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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 6 · 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 = 6 · n = 25
ObjectiveMaximize circle radius
Best known, unproven0.159724477011Source-precision interval: 0.15972447701–0.159724477012 (values inside match)exhibited site certificate 0.159724476Packomania (compiled by Eckard Specht), m=6, n=25. 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.
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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 = 6 · n = 25, 0.159724476
Current leader

0.159724476

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": "0.784309189",
      "y": "0.293699073"
    },
    {
      "turn": "0",
      "x": "1.103758143",
      "y": "0.293699073"
    },
    {
      "turn": "0",
      "x": "1.422226083",
      "y": "0.318715175"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "0.446698181"
    },
    {
      "turn": "0",
      "x": "0.944033666",
      "y": "0.570349983"
    },
    {
      "turn": "0",
      "x": "1.262501606",
      "y": "0.595366084"
    },
    {
      "turn": "0",
      "x": "1.581950560",
      "y": "0.595366084"
    },
    {
      "turn": "0",
      "x": "0.344158417",
      "y": "0.723349091"
    },
    {
      "turn": "0",
      "x": "0.663607371",
      "y": "0.723349091"
    },
    {
      "turn": "0",
      "x": "1.741675037",
      "y": "0.872016994"
    },
    {
      "turn": "0",
      "x": "1.119871044",
      "y": "0.881205474"
    },
    {
      "turn": "0",
      "x": "1.436743253",
      "y": "0.984463526"
    },
    {
      "turn": "0",
      "x": "0.184433940",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.823331848",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.706166104",
      "y": "1.189486283"
    },
    {
      "turn": "0",
      "x": "1.163625802",
      "y": "1.197643707"
    },
    {
      "turn": "0",
      "x": "0.344158417",
      "y": "1.276650909"
    },
    {
      "turn": "0",
      "x": "0.663607371",
      "y": "1.276650909"
    },
    {
      "turn": "0",
      "x": "1.437431452",
      "y": "1.362197955"
    },
    {
      "turn": "0",
      "x": "0.944033666",
      "y": "1.429650017"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "1.553301819"
    },
    {
      "turn": "0",
      "x": "1.422226083",
      "y": "1.681284825"
    },
    {
      "turn": "0",
      "x": "0.784309189",
      "y": "1.706300927"
    },
    {
      "turn": "0",
      "x": "1.103758143",
      "y": "1.706300927"
    }
  ],
  "radius": "0.159724476"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.784309189",
      "y": "0.293699073"
    },
    {
      "turn": "0",
      "x": "1.103758143",
      "y": "0.293699073"
    },
    {
      "turn": "0",
      "x": "1.422226083",
      "y": "0.318715175"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "0.446698181"
    },
    {
      "turn": "0",
      "x": "0.944033666",
      "y": "0.570349983"
    },
    {
      "turn": "0",
      "x": "1.262501606",
      "y": "0.595366084"
    },
    {
      "turn": "0",
      "x": "1.581950560",
      "y": "0.595366084"
    },
    {
      "turn": "0",
      "x": "0.344158417",
      "y": "0.723349091"
    },
    {
      "turn": "0",
      "x": "0.663607371",
      "y": "0.723349091"
    },
    {
      "turn": "0",
      "x": "1.741675037",
      "y": "0.872016994"
    },
    {
      "turn": "0",
      "x": "1.119871044",
      "y": "0.881205474"
    },
    {
      "turn": "0",
      "x": "1.436743253",
      "y": "0.984463526"
    },
    {
      "turn": "0",
      "x": "0.184433940",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.823331848",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.706166104",
      "y": "1.189486283"
    },
    {
      "turn": "0",
      "x": "1.163625802",
      "y": "1.197643707"
    },
    {
      "turn": "0",
      "x": "0.344158417",
      "y": "1.276650909"
    },
    {
      "turn": "0",
      "x": "0.663607371",
      "y": "1.276650909"
    },
    {
      "turn": "0",
      "x": "1.437431452",
      "y": "1.362197955"
    },
    {
      "turn": "0",
      "x": "0.944033666",
      "y": "1.429650017"
    },
    {
      "turn": "0",
      "x": "0.503882894",
      "y": "1.553301819"
    },
    {
      "turn": "0",
      "x": "1.422226083",
      "y": "1.681284825"
    },
    {
      "turn": "0",
      "x": "0.784309189",
      "y": "1.706300927"
    },
    {
      "turn": "0",
      "x": "1.103758143",
      "y": "1.706300927"
    }
  ],
  "radius": "0.159724476"
}

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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