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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 34

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 34
ObjectiveMinimize container circle radius
Best known, unproven2.37504Source-precision interval: 2.37504–2.37505 (values inside match)exhibited site certificate 2.375042541Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.37504+.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). 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.
  • Objectivecontainer circle radius = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
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 Equilateral triangles in a disc n = 34, 2.375042541
Current leader

2.375042541

container circle radius

Matches the best known
Answer sourceThomas Schadt
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 closed unit disc centred at (1,1)

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": "0.200541705",
      "x": "1.166948108",
      "y": "1.826557881"
    },
    {
      "turn": "0.200541705",
      "x": "0.795376300",
      "y": "1.770031036"
    },
    {
      "turn": "-0.337707806",
      "x": "0.571082582",
      "y": "1.676301008"
    },
    {
      "turn": "0.200541705",
      "x": "1.471459241",
      "y": "1.624551129"
    },
    {
      "turn": "-0.337707806",
      "x": "0.988695744",
      "y": "1.622651992"
    },
    {
      "turn": "-0.337707806",
      "x": "1.664778685",
      "y": "1.477172085"
    },
    {
      "turn": "0.200541705",
      "x": "0.540108308",
      "y": "1.435191937"
    },
    {
      "turn": "-0.337707806",
      "x": "0.280824915",
      "y": "1.425191366"
    },
    {
      "turn": "-0.337707806",
      "x": "1.279756130",
      "y": "1.403001680"
    },
    {
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      "x": "0.957721470",
      "y": "1.381542921"
    },
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      "x": "0.733427752",
      "y": "1.287812893"
    },
    {
      "turn": "0.200541705",
      "x": "1.611799267",
      "y": "1.238889923"
    },
    {
      "turn": "0.200541705",
      "x": "0.249850641",
      "y": "1.184082294"
    },
    {
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      "x": "1.248781856",
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    },
    {
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      "x": "1.805118710",
      "y": "1.091510879"
    },
    {
      "turn": "-0.337707806",
      "x": "1.024488138",
      "y": "1.068162581"
    },
    {
      "turn": "0.200541705",
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    {
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    },
    {
      "turn": "-0.337707806",
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    },
    {
      "turn": "0.200541705",
      "x": "1.774144436",
      "y": "0.850401807"
    },
    {
      "turn": "0.097516661",
      "x": "0.160083974",
      "y": "0.834615666"
    },
    {
      "turn": "0.200541705",
      "x": "0.993513864",
      "y": "0.827053510"
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    {
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    {
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      "y": "0.709595637"
    },
    {
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    {
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    {
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      "x": "1.604446470",
      "y": "0.626025450"
    },
    {
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      "x": "0.824698912",
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    },
    {
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      "x": "1.134531252",
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    {
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    },
    {
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    {
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    },
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    },
    {
      "turn": "-0.337707806",
      "x": "0.987044082",
      "y": "0.163779024"
    }
  ],
  "radius": "0.243090496"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 34,
  "sides": 5
}

The current leader's answer

{
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    {
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      "x": "1.166948108",
      "y": "1.826557881"
    },
    {
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    },
    {
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      "y": "1.676301008"
    },
    {
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      "y": "1.624551129"
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    {
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    },
    {
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      "x": "1.664778685",
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    },
    {
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      "x": "0.540108308",
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    },
    {
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      "x": "0.280824915",
      "y": "1.425191366"
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    {
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    {
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      "y": "0.163779024"
    }
  ],
  "radius": "0.243090496"
}

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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