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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 19

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

Instancen = 19
ObjectiveMinimize container square side
Best known, unproven3.12929Source-precision interval: 3.12929–3.1293 (values inside match)exhibited site certificate 3.129293565Compiled by Erich Friedman; Found by Jake Loyd in June 2026. Source expression: s = 3.12929+.

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 = 19, 3.129293565
Current leader

3.129293565

container square side

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 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": "0.201097639",
      "y": "0.907750701"
    },
    {
      "turn": "0.267949192",
      "x": "0.520658583",
      "y": "0.907750701"
    },
    {
      "turn": "0.267949192",
      "x": "0.840219528",
      "y": "0.907750701"
    },
    {
      "turn": "-0.267949192",
      "x": "0.680439056",
      "y": "0.815501403"
    },
    {
      "turn": "-0.267949192",
      "x": "0.360878111",
      "y": "0.815501403"
    },
    {
      "turn": "0.000000000",
      "x": "0.092249299",
      "y": "0.679575408"
    },
    {
      "turn": "-0.577350269",
      "x": "0.907750701",
      "y": "0.655720931"
    },
    {
      "turn": "0.267949192",
      "x": "0.446504208",
      "y": "0.631002806"
    },
    {
      "turn": "-0.577350269",
      "x": "0.184498597",
      "y": "0.519794936"
    },
    {
      "turn": "-0.000000000",
      "x": "0.815501403",
      "y": "0.495940458"
    },
    {
      "turn": "-0.577350269",
      "x": "0.631002806",
      "y": "0.446504208"
    },
    {
      "turn": "-0.217340668",
      "x": "0.384670128",
      "y": "0.374317902"
    },
    {
      "turn": "0.000000000",
      "x": "0.092249299",
      "y": "0.360014464"
    },
    {
      "turn": "0.577350269",
      "x": "0.907750701",
      "y": "0.336159986"
    },
    {
      "turn": "0.319871574",
      "x": "0.329208990",
      "y": "0.183656595"
    },
    {
      "turn": "0.267949192",
      "x": "0.666376391",
      "y": "0.184498597"
    },
    {
      "turn": "-0.241336316",
      "x": "0.154971996",
      "y": "0.100116996"
    },
    {
      "turn": "-0.267949192",
      "x": "0.506595919",
      "y": "0.092249299"
    },
    {
      "turn": "-0.267949192",
      "x": "0.826156863",
      "y": "0.092249299"
    }
  ],
  "radius": "0.184498596"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.201097639",
      "y": "0.907750701"
    },
    {
      "turn": "0.267949192",
      "x": "0.520658583",
      "y": "0.907750701"
    },
    {
      "turn": "0.267949192",
      "x": "0.840219528",
      "y": "0.907750701"
    },
    {
      "turn": "-0.267949192",
      "x": "0.680439056",
      "y": "0.815501403"
    },
    {
      "turn": "-0.267949192",
      "x": "0.360878111",
      "y": "0.815501403"
    },
    {
      "turn": "0.000000000",
      "x": "0.092249299",
      "y": "0.679575408"
    },
    {
      "turn": "-0.577350269",
      "x": "0.907750701",
      "y": "0.655720931"
    },
    {
      "turn": "0.267949192",
      "x": "0.446504208",
      "y": "0.631002806"
    },
    {
      "turn": "-0.577350269",
      "x": "0.184498597",
      "y": "0.519794936"
    },
    {
      "turn": "-0.000000000",
      "x": "0.815501403",
      "y": "0.495940458"
    },
    {
      "turn": "-0.577350269",
      "x": "0.631002806",
      "y": "0.446504208"
    },
    {
      "turn": "-0.217340668",
      "x": "0.384670128",
      "y": "0.374317902"
    },
    {
      "turn": "0.000000000",
      "x": "0.092249299",
      "y": "0.360014464"
    },
    {
      "turn": "0.577350269",
      "x": "0.907750701",
      "y": "0.336159986"
    },
    {
      "turn": "0.319871574",
      "x": "0.329208990",
      "y": "0.183656595"
    },
    {
      "turn": "0.267949192",
      "x": "0.666376391",
      "y": "0.184498597"
    },
    {
      "turn": "-0.241336316",
      "x": "0.154971996",
      "y": "0.100116996"
    },
    {
      "turn": "-0.267949192",
      "x": "0.506595919",
      "y": "0.092249299"
    },
    {
      "turn": "-0.267949192",
      "x": "0.826156863",
      "y": "0.092249299"
    }
  ],
  "radius": "0.184498596"
}

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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