P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · n = 39

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 = 39

ObjectiveMaximize circle radius

Best known, unproven0.129882647187Source-precision interval: 0.129882647186–0.129882647188 (values inside match)exhibited site certificate 0.129882646Packomania (compiled by Eckard Specht), m=6, n=39. 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

Current leader

0.129882646

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": "0.792063818",
      "y": "0.263857243"
    },
    {
      "turn": "0",
      "x": "1.051829112",
      "y": "0.263857243"
    },
    {
      "turn": "0",
      "x": "1.311594406",
      "y": "0.263857243"
    },
    {
      "turn": "0",
      "x": "0.539623504",
      "y": "0.325109968"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    {
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    {
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    {
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    },
    {
      "turn": "0",
      "x": "1.646780078",
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    {
      "turn": "0",
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    {
      "turn": "0",
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    {
      "turn": "0",
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    {
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    {
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "0.409740857",
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    {
      "turn": "0",
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    {
      "turn": "0",
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    {
      "turn": "0",
      "x": "1.773422257",
      "y": "1.132678868"
    },
    {
      "turn": "0",
      "x": "0.599838556",
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    {
      "turn": "0",
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    {
      "turn": "0",
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      "y": "1.224963344"
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    {
      "turn": "0",
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    {
      "turn": "0",
      "x": "1.383774315",
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    },
    {
      "turn": "0",
      "x": "1.643539610",
      "y": "1.357642212"
    },
    {
      "turn": "0",
      "x": "0.668540494",
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    {
      "turn": "0",
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      "y": "1.427549108"
    },
    {
      "turn": "0",
      "x": "0.409740857",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.513656963",
      "y": "1.582605556"
    },
    {
      "turn": "0",
      "x": "0.798423141",
      "y": "1.652512452"
    },
    {
      "turn": "0",
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      "y": "1.674890032"
    },
    {
      "turn": "0",
      "x": "1.044358052",
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    },
    {
      "turn": "0",
      "x": "1.304123346",
      "y": "1.736142757"
    }
  ],
  "radius": "0.129882646"
}

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

Instance parameters

{
  "n": 39,
  "sides": 6
}

The current leader's answer

{
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      "turn": "0",
      "x": "0.792063818",
      "y": "0.263857243"
    },
    {
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    },
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    },
    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    }
  ],
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}

DISCUSSION

Discussion

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