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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 16

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

Instancen = 16
ObjectiveMinimize container square side
Best known, unproven2.9000Source-precision interval: 2.9–2.9001 (values inside match)exhibited site certificate 2.900039516Compiled by Erich Friedman; Found by Ian Watson in April 2026. Source expression: s = 2.9000+.

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 = 16, 2.900039516
Current leader

2.900039516

container square side

Matches the best known
Answer sourceIan Watson
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": "0.371495034",
      "y": "0.900458206"
    },
    {
      "turn": "0.267949192",
      "x": "0.716317924",
      "y": "0.900458206"
    },
    {
      "turn": "0.000000000",
      "x": "0.099541794",
      "y": "0.827588555"
    },
    {
      "turn": "-0.267949192",
      "x": "0.543906479",
      "y": "0.800916411"
    },
    {
      "turn": "-0.577350269",
      "x": "0.900458206",
      "y": "0.692747095"
    },
    {
      "turn": "0.577350269",
      "x": "0.199083589",
      "y": "0.655177110"
    },
    {
      "turn": "0.145970203",
      "x": "0.443297505",
      "y": "0.581958161"
    },
    {
      "turn": "-0.000000000",
      "x": "0.800916411",
      "y": "0.520335650"
    },
    {
      "turn": "0.000000000",
      "x": "0.099541794",
      "y": "0.482765664"
    },
    {
      "turn": "-0.397850817",
      "x": "0.573157138",
      "y": "0.429525618"
    },
    {
      "turn": "-0.577350269",
      "x": "0.900458206",
      "y": "0.347924204"
    },
    {
      "turn": "0.479236593",
      "x": "0.249434079",
      "y": "0.310354219"
    },
    {
      "turn": "-0.076850197",
      "x": "0.124717039",
      "y": "0.155177110"
    },
    {
      "turn": "0.241068678",
      "x": "0.604304102",
      "y": "0.198829903"
    },
    {
      "turn": "-0.267949192",
      "x": "0.421845524",
      "y": "0.099541794"
    },
    {
      "turn": "-0.267949192",
      "x": "0.786762681",
      "y": "0.099541794"
    }
  ],
  "radius": "0.199083587"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.371495034",
      "y": "0.900458206"
    },
    {
      "turn": "0.267949192",
      "x": "0.716317924",
      "y": "0.900458206"
    },
    {
      "turn": "0.000000000",
      "x": "0.099541794",
      "y": "0.827588555"
    },
    {
      "turn": "-0.267949192",
      "x": "0.543906479",
      "y": "0.800916411"
    },
    {
      "turn": "-0.577350269",
      "x": "0.900458206",
      "y": "0.692747095"
    },
    {
      "turn": "0.577350269",
      "x": "0.199083589",
      "y": "0.655177110"
    },
    {
      "turn": "0.145970203",
      "x": "0.443297505",
      "y": "0.581958161"
    },
    {
      "turn": "-0.000000000",
      "x": "0.800916411",
      "y": "0.520335650"
    },
    {
      "turn": "0.000000000",
      "x": "0.099541794",
      "y": "0.482765664"
    },
    {
      "turn": "-0.397850817",
      "x": "0.573157138",
      "y": "0.429525618"
    },
    {
      "turn": "-0.577350269",
      "x": "0.900458206",
      "y": "0.347924204"
    },
    {
      "turn": "0.479236593",
      "x": "0.249434079",
      "y": "0.310354219"
    },
    {
      "turn": "-0.076850197",
      "x": "0.124717039",
      "y": "0.155177110"
    },
    {
      "turn": "0.241068678",
      "x": "0.604304102",
      "y": "0.198829903"
    },
    {
      "turn": "-0.267949192",
      "x": "0.421845524",
      "y": "0.099541794"
    },
    {
      "turn": "-0.267949192",
      "x": "0.786762681",
      "y": "0.099541794"
    }
  ],
  "radius": "0.199083587"
}

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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