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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 28

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

Instancen = 28
ObjectiveMinimize container square side
Best known, unproven3.75707Source-precision interval: 3.75707–3.75708 (values inside match)exhibited site certificate 3.757072811Compiled by Erich Friedman; 28. Found by Jake Loyd in June 2026. Source expression: s = 3.75707+.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]². 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 square side = 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 square n = 28, 3.757072811
Current leader

3.757072811

container square side

Matches the best known
Answer source28. Found by Jake 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 unit square [0,1]²

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": "0.267949192",
      "x": "0.286752568",
      "y": "0.923164881"
    },
    {
      "turn": "0.267949192",
      "x": "0.552917229",
      "y": "0.923164881"
    },
    {
      "turn": "0.267949192",
      "x": "0.819081889",
      "y": "0.923164881"
    },
    {
      "turn": "-0.000000000",
      "x": "0.076835119",
      "y": "0.866917670"
    },
    {
      "turn": "-0.267949192",
      "x": "0.419834899",
      "y": "0.846329762"
    },
    {
      "turn": "-0.267949192",
      "x": "0.685999559",
      "y": "0.846329762"
    },
    {
      "turn": "-0.531309333",
      "x": "0.913989222",
      "y": "0.740877702"
    },
    {
      "turn": "-0.577350269",
      "x": "0.153670238",
      "y": "0.733835340"
    },
    {
      "turn": "0.267949192",
      "x": "0.370968818",
      "y": "0.692659523"
    },
    {
      "turn": "0.267949192",
      "x": "0.637133478",
      "y": "0.692659523"
    },
    {
      "turn": "-0.267949192",
      "x": "0.504051148",
      "y": "0.615824404"
    },
    {
      "turn": "-0.000000000",
      "x": "0.827597393",
      "y": "0.602717802"
    },
    {
      "turn": "0.000000000",
      "x": "0.076835119",
      "y": "0.600753010"
    },
    {
      "turn": "-0.267949192",
      "x": "0.282907436",
      "y": "0.537845834"
    },
    {
      "turn": "-0.577350269",
      "x": "0.923164881",
      "y": "0.480450610"
    },
    {
      "turn": "0.267949192",
      "x": "0.504051148",
      "y": "0.462154166"
    },
    {
      "turn": "-0.267949192",
      "x": "0.636412312",
      "y": "0.384069952"
    },
    {
      "turn": "-0.267949192",
      "x": "0.133082330",
      "y": "0.333841625"
    },
    {
      "turn": "-0.000000000",
      "x": "0.846329762",
      "y": "0.347368280"
    },
    {
      "turn": "0.267949192",
      "x": "0.281465105",
      "y": "0.384175596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.414547435",
      "y": "0.307340477"
    },
    {
      "turn": "-0.577350269",
      "x": "0.923164881",
      "y": "0.214285950"
    },
    {
      "turn": "0.267949192",
      "x": "0.636412312",
      "y": "0.230399714"
    },
    {
      "turn": "0.267949192",
      "x": "0.133082330",
      "y": "0.180171386"
    },
    {
      "turn": "-0.168440884",
      "x": "0.766302720",
      "y": "0.100568753"
    },
    {
      "turn": "0.267949192",
      "x": "0.383946545",
      "y": "0.153670238"
    },
    {
      "turn": "-0.267949192",
      "x": "0.250864215",
      "y": "0.076835119"
    },
    {
      "turn": "-0.267949192",
      "x": "0.517028876",
      "y": "0.076835119"
    }
  ],
  "radius": "0.153670237"
}
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": "0.267949192",
      "x": "0.286752568",
      "y": "0.923164881"
    },
    {
      "turn": "0.267949192",
      "x": "0.552917229",
      "y": "0.923164881"
    },
    {
      "turn": "0.267949192",
      "x": "0.819081889",
      "y": "0.923164881"
    },
    {
      "turn": "-0.000000000",
      "x": "0.076835119",
      "y": "0.866917670"
    },
    {
      "turn": "-0.267949192",
      "x": "0.419834899",
      "y": "0.846329762"
    },
    {
      "turn": "-0.267949192",
      "x": "0.685999559",
      "y": "0.846329762"
    },
    {
      "turn": "-0.531309333",
      "x": "0.913989222",
      "y": "0.740877702"
    },
    {
      "turn": "-0.577350269",
      "x": "0.153670238",
      "y": "0.733835340"
    },
    {
      "turn": "0.267949192",
      "x": "0.370968818",
      "y": "0.692659523"
    },
    {
      "turn": "0.267949192",
      "x": "0.637133478",
      "y": "0.692659523"
    },
    {
      "turn": "-0.267949192",
      "x": "0.504051148",
      "y": "0.615824404"
    },
    {
      "turn": "-0.000000000",
      "x": "0.827597393",
      "y": "0.602717802"
    },
    {
      "turn": "0.000000000",
      "x": "0.076835119",
      "y": "0.600753010"
    },
    {
      "turn": "-0.267949192",
      "x": "0.282907436",
      "y": "0.537845834"
    },
    {
      "turn": "-0.577350269",
      "x": "0.923164881",
      "y": "0.480450610"
    },
    {
      "turn": "0.267949192",
      "x": "0.504051148",
      "y": "0.462154166"
    },
    {
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      "x": "0.636412312",
      "y": "0.384069952"
    },
    {
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      "x": "0.133082330",
      "y": "0.333841625"
    },
    {
      "turn": "-0.000000000",
      "x": "0.846329762",
      "y": "0.347368280"
    },
    {
      "turn": "0.267949192",
      "x": "0.281465105",
      "y": "0.384175596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.414547435",
      "y": "0.307340477"
    },
    {
      "turn": "-0.577350269",
      "x": "0.923164881",
      "y": "0.214285950"
    },
    {
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      "x": "0.636412312",
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    {
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    {
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    },
    {
      "turn": "0.267949192",
      "x": "0.383946545",
      "y": "0.153670238"
    },
    {
      "turn": "-0.267949192",
      "x": "0.250864215",
      "y": "0.076835119"
    },
    {
      "turn": "-0.267949192",
      "x": "0.517028876",
      "y": "0.076835119"
    }
  ],
  "radius": "0.153670237"
}

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

DISCUSSION

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