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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 29

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

Instancen = 29
ObjectiveMinimize container circle radius
Best known, unproven2.20935Source-precision interval: 2.20935–2.20936 (values inside match)exhibited site certificate 2.209353136Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: r = 2.20935+.

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 = 29, 2.209353136
Current leader

2.209353136

container circle radius

Matches the best known
Answer sourceJake Loyd
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": "0.357898083",
      "x": "0.833888532",
      "y": "1.819761002"
    },
    {
      "turn": "-0.181871594",
      "x": "1.083586224",
      "y": "1.741336527"
    },
    {
      "turn": "-0.028216365",
      "x": "1.462212158",
      "y": "1.705460535"
    },
    {
      "turn": "0.391515212",
      "x": "0.510698516",
      "y": "1.596392483"
    },
    {
      "turn": "-0.151573230",
      "x": "0.771771809",
      "y": "1.482739074"
    },
    {
      "turn": "0.357898083",
      "x": "1.126196445",
      "y": "1.483512923"
    },
    {
      "turn": "-0.151573230",
      "x": "0.318842424",
      "y": "1.418966775"
    },
    {
      "turn": "0.106398833",
      "x": "1.649158281",
      "y": "1.371214360"
    },
    {
      "turn": "-0.432457393",
      "x": "1.401336293",
      "y": "1.264936601"
    },
    {
      "turn": "0.391515212",
      "x": "0.829498933",
      "y": "1.227873970"
    },
    {
      "turn": "-0.443695491",
      "x": "1.824512815",
      "y": "1.177463414"
    },
    {
      "turn": "0.391515212",
      "x": "0.401013804",
      "y": "1.169638312"
    },
    {
      "turn": "-0.151573230",
      "x": "1.064132206",
      "y": "1.102251669"
    },
    {
      "turn": "-0.151573230",
      "x": "0.637642842",
      "y": "1.050448262"
    },
    {
      "turn": "-0.151573230",
      "x": "0.209157712",
      "y": "0.992212605"
    },
    {
      "turn": "0.218849942",
      "x": "1.491184027",
      "y": "0.951588144"
    },
    {
      "turn": "0.391515212",
      "x": "1.121859330",
      "y": "0.847386566"
    },
    {
      "turn": "-0.291228043",
      "x": "1.722797364",
      "y": "0.808243527"
    },
    {
      "turn": "0.391515212",
      "x": "0.695369966",
      "y": "0.795583159"
    },
    {
      "turn": "0.391515212",
      "x": "0.266884836",
      "y": "0.737347501"
    },
    {
      "turn": "-0.151573230",
      "x": "0.930003238",
      "y": "0.669960858"
    },
    {
      "turn": "-0.151573230",
      "x": "0.503513874",
      "y": "0.618157451"
    },
    {
      "turn": "-0.151573230",
      "x": "1.321960961",
      "y": "0.610470547"
    },
    {
      "turn": "0.244938177",
      "x": "1.711516802",
      "y": "0.547166161"
    },
    {
      "turn": "0.391515212",
      "x": "0.987730363",
      "y": "0.415095755"
    },
    {
      "turn": "0.391515212",
      "x": "1.379688085",
      "y": "0.355605444"
    },
    {
      "turn": "0.391515212",
      "x": "0.512038406",
      "y": "0.352147926"
    },
    {
      "turn": "-0.151573230",
      "x": "0.761621622",
      "y": "0.274708531"
    },
    {
      "turn": "-0.151573230",
      "x": "1.187831993",
      "y": "0.178179736"
    }
  ],
  "radius": "0.261320954"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 29,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.357898083",
      "x": "0.833888532",
      "y": "1.819761002"
    },
    {
      "turn": "-0.181871594",
      "x": "1.083586224",
      "y": "1.741336527"
    },
    {
      "turn": "-0.028216365",
      "x": "1.462212158",
      "y": "1.705460535"
    },
    {
      "turn": "0.391515212",
      "x": "0.510698516",
      "y": "1.596392483"
    },
    {
      "turn": "-0.151573230",
      "x": "0.771771809",
      "y": "1.482739074"
    },
    {
      "turn": "0.357898083",
      "x": "1.126196445",
      "y": "1.483512923"
    },
    {
      "turn": "-0.151573230",
      "x": "0.318842424",
      "y": "1.418966775"
    },
    {
      "turn": "0.106398833",
      "x": "1.649158281",
      "y": "1.371214360"
    },
    {
      "turn": "-0.432457393",
      "x": "1.401336293",
      "y": "1.264936601"
    },
    {
      "turn": "0.391515212",
      "x": "0.829498933",
      "y": "1.227873970"
    },
    {
      "turn": "-0.443695491",
      "x": "1.824512815",
      "y": "1.177463414"
    },
    {
      "turn": "0.391515212",
      "x": "0.401013804",
      "y": "1.169638312"
    },
    {
      "turn": "-0.151573230",
      "x": "1.064132206",
      "y": "1.102251669"
    },
    {
      "turn": "-0.151573230",
      "x": "0.637642842",
      "y": "1.050448262"
    },
    {
      "turn": "-0.151573230",
      "x": "0.209157712",
      "y": "0.992212605"
    },
    {
      "turn": "0.218849942",
      "x": "1.491184027",
      "y": "0.951588144"
    },
    {
      "turn": "0.391515212",
      "x": "1.121859330",
      "y": "0.847386566"
    },
    {
      "turn": "-0.291228043",
      "x": "1.722797364",
      "y": "0.808243527"
    },
    {
      "turn": "0.391515212",
      "x": "0.695369966",
      "y": "0.795583159"
    },
    {
      "turn": "0.391515212",
      "x": "0.266884836",
      "y": "0.737347501"
    },
    {
      "turn": "-0.151573230",
      "x": "0.930003238",
      "y": "0.669960858"
    },
    {
      "turn": "-0.151573230",
      "x": "0.503513874",
      "y": "0.618157451"
    },
    {
      "turn": "-0.151573230",
      "x": "1.321960961",
      "y": "0.610470547"
    },
    {
      "turn": "0.244938177",
      "x": "1.711516802",
      "y": "0.547166161"
    },
    {
      "turn": "0.391515212",
      "x": "0.987730363",
      "y": "0.415095755"
    },
    {
      "turn": "0.391515212",
      "x": "1.379688085",
      "y": "0.355605444"
    },
    {
      "turn": "0.391515212",
      "x": "0.512038406",
      "y": "0.352147926"
    },
    {
      "turn": "-0.151573230",
      "x": "0.761621622",
      "y": "0.274708531"
    },
    {
      "turn": "-0.151573230",
      "x": "1.187831993",
      "y": "0.178179736"
    }
  ],
  "radius": "0.261320954"
}

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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