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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 19

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 = 19
ObjectiveMaximize circle radius
Best known, unproven0.173441904330Source-precision interval: 0.173441904329–0.173441904331 (values inside match)exhibited site certificate 0.173441903Packomania (compiled by Eckard Specht), m=5, n=19. 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
The record-holding arrangement for Equal circles in regular polygons m = 5 · n = 19, 0.173441903
Current leader

0.173441903

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": "0.364424910",
      "y": "1.373283093"
    },
    {
      "turn": "0",
      "x": "0.364424910",
      "y": "0.973779480"
    },
    {
      "turn": "0",
      "x": "0.365410135",
      "y": "0.560948806"
    },
    {
      "turn": "0",
      "x": "0.644467558",
      "y": "0.769073612"
    },
    {
      "turn": "0",
      "x": "0.648022499",
      "y": "1.173531286"
    },
    {
      "turn": "0",
      "x": "0.658468188",
      "y": "1.557312785"
    },
    {
      "turn": "0",
      "x": "0.705111317",
      "y": "0.427531944"
    },
    {
      "turn": "0",
      "x": "0.942065777",
      "y": "1.357560978"
    },
    {
      "turn": "0",
      "x": "0.946552577",
      "y": "0.996872946"
    },
    {
      "turn": "0",
      "x": "0.970573198",
      "y": "0.650821814"
    },
    {
      "turn": "0",
      "x": "1.054089879",
      "y": "1.685858065"
    },
    {
      "turn": "0",
      "x": "1.054089879",
      "y": "0.314141935"
    },
    {
      "turn": "0",
      "x": "1.246528358",
      "y": "1.191335489"
    },
    {
      "turn": "0",
      "x": "1.258251960",
      "y": "0.844649848"
    },
    {
      "turn": "0",
      "x": "1.377827642",
      "y": "1.561269793"
    },
    {
      "turn": "0",
      "x": "1.377827642",
      "y": "0.438730207"
    },
    {
      "turn": "0",
      "x": "1.581720829",
      "y": "1.280634896"
    },
    {
      "turn": "0",
      "x": "1.581720829",
      "y": "0.719365104"
    },
    {
      "turn": "0",
      "x": "1.785614016",
      "y": "1.000000000"
    }
  ],
  "radius": "0.173441903"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.364424910",
      "y": "1.373283093"
    },
    {
      "turn": "0",
      "x": "0.364424910",
      "y": "0.973779480"
    },
    {
      "turn": "0",
      "x": "0.365410135",
      "y": "0.560948806"
    },
    {
      "turn": "0",
      "x": "0.644467558",
      "y": "0.769073612"
    },
    {
      "turn": "0",
      "x": "0.648022499",
      "y": "1.173531286"
    },
    {
      "turn": "0",
      "x": "0.658468188",
      "y": "1.557312785"
    },
    {
      "turn": "0",
      "x": "0.705111317",
      "y": "0.427531944"
    },
    {
      "turn": "0",
      "x": "0.942065777",
      "y": "1.357560978"
    },
    {
      "turn": "0",
      "x": "0.946552577",
      "y": "0.996872946"
    },
    {
      "turn": "0",
      "x": "0.970573198",
      "y": "0.650821814"
    },
    {
      "turn": "0",
      "x": "1.054089879",
      "y": "1.685858065"
    },
    {
      "turn": "0",
      "x": "1.054089879",
      "y": "0.314141935"
    },
    {
      "turn": "0",
      "x": "1.246528358",
      "y": "1.191335489"
    },
    {
      "turn": "0",
      "x": "1.258251960",
      "y": "0.844649848"
    },
    {
      "turn": "0",
      "x": "1.377827642",
      "y": "1.561269793"
    },
    {
      "turn": "0",
      "x": "1.377827642",
      "y": "0.438730207"
    },
    {
      "turn": "0",
      "x": "1.581720829",
      "y": "1.280634896"
    },
    {
      "turn": "0",
      "x": "1.581720829",
      "y": "0.719365104"
    },
    {
      "turn": "0",
      "x": "1.785614016",
      "y": "1.000000000"
    }
  ],
  "radius": "0.173441903"
}

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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