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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 23

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 = 23
ObjectiveMaximize circle radius
Best known, unproven0.158923367389Source-precision interval: 0.158923367388–0.15892336739 (values inside match)exhibited site certificate 0.158923366Packomania (compiled by Eckard Specht), m=5, n=23. 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.
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 = 5 · n = 23, 0.158923366
Current leader

0.158923366

circle radius

Answer sourceEckard Specht
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": "0.349906373",
      "y": "1.472320667"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.041966153"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "0.527679333"
    },
    {
      "turn": "0",
      "x": "0.536731996",
      "y": "0.784822743"
    },
    {
      "turn": "0",
      "x": "0.583840754",
      "y": "1.257143410"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "1.567325156"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "0.432674844"
    },
    {
      "turn": "0",
      "x": "0.770666378",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.840048158",
      "y": "0.689818254"
    },
    {
      "turn": "0",
      "x": "0.899946292",
      "y": "1.290367441"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "1.668749677"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "0.331250323"
    },
    {
      "turn": "0",
      "x": "1.085892280",
      "y": "0.959266998"
    },
    {
      "turn": "0",
      "x": "1.141278325",
      "y": "0.588393733"
    },
    {
      "turn": "0",
      "x": "1.182401875",
      "y": "1.436125426"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "1.743481255"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "0.256518745"
    },
    {
      "turn": "0",
      "x": "1.348193931",
      "y": "1.138778532"
    },
    {
      "turn": "0",
      "x": "1.372504654",
      "y": "0.821862872"
    },
    {
      "turn": "0",
      "x": "1.450214787",
      "y": "0.513662155"
    },
    {
      "turn": "0",
      "x": "1.499741088",
      "y": "1.418170740"
    },
    {
      "turn": "0",
      "x": "1.686566711",
      "y": "1.161027330"
    },
    {
      "turn": "0",
      "x": "1.689634706",
      "y": "0.843195402"
    }
  ],
  "radius": "0.158923366"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.472320667"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "1.041966153"
    },
    {
      "turn": "0",
      "x": "0.349906373",
      "y": "0.527679333"
    },
    {
      "turn": "0",
      "x": "0.536731996",
      "y": "0.784822743"
    },
    {
      "turn": "0",
      "x": "0.583840754",
      "y": "1.257143410"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "1.567325156"
    },
    {
      "turn": "0",
      "x": "0.653222535",
      "y": "0.432674844"
    },
    {
      "turn": "0",
      "x": "0.770666378",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.840048158",
      "y": "0.689818254"
    },
    {
      "turn": "0",
      "x": "0.899946292",
      "y": "1.290367441"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "1.668749677"
    },
    {
      "turn": "0",
      "x": "0.954452702",
      "y": "0.331250323"
    },
    {
      "turn": "0",
      "x": "1.085892280",
      "y": "0.959266998"
    },
    {
      "turn": "0",
      "x": "1.141278325",
      "y": "0.588393733"
    },
    {
      "turn": "0",
      "x": "1.182401875",
      "y": "1.436125426"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "1.743481255"
    },
    {
      "turn": "0",
      "x": "1.263389164",
      "y": "0.256518745"
    },
    {
      "turn": "0",
      "x": "1.348193931",
      "y": "1.138778532"
    },
    {
      "turn": "0",
      "x": "1.372504654",
      "y": "0.821862872"
    },
    {
      "turn": "0",
      "x": "1.450214787",
      "y": "0.513662155"
    },
    {
      "turn": "0",
      "x": "1.499741088",
      "y": "1.418170740"
    },
    {
      "turn": "0",
      "x": "1.686566711",
      "y": "1.161027330"
    },
    {
      "turn": "0",
      "x": "1.689634706",
      "y": "0.843195402"
    }
  ],
  "radius": "0.158923366"
}

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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