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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 40

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

Instancen = 40
ObjectiveMinimize container circle radius
Best known, unproven2.55675Source-precision interval: 2.55675–2.55676 (values inside match)exhibited site certificate 2.556750012Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.55675+.

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 = 40, 2.556750012
Current leader

2.556750012

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

{
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      "x": "0.984974908",
      "y": "1.796592872"
    },
    {
      "turn": "-0.574546492",
      "x": "0.759162783",
      "y": "1.795642023"
    },
    {
      "turn": "0.002022616",
      "x": "1.323755526",
      "y": "1.782586425"
    },
    {
      "turn": "0.002154826",
      "x": "0.647041117",
      "y": "1.599607618"
    },
    {
      "turn": "-0.545690442",
      "x": "0.412790876",
      "y": "1.593462975"
    },
    {
      "turn": "-0.574546492",
      "x": "1.040980457",
      "y": "1.567857433"
    },
    {
      "turn": "0.002022616",
      "x": "1.605575844",
      "y": "1.554778703"
    },
    {
      "turn": "-0.574656593",
      "x": "1.379763565",
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    {
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    {
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    {
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    {
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    {
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    {
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  ],
  "radius": "0.225814126"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 40,
  "sides": 5
}

The current leader's answer

{
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      "x": "0.984974908",
      "y": "1.796592872"
    },
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      "x": "1.605575844",
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    {
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      "x": "1.379763565",
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  ],
  "radius": "0.225814126"
}

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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