P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 29

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 = 29

ObjectiveMaximize circle radius

Best known, unproven0.154929849312Source-precision interval: 0.154929849311–0.154929849313 (values inside match)exhibited site certificate 0.154929848Packomania (compiled by Eckard Specht), m=8, n=29. Stefan Zorcic, private communication, July 2026. 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

Current leader

0.154929848

circle radius

Answer sourceStefan Zorcic

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": "1.446916492",
      "y": "0.861648961"
    },
    {
      "turn": "0",
      "x": "0.671545926",
      "y": "0.303744993"
    },
    {
      "turn": "0",
      "x": "0.474903666",
      "y": "1.040444807"
    },
    {
      "turn": "0",
      "x": "1.428526246",
      "y": "1.170962445"
    },
    {
      "turn": "0",
      "x": "1.821137005",
      "y": "0.973311854"
    },
    {
      "turn": "0",
      "x": "0.592180803",
      "y": "1.335603241"
    },
    {
      "turn": "0",
      "x": "1.286273034",
      "y": "0.286273034"
    },
    {
      "turn": "0",
      "x": "0.982788579",
      "y": "1.825175935"
    },
    {
      "turn": "0",
      "x": "1.027549884",
      "y": "0.765251165"
    },
    {
      "turn": "0",
      "x": "1.702333166",
      "y": "0.686219900"
    },
    {
      "turn": "0",
      "x": "0.167694860",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.356325493",
      "y": "0.754171773"
    },
    {
      "turn": "0",
      "x": "1.560116939",
      "y": "1.600297107"
    },
    {
      "turn": "0",
      "x": "0.286273034",
      "y": "1.286273034"
    },
    {
      "turn": "0",
      "x": "1.698492839",
      "y": "1.323051470"
    },
    {
      "turn": "0",
      "x": "0.657236201",
      "y": "0.615271080"
    },
    {
      "turn": "0",
      "x": "0.932898518",
      "y": "0.470201760"
    },
    {
      "turn": "0",
      "x": "0.942343772",
      "y": "1.517967129"
    },
    {
      "turn": "0",
      "x": "1.233399099",
      "y": "1.411666474"
    },
    {
      "turn": "0",
      "x": "1.572546067",
      "y": "0.404851207"
    },
    {
      "turn": "0",
      "x": "0.396770300",
      "y": "0.446962967"
    },
    {
      "turn": "0",
      "x": "1.287187193",
      "y": "0.596131384"
    },
    {
      "turn": "0",
      "x": "0.877585468",
      "y": "1.214949977"
    },
    {
      "turn": "0",
      "x": "0.763234634",
      "y": "0.926962269"
    },
    {
      "turn": "0",
      "x": "1.141900718",
      "y": "1.053238873"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.167694860"
    },
    {
      "turn": "0",
      "x": "0.410753069",
      "y": "1.586794423"
    },
    {
      "turn": "0",
      "x": "0.696515545",
      "y": "1.706597762"
    },
    {
      "turn": "0",
      "x": "1.273843906",
      "y": "1.718875280"
    }
  ],
  "radius": "0.154929848"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.446916492",
      "y": "0.861648961"
    },
    {
      "turn": "0",
      "x": "0.671545926",
      "y": "0.303744993"
    },
    {
      "turn": "0",
      "x": "0.474903666",
      "y": "1.040444807"
    },
    {
      "turn": "0",
      "x": "1.428526246",
      "y": "1.170962445"
    },
    {
      "turn": "0",
      "x": "1.821137005",
      "y": "0.973311854"
    },
    {
      "turn": "0",
      "x": "0.592180803",
      "y": "1.335603241"
    },
    {
      "turn": "0",
      "x": "1.286273034",
      "y": "0.286273034"
    },
    {
      "turn": "0",
      "x": "0.982788579",
      "y": "1.825175935"
    },
    {
      "turn": "0",
      "x": "1.027549884",
      "y": "0.765251165"
    },
    {
      "turn": "0",
      "x": "1.702333166",
      "y": "0.686219900"
    },
    {
      "turn": "0",
      "x": "0.167694860",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.356325493",
      "y": "0.754171773"
    },
    {
      "turn": "0",
      "x": "1.560116939",
      "y": "1.600297107"
    },
    {
      "turn": "0",
      "x": "0.286273034",
      "y": "1.286273034"
    },
    {
      "turn": "0",
      "x": "1.698492839",
      "y": "1.323051470"
    },
    {
      "turn": "0",
      "x": "0.657236201",
      "y": "0.615271080"
    },
    {
      "turn": "0",
      "x": "0.932898518",
      "y": "0.470201760"
    },
    {
      "turn": "0",
      "x": "0.942343772",
      "y": "1.517967129"
    },
    {
      "turn": "0",
      "x": "1.233399099",
      "y": "1.411666474"
    },
    {
      "turn": "0",
      "x": "1.572546067",
      "y": "0.404851207"
    },
    {
      "turn": "0",
      "x": "0.396770300",
      "y": "0.446962967"
    },
    {
      "turn": "0",
      "x": "1.287187193",
      "y": "0.596131384"
    },
    {
      "turn": "0",
      "x": "0.877585468",
      "y": "1.214949977"
    },
    {
      "turn": "0",
      "x": "0.763234634",
      "y": "0.926962269"
    },
    {
      "turn": "0",
      "x": "1.141900718",
      "y": "1.053238873"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.167694860"
    },
    {
      "turn": "0",
      "x": "0.410753069",
      "y": "1.586794423"
    },
    {
      "turn": "0",
      "x": "0.696515545",
      "y": "1.706597762"
    },
    {
      "turn": "0",
      "x": "1.273843906",
      "y": "1.718875280"
    }
  ],
  "radius": "0.154929848"
}

DISCUSSION

Discussion

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