"
    },
    {
      "turn": "0",
      "x": "1.822723908",
      "y": "1.006236761"
    },
    {
      "turn": "0",
      "x": "1.011601380",
      "y": "1.449753794"
    },
    {
      "turn": "0",
      "x": "1.309820543",
      "y": "1.326227372"
    },
    {
      "turn": "0",
      "x": "0.742684983",
      "y": "1.628296273"
    },
    {
      "turn": "0",
      "x": "1.626221749",
      "y": "1.262323361"
    },
    {
      "turn": "0",
      "x": "1.267687981",
      "y": "1.646255954"
    },
    {
      "turn": "0",
      "x": "0.998771583",
      "y": "1.824798432"
    },
    {
      "turn": "0",
      "x": "1.584089187",
      "y": "1.582351943"
    }
  ],
  "radius": "0.161395047"
}

DISCUSSION

Discussion

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