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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 25

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 = 25
ObjectiveMaximize circle radius
Best known, unproven0.153122210789Source-precision interval: 0.153122210788–0.15312221079 (values inside match)exhibited site certificate 0.153122210Packomania (compiled by Eckard Specht), m=5, n=25. 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 = 25, 0.15312221
Current leader

0.15312221

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": "0.344105216",
      "y": "1.466012230"
    },
    {
      "turn": "0",
      "x": "0.344105216",
      "y": "1.047745054"
    },
    {
      "turn": "0",
      "x": "0.344105216",
      "y": "0.653003990"
    },
    {
      "turn": "0",
      "x": "0.567821081",
      "y": "1.256878642"
    },
    {
      "turn": "0",
      "x": "0.572682640",
      "y": "0.449195239"
    },
    {
      "turn": "0",
      "x": "0.578264391",
      "y": "0.850374522"
    },
    {
      "turn": "0",
      "x": "0.632112698",
      "y": "1.570114758"
    },
    {
      "turn": "0",
      "x": "0.801980255",
      "y": "1.059508110"
    },
    {
      "turn": "0",
      "x": "0.806841815",
      "y": "0.646565771"
    },
    {
      "turn": "0",
      "x": "0.855828563",
      "y": "1.360981169"
    },
    {
      "turn": "0",
      "x": "0.880269078",
      "y": "0.349254347"
    },
    {
      "turn": "0",
      "x": "0.923368451",
      "y": "1.664749488"
    },
    {
      "turn": "0",
      "x": "1.032070979",
      "y": "0.854068708"
    },
    {
      "turn": "0",
      "x": "1.093236008",
      "y": "1.154142841"
    },
    {
      "turn": "0",
      "x": "1.128482420",
      "y": "0.528630440"
    },
    {
      "turn": "0",
      "x": "1.147084315",
      "y": "1.455615900"
    },
    {
      "turn": "0",
      "x": "1.224598846",
      "y": "0.237704760"
    },
    {
      "turn": "0",
      "x": "1.225168845",
      "y": "1.761365163"
    },
    {
      "turn": "0",
      "x": "1.324696771",
      "y": "0.763759285"
    },
    {
      "turn": "0",
      "x": "1.385861799",
      "y": "1.063833418"
    },
    {
      "turn": "0",
      "x": "1.430217149",
      "y": "0.476268250"
    },
    {
      "turn": "0",
      "x": "1.450718629",
      "y": "1.495513883"
    },
    {
      "turn": "0",
      "x": "1.630724584",
      "y": "1.247756941"
    },
    {
      "turn": "0",
      "x": "1.630724584",
      "y": "0.752243059"
    },
    {
      "turn": "0",
      "x": "1.810730539",
      "y": "1.000000000"
    }
  ],
  "radius": "0.153122210"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.344105216",
      "y": "1.466012230"
    },
    {
      "turn": "0",
      "x": "0.344105216",
      "y": "1.047745054"
    },
    {
      "turn": "0",
      "x": "0.344105216",
      "y": "0.653003990"
    },
    {
      "turn": "0",
      "x": "0.567821081",
      "y": "1.256878642"
    },
    {
      "turn": "0",
      "x": "0.572682640",
      "y": "0.449195239"
    },
    {
      "turn": "0",
      "x": "0.578264391",
      "y": "0.850374522"
    },
    {
      "turn": "0",
      "x": "0.632112698",
      "y": "1.570114758"
    },
    {
      "turn": "0",
      "x": "0.801980255",
      "y": "1.059508110"
    },
    {
      "turn": "0",
      "x": "0.806841815",
      "y": "0.646565771"
    },
    {
      "turn": "0",
      "x": "0.855828563",
      "y": "1.360981169"
    },
    {
      "turn": "0",
      "x": "0.880269078",
      "y": "0.349254347"
    },
    {
      "turn": "0",
      "x": "0.923368451",
      "y": "1.664749488"
    },
    {
      "turn": "0",
      "x": "1.032070979",
      "y": "0.854068708"
    },
    {
      "turn": "0",
      "x": "1.093236008",
      "y": "1.154142841"
    },
    {
      "turn": "0",
      "x": "1.128482420",
      "y": "0.528630440"
    },
    {
      "turn": "0",
      "x": "1.147084315",
      "y": "1.455615900"
    },
    {
      "turn": "0",
      "x": "1.224598846",
      "y": "0.237704760"
    },
    {
      "turn": "0",
      "x": "1.225168845",
      "y": "1.761365163"
    },
    {
      "turn": "0",
      "x": "1.324696771",
      "y": "0.763759285"
    },
    {
      "turn": "0",
      "x": "1.385861799",
      "y": "1.063833418"
    },
    {
      "turn": "0",
      "x": "1.430217149",
      "y": "0.476268250"
    },
    {
      "turn": "0",
      "x": "1.450718629",
      "y": "1.495513883"
    },
    {
      "turn": "0",
      "x": "1.630724584",
      "y": "1.247756941"
    },
    {
      "turn": "0",
      "x": "1.630724584",
      "y": "0.752243059"
    },
    {
      "turn": "0",
      "x": "1.810730539",
      "y": "1.000000000"
    }
  ],
  "radius": "0.153122210"
}

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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