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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 33

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

Instancen = 33
ObjectiveMinimize container circle radius
Best known, unproven2.34169Source-precision interval: 2.34169–2.3417 (values inside match)exhibited site certificate 2.341693803Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.34169+.

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 = 33, 2.341693803
Current leader

2.341693803

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": "0.172925527",
      "x": "1.209984722",
      "y": "1.814112101"
    },
    {
      "turn": "-0.367712810",
      "x": "0.977749600",
      "y": "1.731317513"
    },
    {
      "turn": "0.183515780",
      "x": "0.568965132",
      "y": "1.627107576"
    },
    {
      "turn": "0.172925527",
      "x": "1.506187025",
      "y": "1.591544149"
    },
    {
      "turn": "-0.356104219",
      "x": "0.338478455",
      "y": "1.539563374"
    },
    {
      "turn": "-0.367712810",
      "x": "1.273951903",
      "y": "1.508749561"
    },
    {
      "turn": "0.172925527",
      "x": "0.933334255",
      "y": "1.488798704"
    },
    {
      "turn": "-0.367712810",
      "x": "1.694006803",
      "y": "1.431819928"
    },
    {
      "turn": "-0.367712810",
      "x": "0.703818928",
      "y": "1.398375212"
    },
    {
      "turn": "0.172925527",
      "x": "1.255569156",
      "y": "1.261463094"
    },
    {
      "turn": "0.172925527",
      "x": "0.497383399",
      "y": "1.185529080"
    },
    {
      "turn": "0.172925527",
      "x": "1.675624055",
      "y": "1.184533460"
    },
    {
      "turn": "-0.367712810",
      "x": "0.237107331",
      "y": "1.181388094"
    },
    {
      "turn": "-0.367712810",
      "x": "1.023334034",
      "y": "1.178668506"
    },
    {
      "turn": "-0.367712810",
      "x": "1.443388933",
      "y": "1.101738872"
    },
    {
      "turn": "-0.367712810",
      "x": "0.685203177",
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    },
    {
      "turn": "0.172925527",
      "x": "0.192691986",
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    },
    {
      "turn": "0.172925527",
      "x": "0.978918689",
      "y": "0.936149696"
    },
    {
      "turn": "-0.367712810",
      "x": "1.786428684",
      "y": "0.934247094"
    },
    {
      "turn": "0.172925527",
      "x": "1.398973589",
      "y": "0.859220063"
    },
    {
      "turn": "0.172925527",
      "x": "0.640787832",
      "y": "0.783286049"
    },
    {
      "turn": "-0.367712810",
      "x": "0.380511764",
      "y": "0.779145063"
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    {
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    {
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      "x": "1.742013340",
      "y": "0.691728285"
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    {
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      "x": "0.828607609",
      "y": "0.623561828"
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    {
      "turn": "-0.367712810",
      "x": "1.509778218",
      "y": "0.608933697"
    },
    {
      "turn": "0.172925527",
      "x": "0.336096419",
      "y": "0.536626254"
    },
    {
      "turn": "0.172925527",
      "x": "1.122323122",
      "y": "0.533906666"
    },
    {
      "turn": "0.172925527",
      "x": "0.784192265",
      "y": "0.381043019"
    },
    {
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      "x": "0.523916197",
      "y": "0.376902032"
    },
    {
      "turn": "0.172925527",
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    {
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      "x": "1.278785359",
      "y": "0.275258480"
    },
    {
      "turn": "-0.367712810",
      "x": "0.925067245",
      "y": "0.166116391"
    }
  ],
  "radius": "0.246552418"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 33,
  "sides": 5
}

The current leader's answer

{
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      "x": "1.209984722",
      "y": "1.814112101"
    },
    {
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    },
    {
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      "x": "0.568965132",
      "y": "1.627107576"
    },
    {
      "turn": "0.172925527",
      "x": "1.506187025",
      "y": "1.591544149"
    },
    {
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      "x": "0.338478455",
      "y": "1.539563374"
    },
    {
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      "x": "1.273951903",
      "y": "1.508749561"
    },
    {
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      "x": "0.933334255",
      "y": "1.488798704"
    },
    {
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      "x": "1.694006803",
      "y": "1.431819928"
    },
    {
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    {
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      "x": "0.925067245",
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    }
  ],
  "radius": "0.246552418"
}

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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