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

Equal circles in regular polygons · m = 6 · n = 34

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 = 34
ObjectiveMaximize circle radius
Best known, unproven0.140313381692Source-precision interval: 0.140313381691–0.140313381693 (values inside match)exhibited site certificate 0.140313381Packomania (compiled by Eckard Specht), m=6, n=34. 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 = 6 · n = 34, 0.140313381
Current leader

0.140313381

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": "0.588654230",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "1.149907757",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "1.429897828",
      "y": "0.293180837"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "0.513940188"
    },
    {
      "turn": "0",
      "x": "0.728967612",
      "y": "0.517317884"
    },
    {
      "turn": "0",
      "x": "1.009594375",
      "y": "0.517317884"
    },
    {
      "turn": "0",
      "x": "1.289584447",
      "y": "0.536210743"
    },
    {
      "turn": "0",
      "x": "1.570211210",
      "y": "0.536210743"
    },
    {
      "turn": "0",
      "x": "0.302333319",
      "y": "0.756970094"
    },
    {
      "turn": "0",
      "x": "0.582960082",
      "y": "0.756970094"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "0.760347790"
    },
    {
      "turn": "0",
      "x": "1.220028367",
      "y": "0.808080808"
    },
    {
      "turn": "0",
      "x": "1.631374137",
      "y": "0.810091149"
    },
    {
      "turn": "0",
      "x": "0.162019937",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.723273464",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.015288523",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.424768211",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.837980063",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.631374137",
      "y": "1.189908851"
    },
    {
      "turn": "0",
      "x": "1.220028367",
      "y": "1.191919192"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "1.239652210"
    },
    {
      "turn": "0",
      "x": "0.302333319",
      "y": "1.243029906"
    },
    {
      "turn": "0",
      "x": "0.582960082",
      "y": "1.243029906"
    },
    {
      "turn": "0",
      "x": "1.289584447",
      "y": "1.463789257"
    },
    {
      "turn": "0",
      "x": "1.570211210",
      "y": "1.463789257"
    },
    {
      "turn": "0",
      "x": "0.728967612",
      "y": "1.482682116"
    },
    {
      "turn": "0",
      "x": "1.009594375",
      "y": "1.482682116"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "1.486059812"
    },
    {
      "turn": "0",
      "x": "1.429897828",
      "y": "1.706819163"
    },
    {
      "turn": "0",
      "x": "0.588654230",
      "y": "1.725712022"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "1.725712022"
    },
    {
      "turn": "0",
      "x": "1.149907757",
      "y": "1.725712022"
    }
  ],
  "radius": "0.140313381"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 34,
  "sides": 6
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.588654230",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "1.149907757",
      "y": "0.274287978"
    },
    {
      "turn": "0",
      "x": "1.429897828",
      "y": "0.293180837"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "0.513940188"
    },
    {
      "turn": "0",
      "x": "0.728967612",
      "y": "0.517317884"
    },
    {
      "turn": "0",
      "x": "1.009594375",
      "y": "0.517317884"
    },
    {
      "turn": "0",
      "x": "1.289584447",
      "y": "0.536210743"
    },
    {
      "turn": "0",
      "x": "1.570211210",
      "y": "0.536210743"
    },
    {
      "turn": "0",
      "x": "0.302333319",
      "y": "0.756970094"
    },
    {
      "turn": "0",
      "x": "0.582960082",
      "y": "0.756970094"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "0.760347790"
    },
    {
      "turn": "0",
      "x": "1.220028367",
      "y": "0.808080808"
    },
    {
      "turn": "0",
      "x": "1.631374137",
      "y": "0.810091149"
    },
    {
      "turn": "0",
      "x": "0.162019937",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.723273464",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.015288523",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.424768211",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.837980063",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.631374137",
      "y": "1.189908851"
    },
    {
      "turn": "0",
      "x": "1.220028367",
      "y": "1.191919192"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "1.239652210"
    },
    {
      "turn": "0",
      "x": "0.302333319",
      "y": "1.243029906"
    },
    {
      "turn": "0",
      "x": "0.582960082",
      "y": "1.243029906"
    },
    {
      "turn": "0",
      "x": "1.289584447",
      "y": "1.463789257"
    },
    {
      "turn": "0",
      "x": "1.570211210",
      "y": "1.463789257"
    },
    {
      "turn": "0",
      "x": "0.728967612",
      "y": "1.482682116"
    },
    {
      "turn": "0",
      "x": "1.009594375",
      "y": "1.482682116"
    },
    {
      "turn": "0",
      "x": "0.442646701",
      "y": "1.486059812"
    },
    {
      "turn": "0",
      "x": "1.429897828",
      "y": "1.706819163"
    },
    {
      "turn": "0",
      "x": "0.588654230",
      "y": "1.725712022"
    },
    {
      "turn": "0",
      "x": "0.869280993",
      "y": "1.725712022"
    },
    {
      "turn": "0",
      "x": "1.149907757",
      "y": "1.725712022"
    }
  ],
  "radius": "0.140313381"
}

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?