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

Equal circles in regular polygons · m = 8 · n = 38

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 = 8 · n = 38
ObjectiveMaximize circle radius
Best known, unproven0.136097981333Source-precision interval: 0.136097981332–0.136097981334 (values inside match)exhibited site certificate 0.136097980Packomania (compiled by Eckard Specht), m=8, n=38. 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 = 8 · n = 38, 0.13609798
Current leader

0.13609798

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": "0.573571172",
      "y": "0.323943997"
    },
    {
      "turn": "0",
      "x": "0.825047451",
      "y": "0.219779112"
    },
    {
      "turn": "0",
      "x": "0.358328126",
      "y": "0.490560543"
    },
    {
      "turn": "0",
      "x": "1.095233058",
      "y": "0.186758218"
    },
    {
      "turn": "0",
      "x": "0.857830114",
      "y": "0.489993729"
    },
    {
      "turn": "0",
      "x": "0.254163241",
      "y": "0.742036822"
    },
    {
      "turn": "0",
      "x": "0.608201941",
      "y": "0.598512933"
    },
    {
      "turn": "0",
      "x": "1.349181389",
      "y": "0.291947061"
    },
    {
      "turn": "0",
      "x": "1.132295322",
      "y": "0.456419173"
    },
    {
      "turn": "0",
      "x": "0.149998355",
      "y": "0.993513100"
    },
    {
      "turn": "0",
      "x": "0.504037056",
      "y": "0.849989212"
    },
    {
      "turn": "0",
      "x": "1.602941896",
      "y": "0.397058104"
    },
    {
      "turn": "0",
      "x": "1.023532517",
      "y": "0.705941305"
    },
    {
      "turn": "0",
      "x": "0.773904344",
      "y": "0.814460509"
    },
    {
      "turn": "0",
      "x": "1.386126997",
      "y": "0.561624023"
    },
    {
      "turn": "0",
      "x": "0.399872171",
      "y": "1.101465490"
    },
    {
      "turn": "0",
      "x": "1.275008796",
      "y": "0.810106190"
    },
    {
      "turn": "0",
      "x": "0.669739459",
      "y": "1.065936788"
    },
    {
      "turn": "0",
      "x": "1.703388593",
      "y": "0.650042497"
    },
    {
      "turn": "0",
      "x": "0.994315783",
      "y": "0.976564705"
    },
    {
      "turn": "0",
      "x": "0.293040161",
      "y": "1.351820366"
    },
    {
      "turn": "0",
      "x": "1.540887355",
      "y": "0.868409138"
    },
    {
      "turn": "0",
      "x": "0.563269848",
      "y": "1.319162163"
    },
    {
      "turn": "0",
      "x": "0.888813639",
      "y": "1.227482898"
    },
    {
      "turn": "0",
      "x": "1.245792062",
      "y": "1.080729590"
    },
    {
      "turn": "0",
      "x": "1.811249032",
      "y": "0.899956018"
    },
    {
      "turn": "0",
      "x": "1.511670621",
      "y": "1.139032538"
    },
    {
      "turn": "0",
      "x": "0.783860544",
      "y": "1.478631241"
    },
    {
      "turn": "0",
      "x": "1.140289918",
      "y": "1.331647783"
    },
    {
      "turn": "0",
      "x": "0.462549601",
      "y": "1.628934058"
    },
    {
      "turn": "0",
      "x": "1.782032298",
      "y": "1.170579418"
    },
    {
      "turn": "0",
      "x": "1.406168477",
      "y": "1.389950731"
    },
    {
      "turn": "0",
      "x": "1.035336823",
      "y": "1.582796126"
    },
    {
      "turn": "0",
      "x": "0.748523721",
      "y": "1.748523721"
    },
    {
      "turn": "0",
      "x": "1.677603467",
      "y": "1.422520332"
    },
    {
      "turn": "0",
      "x": "1.251471306",
      "y": "1.748511716"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.852688607"
    },
    {
      "turn": "0",
      "x": "1.502952557",
      "y": "1.644358836"
    }
  ],
  "radius": "0.136097980"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 38,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.573571172",
      "y": "0.323943997"
    },
    {
      "turn": "0",
      "x": "0.825047451",
      "y": "0.219779112"
    },
    {
      "turn": "0",
      "x": "0.358328126",
      "y": "0.490560543"
    },
    {
      "turn": "0",
      "x": "1.095233058",
      "y": "0.186758218"
    },
    {
      "turn": "0",
      "x": "0.857830114",
      "y": "0.489993729"
    },
    {
      "turn": "0",
      "x": "0.254163241",
      "y": "0.742036822"
    },
    {
      "turn": "0",
      "x": "0.608201941",
      "y": "0.598512933"
    },
    {
      "turn": "0",
      "x": "1.349181389",
      "y": "0.291947061"
    },
    {
      "turn": "0",
      "x": "1.132295322",
      "y": "0.456419173"
    },
    {
      "turn": "0",
      "x": "0.149998355",
      "y": "0.993513100"
    },
    {
      "turn": "0",
      "x": "0.504037056",
      "y": "0.849989212"
    },
    {
      "turn": "0",
      "x": "1.602941896",
      "y": "0.397058104"
    },
    {
      "turn": "0",
      "x": "1.023532517",
      "y": "0.705941305"
    },
    {
      "turn": "0",
      "x": "0.773904344",
      "y": "0.814460509"
    },
    {
      "turn": "0",
      "x": "1.386126997",
      "y": "0.561624023"
    },
    {
      "turn": "0",
      "x": "0.399872171",
      "y": "1.101465490"
    },
    {
      "turn": "0",
      "x": "1.275008796",
      "y": "0.810106190"
    },
    {
      "turn": "0",
      "x": "0.669739459",
      "y": "1.065936788"
    },
    {
      "turn": "0",
      "x": "1.703388593",
      "y": "0.650042497"
    },
    {
      "turn": "0",
      "x": "0.994315783",
      "y": "0.976564705"
    },
    {
      "turn": "0",
      "x": "0.293040161",
      "y": "1.351820366"
    },
    {
      "turn": "0",
      "x": "1.540887355",
      "y": "0.868409138"
    },
    {
      "turn": "0",
      "x": "0.563269848",
      "y": "1.319162163"
    },
    {
      "turn": "0",
      "x": "0.888813639",
      "y": "1.227482898"
    },
    {
      "turn": "0",
      "x": "1.245792062",
      "y": "1.080729590"
    },
    {
      "turn": "0",
      "x": "1.811249032",
      "y": "0.899956018"
    },
    {
      "turn": "0",
      "x": "1.511670621",
      "y": "1.139032538"
    },
    {
      "turn": "0",
      "x": "0.783860544",
      "y": "1.478631241"
    },
    {
      "turn": "0",
      "x": "1.140289918",
      "y": "1.331647783"
    },
    {
      "turn": "0",
      "x": "0.462549601",
      "y": "1.628934058"
    },
    {
      "turn": "0",
      "x": "1.782032298",
      "y": "1.170579418"
    },
    {
      "turn": "0",
      "x": "1.406168477",
      "y": "1.389950731"
    },
    {
      "turn": "0",
      "x": "1.035336823",
      "y": "1.582796126"
    },
    {
      "turn": "0",
      "x": "0.748523721",
      "y": "1.748523721"
    },
    {
      "turn": "0",
      "x": "1.677603467",
      "y": "1.422520332"
    },
    {
      "turn": "0",
      "x": "1.251471306",
      "y": "1.748511716"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.852688607"
    },
    {
      "turn": "0",
      "x": "1.502952557",
      "y": "1.644358836"
    }
  ],
  "radius": "0.136097980"
}

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?