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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 28

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

Instancen = 28
ObjectiveMinimize container circle radius
Best known, unproven2.18805Source-precision interval: 2.18805–2.18806 (values inside match)exhibited site certificate 2.188057309Compiled by Erich Friedman; Found by Thomas Schadt in August 2026. Source expression: r = 2.18805+.

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 = 28, 2.188057309
Current leader

2.188057309

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": "0.382247798",
      "x": "0.864546531",
      "y": "1.822123388"
    },
    {
      "turn": "-0.159829549",
      "x": "1.115265579",
      "y": "1.739877759"
    },
    {
      "turn": "-0.057010629",
      "x": "1.500913449",
      "y": "1.676306758"
    },
    {
      "turn": "0.245556389",
      "x": "0.546949458",
      "y": "1.595652307"
    },
    {
      "turn": "-0.298461953",
      "x": "0.309281182",
      "y": "1.473622703"
    },
    {
      "turn": "0.442982540",
      "x": "1.188690398",
      "y": "1.462206592"
    },
    {
      "turn": "-0.283137199",
      "x": "0.773353995",
      "y": "1.454110798"
    },
    {
      "turn": "0.503758381",
      "x": "1.606019824",
      "y": "1.434279830"
    },
    {
      "turn": "-0.107001455",
      "x": "1.407961835",
      "y": "1.227983311"
    },
    {
      "turn": "0.252875654",
      "x": "0.469749353",
      "y": "1.198714960"
    },
    {
      "turn": "0.252875654",
      "x": "0.926593652",
      "y": "1.185813846"
    },
    {
      "turn": "-0.485401586",
      "x": "1.832970723",
      "y": "1.120270072"
    },
    {
      "turn": "-0.283137199",
      "x": "0.233107022",
      "y": "1.081607117"
    },
    {
      "turn": "-0.283137199",
      "x": "0.694447272",
      "y": "1.060384813"
    },
    {
      "turn": "-0.283137199",
      "x": "1.109405887",
      "y": "0.979446283"
    },
    {
      "turn": "0.252875654",
      "x": "1.455365525",
      "y": "0.894227626"
    },
    {
      "turn": "0.252875654",
      "x": "0.225658560",
      "y": "0.817847938"
    },
    {
      "turn": "0.252875654",
      "x": "0.686998810",
      "y": "0.796625634"
    },
    {
      "turn": "-0.335004537",
      "x": "1.704115325",
      "y": "0.754886541"
    },
    {
      "turn": "0.252875654",
      "x": "1.101957426",
      "y": "0.715687104"
    },
    {
      "turn": "-0.283137199",
      "x": "0.450356479",
      "y": "0.679517791"
    },
    {
      "turn": "-0.283137199",
      "x": "0.869811045",
      "y": "0.590258071"
    },
    {
      "turn": "-0.283137199",
      "x": "1.326655344",
      "y": "0.577356957"
    },
    {
      "turn": "0.203069125",
      "x": "1.671765619",
      "y": "0.493012754"
    },
    {
      "turn": "0.492597609",
      "x": "0.487072746",
      "y": "0.332758854"
    },
    {
      "turn": "0.252875654",
      "x": "0.862362584",
      "y": "0.326498893"
    },
    {
      "turn": "0.252875654",
      "x": "1.319206883",
      "y": "0.313597778"
    },
    {
      "turn": "-0.283137199",
      "x": "1.087060502",
      "y": "0.188168746"
    }
  ],
  "radius": "0.263864327"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 28,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.382247798",
      "x": "0.864546531",
      "y": "1.822123388"
    },
    {
      "turn": "-0.159829549",
      "x": "1.115265579",
      "y": "1.739877759"
    },
    {
      "turn": "-0.057010629",
      "x": "1.500913449",
      "y": "1.676306758"
    },
    {
      "turn": "0.245556389",
      "x": "0.546949458",
      "y": "1.595652307"
    },
    {
      "turn": "-0.298461953",
      "x": "0.309281182",
      "y": "1.473622703"
    },
    {
      "turn": "0.442982540",
      "x": "1.188690398",
      "y": "1.462206592"
    },
    {
      "turn": "-0.283137199",
      "x": "0.773353995",
      "y": "1.454110798"
    },
    {
      "turn": "0.503758381",
      "x": "1.606019824",
      "y": "1.434279830"
    },
    {
      "turn": "-0.107001455",
      "x": "1.407961835",
      "y": "1.227983311"
    },
    {
      "turn": "0.252875654",
      "x": "0.469749353",
      "y": "1.198714960"
    },
    {
      "turn": "0.252875654",
      "x": "0.926593652",
      "y": "1.185813846"
    },
    {
      "turn": "-0.485401586",
      "x": "1.832970723",
      "y": "1.120270072"
    },
    {
      "turn": "-0.283137199",
      "x": "0.233107022",
      "y": "1.081607117"
    },
    {
      "turn": "-0.283137199",
      "x": "0.694447272",
      "y": "1.060384813"
    },
    {
      "turn": "-0.283137199",
      "x": "1.109405887",
      "y": "0.979446283"
    },
    {
      "turn": "0.252875654",
      "x": "1.455365525",
      "y": "0.894227626"
    },
    {
      "turn": "0.252875654",
      "x": "0.225658560",
      "y": "0.817847938"
    },
    {
      "turn": "0.252875654",
      "x": "0.686998810",
      "y": "0.796625634"
    },
    {
      "turn": "-0.335004537",
      "x": "1.704115325",
      "y": "0.754886541"
    },
    {
      "turn": "0.252875654",
      "x": "1.101957426",
      "y": "0.715687104"
    },
    {
      "turn": "-0.283137199",
      "x": "0.450356479",
      "y": "0.679517791"
    },
    {
      "turn": "-0.283137199",
      "x": "0.869811045",
      "y": "0.590258071"
    },
    {
      "turn": "-0.283137199",
      "x": "1.326655344",
      "y": "0.577356957"
    },
    {
      "turn": "0.203069125",
      "x": "1.671765619",
      "y": "0.493012754"
    },
    {
      "turn": "0.492597609",
      "x": "0.487072746",
      "y": "0.332758854"
    },
    {
      "turn": "0.252875654",
      "x": "0.862362584",
      "y": "0.326498893"
    },
    {
      "turn": "0.252875654",
      "x": "1.319206883",
      "y": "0.313597778"
    },
    {
      "turn": "-0.283137199",
      "x": "1.087060502",
      "y": "0.188168746"
    }
  ],
  "radius": "0.263864327"
}

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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