P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 9

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 = 9

ObjectiveMaximize circle radius

Best known, unproven0.243640224399Source-precision interval: 0.243640224398–0.2436402244 (values inside match)exhibited site certificate 0.243640224Packomania (compiled by Eckard Specht), m=5, n=9. 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

Current leader

0.243640224

circle radius

Answer sourceEckard Specht

Solution methodPublished reference construction

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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": "0.434623230",
      "y": "1.410770268"
    },
    {
      "turn": "0",
      "x": "0.434623230",
      "y": "0.589229732"
    },
    {
      "turn": "0",
      "x": "0.696749198",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.898054476",
      "y": "1.561348207"
    },
    {
      "turn": "0",
      "x": "0.898054476",
      "y": "0.438651793"
    },
    {
      "turn": "0",
      "x": "1.184029646",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.372309614",
      "y": "1.449436191"
    },
    {
      "turn": "0",
      "x": "1.372309614",
      "y": "0.550563809"
    },
    {
      "turn": "0",
      "x": "1.691478798",
      "y": "1.000000000"
    }
  ],
  "radius": "0.243640224"
}

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

Instance parameters

{
  "n": 9,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.434623230",
      "y": "1.410770268"
    },
    {
      "turn": "0",
      "x": "0.434623230",
      "y": "0.589229732"
    },
    {
      "turn": "0",
      "x": "0.696749198",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.898054476",
      "y": "1.561348207"
    },
    {
      "turn": "0",
      "x": "0.898054476",
      "y": "0.438651793"
    },
    {
      "turn": "0",
      "x": "1.184029646",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.372309614",
      "y": "1.449436191"
    },
    {
      "turn": "0",
      "x": "1.372309614",
      "y": "0.550563809"
    },
    {
      "turn": "0",
      "x": "1.691478798",
      "y": "1.000000000"
    }
  ],
  "radius": "0.243640224"
}

DISCUSSION

Discussion

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