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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 21

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

Instancen = 21
ObjectiveMinimize container square side
Best known, unproven3.31102Source-precision interval: 3.31102–3.31103 (values inside match)exhibited site certificate 3.31102025Compiled by Erich Friedman; Found by Thomas Schadt in June 2026. Source expression: s = 3.31102+.

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 = 21, 3.31102025
Current leader

3.31102025

container square side

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 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": "0.302021713",
      "y": "0.912813841"
    },
    {
      "turn": "0.267949192",
      "x": "0.604043427",
      "y": "0.912813841"
    },
    {
      "turn": "0.418134901",
      "x": "0.877527142",
      "y": "0.875878672"
    },
    {
      "turn": "-0.267949192",
      "x": "0.151010857",
      "y": "0.825627682"
    },
    {
      "turn": "-0.267949192",
      "x": "0.453032570",
      "y": "0.825627682"
    },
    {
      "turn": "-0.128253623",
      "x": "0.755054283",
      "y": "0.751757344"
    },
    {
      "turn": "0.564979236",
      "x": "0.910011926",
      "y": "0.600523330"
    },
    {
      "turn": "0.267949192",
      "x": "0.336477588",
      "y": "0.651255365"
    },
    {
      "turn": "0.018572397",
      "x": "0.092733365",
      "y": "0.590772204"
    },
    {
      "turn": "-0.018572397",
      "x": "0.560664019",
      "y": "0.569846697"
    },
    {
      "turn": "-0.009328245",
      "x": "0.820023852",
      "y": "0.451165198"
    },
    {
      "turn": "-0.552849787",
      "x": "0.185466731",
      "y": "0.443102884"
    },
    {
      "turn": "0.552849787",
      "x": "0.642182718",
      "y": "0.415702586"
    },
    {
      "turn": "-0.267949192",
      "x": "0.409653162",
      "y": "0.348744635"
    },
    {
      "turn": "0.564979236",
      "x": "0.910011926",
      "y": "0.301668396"
    },
    {
      "turn": "0.018572397",
      "x": "0.092733365",
      "y": "0.294889620"
    },
    {
      "turn": "0.267949192",
      "x": "0.354609014",
      "y": "0.174372318"
    },
    {
      "turn": "0.267949192",
      "x": "0.656630728",
      "y": "0.174372318"
    },
    {
      "turn": "-0.267949192",
      "x": "0.807641584",
      "y": "0.087186159"
    },
    {
      "turn": "-0.267949192",
      "x": "0.203598158",
      "y": "0.087186159"
    },
    {
      "turn": "-0.267949192",
      "x": "0.505619871",
      "y": "0.087186159"
    }
  ],
  "radius": "0.174372316"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.302021713",
      "y": "0.912813841"
    },
    {
      "turn": "0.267949192",
      "x": "0.604043427",
      "y": "0.912813841"
    },
    {
      "turn": "0.418134901",
      "x": "0.877527142",
      "y": "0.875878672"
    },
    {
      "turn": "-0.267949192",
      "x": "0.151010857",
      "y": "0.825627682"
    },
    {
      "turn": "-0.267949192",
      "x": "0.453032570",
      "y": "0.825627682"
    },
    {
      "turn": "-0.128253623",
      "x": "0.755054283",
      "y": "0.751757344"
    },
    {
      "turn": "0.564979236",
      "x": "0.910011926",
      "y": "0.600523330"
    },
    {
      "turn": "0.267949192",
      "x": "0.336477588",
      "y": "0.651255365"
    },
    {
      "turn": "0.018572397",
      "x": "0.092733365",
      "y": "0.590772204"
    },
    {
      "turn": "-0.018572397",
      "x": "0.560664019",
      "y": "0.569846697"
    },
    {
      "turn": "-0.009328245",
      "x": "0.820023852",
      "y": "0.451165198"
    },
    {
      "turn": "-0.552849787",
      "x": "0.185466731",
      "y": "0.443102884"
    },
    {
      "turn": "0.552849787",
      "x": "0.642182718",
      "y": "0.415702586"
    },
    {
      "turn": "-0.267949192",
      "x": "0.409653162",
      "y": "0.348744635"
    },
    {
      "turn": "0.564979236",
      "x": "0.910011926",
      "y": "0.301668396"
    },
    {
      "turn": "0.018572397",
      "x": "0.092733365",
      "y": "0.294889620"
    },
    {
      "turn": "0.267949192",
      "x": "0.354609014",
      "y": "0.174372318"
    },
    {
      "turn": "0.267949192",
      "x": "0.656630728",
      "y": "0.174372318"
    },
    {
      "turn": "-0.267949192",
      "x": "0.807641584",
      "y": "0.087186159"
    },
    {
      "turn": "-0.267949192",
      "x": "0.203598158",
      "y": "0.087186159"
    },
    {
      "turn": "-0.267949192",
      "x": "0.505619871",
      "y": "0.087186159"
    }
  ],
  "radius": "0.174372316"
}

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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