MinMax Arena
中/EN
P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 35

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 = 35
ObjectiveMaximize circle radius
Best known, unproven0.130170592942Source-precision interval: 0.130170592941–0.130170592943 (values inside match)exhibited site certificate 0.130170592Packomania (compiled by Eckard Specht), m=5, n=35. Paolo Amore, Circle packing in regular polygons , Phys. Fluids 35 , 027130 (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 = 35, 0.130170592
Current leader

0.130170592

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
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": "0.321153599",
      "y": "1.420770104"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "1.076955627"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "0.816614441"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "0.556273255"
    },
    {
      "turn": "0",
      "x": "0.516666858",
      "y": "1.248862865"
    },
    {
      "turn": "0",
      "x": "0.538635809",
      "y": "1.563875034"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.953072438"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.692731252"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.432390066"
    },
    {
      "turn": "0",
      "x": "0.736173866",
      "y": "1.388842172"
    },
    {
      "turn": "0",
      "x": "0.748579453",
      "y": "1.128282220"
    },
    {
      "turn": "0",
      "x": "0.786234990",
      "y": "1.644324885"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.872622587"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.612281402"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.351940216"
    },
    {
      "turn": "0",
      "x": "0.983773048",
      "y": "1.469292022"
    },
    {
      "turn": "0",
      "x": "0.998939153",
      "y": "1.037826523"
    },
    {
      "turn": "0",
      "x": "1.023191909",
      "y": "0.482110809"
    },
    {
      "turn": "0",
      "x": "1.033834172",
      "y": "1.724774736"
    },
    {
      "turn": "0",
      "x": "1.050021707",
      "y": "0.762794150"
    },
    {
      "turn": "0",
      "x": "1.129082209",
      "y": "0.244277301"
    },
    {
      "turn": "0",
      "x": "1.136797757",
      "y": "1.258671579"
    },
    {
      "turn": "0",
      "x": "1.242688057",
      "y": "1.496505086"
    },
    {
      "turn": "0",
      "x": "1.257387391",
      "y": "1.006488981"
    },
    {
      "turn": "0",
      "x": "1.261045820",
      "y": "0.587955269"
    },
    {
      "turn": "0",
      "x": "1.292749181",
      "y": "1.751987800"
    },
    {
      "turn": "0",
      "x": "1.366936120",
      "y": "0.350121762"
    },
    {
      "turn": "0",
      "x": "1.395245995",
      "y": "1.227334036"
    },
    {
      "turn": "0",
      "x": "1.414070530",
      "y": "0.798575713"
    },
    {
      "turn": "0",
      "x": "1.501136295",
      "y": "1.465167544"
    },
    {
      "turn": "0",
      "x": "1.519960830",
      "y": "0.560742205"
    },
    {
      "turn": "0",
      "x": "1.567095239",
      "y": "1.009196157"
    },
    {
      "turn": "0",
      "x": "1.654161005",
      "y": "1.254547101"
    },
    {
      "turn": "0",
      "x": "1.672985539",
      "y": "0.771362649"
    },
    {
      "turn": "0",
      "x": "1.837407658",
      "y": "0.999999993"
    }
  ],
  "radius": "0.130170592"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "1.420770104"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "1.076955627"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "0.816614441"
    },
    {
      "turn": "0",
      "x": "0.321153599",
      "y": "0.556273255"
    },
    {
      "turn": "0",
      "x": "0.516666858",
      "y": "1.248862865"
    },
    {
      "turn": "0",
      "x": "0.538635809",
      "y": "1.563875034"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.953072438"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.692731252"
    },
    {
      "turn": "0",
      "x": "0.550130647",
      "y": "0.432390066"
    },
    {
      "turn": "0",
      "x": "0.736173866",
      "y": "1.388842172"
    },
    {
      "turn": "0",
      "x": "0.748579453",
      "y": "1.128282220"
    },
    {
      "turn": "0",
      "x": "0.786234990",
      "y": "1.644324885"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.872622587"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.612281402"
    },
    {
      "turn": "0",
      "x": "0.797729828",
      "y": "0.351940216"
    },
    {
      "turn": "0",
      "x": "0.983773048",
      "y": "1.469292022"
    },
    {
      "turn": "0",
      "x": "0.998939153",
      "y": "1.037826523"
    },
    {
      "turn": "0",
      "x": "1.023191909",
      "y": "0.482110809"
    },
    {
      "turn": "0",
      "x": "1.033834172",
      "y": "1.724774736"
    },
    {
      "turn": "0",
      "x": "1.050021707",
      "y": "0.762794150"
    },
    {
      "turn": "0",
      "x": "1.129082209",
      "y": "0.244277301"
    },
    {
      "turn": "0",
      "x": "1.136797757",
      "y": "1.258671579"
    },
    {
      "turn": "0",
      "x": "1.242688057",
      "y": "1.496505086"
    },
    {
      "turn": "0",
      "x": "1.257387391",
      "y": "1.006488981"
    },
    {
      "turn": "0",
      "x": "1.261045820",
      "y": "0.587955269"
    },
    {
      "turn": "0",
      "x": "1.292749181",
      "y": "1.751987800"
    },
    {
      "turn": "0",
      "x": "1.366936120",
      "y": "0.350121762"
    },
    {
      "turn": "0",
      "x": "1.395245995",
      "y": "1.227334036"
    },
    {
      "turn": "0",
      "x": "1.414070530",
      "y": "0.798575713"
    },
    {
      "turn": "0",
      "x": "1.501136295",
      "y": "1.465167544"
    },
    {
      "turn": "0",
      "x": "1.519960830",
      "y": "0.560742205"
    },
    {
      "turn": "0",
      "x": "1.567095239",
      "y": "1.009196157"
    },
    {
      "turn": "0",
      "x": "1.654161005",
      "y": "1.254547101"
    },
    {
      "turn": "0",
      "x": "1.672985539",
      "y": "0.771362649"
    },
    {
      "turn": "0",
      "x": "1.837407658",
      "y": "0.999999993"
    }
  ],
  "radius": "0.130170592"
}

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

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?