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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 25

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

Instancen = 25
ObjectiveMinimize container square side
Best known, unproven3.537Source-precision interval: 3.537–3.538 (values inside match)exhibited site certificate 3.537363534Compiled by Erich Friedman; 25. Found by David W. Cantrell in August 2012. Source expression: s = 3.537+.

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 = 25, 3.537363534
Current leader

3.537363534

container square side

Matches the best known
Answer source25. Found by David W. Cantrell
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": "0.302671216",
      "y": "0.918392573"
    },
    {
      "turn": "0.267949192",
      "x": "0.585367637",
      "y": "0.918392573"
    },
    {
      "turn": "0.297688075",
      "x": "0.863357924",
      "y": "0.910736167"
    },
    {
      "turn": "0.000000000",
      "x": "0.081607427",
      "y": "0.855375011"
    },
    {
      "turn": "-0.267949192",
      "x": "0.444019427",
      "y": "0.836785145"
    },
    {
      "turn": "-0.254547792",
      "x": "0.721826787",
      "y": "0.821276276"
    },
    {
      "turn": "-0.577350269",
      "x": "0.163214855",
      "y": "0.714026800"
    },
    {
      "turn": "-0.577350269",
      "x": "0.918392573",
      "y": "0.662584780"
    },
    {
      "turn": "0.267949192",
      "x": "0.386170493",
      "y": "0.673570291"
    },
    {
      "turn": "0.281440998",
      "x": "0.658912322",
      "y": "0.656431173"
    },
    {
      "turn": "-0.267949192",
      "x": "0.515561276",
      "y": "0.571251992"
    },
    {
      "turn": "0.000000000",
      "x": "0.081607427",
      "y": "0.572678590"
    },
    {
      "turn": "0.000000000",
      "x": "0.836785145",
      "y": "0.521236569"
    },
    {
      "turn": "0.372893868",
      "x": "0.246731708",
      "y": "0.431330379"
    },
    {
      "turn": "-0.577350269",
      "x": "0.918392573",
      "y": "0.379888359"
    },
    {
      "turn": "0.267949192",
      "x": "0.545143765",
      "y": "0.408037137"
    },
    {
      "turn": "-0.267949192",
      "x": "0.689150959",
      "y": "0.331035204"
    },
    {
      "turn": "-0.267949192",
      "x": "0.403795555",
      "y": "0.326429709"
    },
    {
      "turn": "-0.168236666",
      "x": "0.123365854",
      "y": "0.324466190"
    },
    {
      "turn": "0.250040896",
      "x": "0.853566454",
      "y": "0.163122927"
    },
    {
      "turn": "0.267949192",
      "x": "0.282696421",
      "y": "0.163214855"
    },
    {
      "turn": "0.267949192",
      "x": "0.565392842",
      "y": "0.163214855"
    },
    {
      "turn": "-0.267949192",
      "x": "0.141348210",
      "y": "0.081607427"
    },
    {
      "turn": "-0.267949192",
      "x": "0.424044631",
      "y": "0.081607427"
    },
    {
      "turn": "-0.267949192",
      "x": "0.706741052",
      "y": "0.081607427"
    }
  ],
  "radius": "0.163214853"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.302671216",
      "y": "0.918392573"
    },
    {
      "turn": "0.267949192",
      "x": "0.585367637",
      "y": "0.918392573"
    },
    {
      "turn": "0.297688075",
      "x": "0.863357924",
      "y": "0.910736167"
    },
    {
      "turn": "0.000000000",
      "x": "0.081607427",
      "y": "0.855375011"
    },
    {
      "turn": "-0.267949192",
      "x": "0.444019427",
      "y": "0.836785145"
    },
    {
      "turn": "-0.254547792",
      "x": "0.721826787",
      "y": "0.821276276"
    },
    {
      "turn": "-0.577350269",
      "x": "0.163214855",
      "y": "0.714026800"
    },
    {
      "turn": "-0.577350269",
      "x": "0.918392573",
      "y": "0.662584780"
    },
    {
      "turn": "0.267949192",
      "x": "0.386170493",
      "y": "0.673570291"
    },
    {
      "turn": "0.281440998",
      "x": "0.658912322",
      "y": "0.656431173"
    },
    {
      "turn": "-0.267949192",
      "x": "0.515561276",
      "y": "0.571251992"
    },
    {
      "turn": "0.000000000",
      "x": "0.081607427",
      "y": "0.572678590"
    },
    {
      "turn": "0.000000000",
      "x": "0.836785145",
      "y": "0.521236569"
    },
    {
      "turn": "0.372893868",
      "x": "0.246731708",
      "y": "0.431330379"
    },
    {
      "turn": "-0.577350269",
      "x": "0.918392573",
      "y": "0.379888359"
    },
    {
      "turn": "0.267949192",
      "x": "0.545143765",
      "y": "0.408037137"
    },
    {
      "turn": "-0.267949192",
      "x": "0.689150959",
      "y": "0.331035204"
    },
    {
      "turn": "-0.267949192",
      "x": "0.403795555",
      "y": "0.326429709"
    },
    {
      "turn": "-0.168236666",
      "x": "0.123365854",
      "y": "0.324466190"
    },
    {
      "turn": "0.250040896",
      "x": "0.853566454",
      "y": "0.163122927"
    },
    {
      "turn": "0.267949192",
      "x": "0.282696421",
      "y": "0.163214855"
    },
    {
      "turn": "0.267949192",
      "x": "0.565392842",
      "y": "0.163214855"
    },
    {
      "turn": "-0.267949192",
      "x": "0.141348210",
      "y": "0.081607427"
    },
    {
      "turn": "-0.267949192",
      "x": "0.424044631",
      "y": "0.081607427"
    },
    {
      "turn": "-0.267949192",
      "x": "0.706741052",
      "y": "0.081607427"
    }
  ],
  "radius": "0.163214853"
}

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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