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P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 26

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

Instancen = 26
ObjectiveMinimize container triangle side
Best known, unproven8.618Source-precision interval: 8.618–8.619 (values inside match)exhibited site certificate 8.618802344Compiled by Erich Friedman; Found by Erich Friedman in 1997. Source expression: s = 4 + 8 / √3 = 8.618+.

Formal definition

  • ContainerThe editor and certificate normalize to the equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2). 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 triangle side = 1/(2 sin(pi/4) 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 Squares in an equilateral triangle n = 26, 8.618802344
Current leader

8.618802344

container triangle side

Matches the best known
Answer sourceErich Friedman
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 equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2)

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": "0.131652498",
      "x": "0.438702368",
      "y": "0.179727772"
    },
    {
      "turn": "0.131652498",
      "x": "0.304727772",
      "y": "0.179727772"
    },
    {
      "turn": "0.131652498",
      "x": "0.112740474",
      "y": "0.079246825"
    },
    {
      "turn": "-0.131652498",
      "x": "0.445272228",
      "y": "0.612740474"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512259526",
      "y": "0.728765877"
    },
    {
      "turn": "-0.131652498",
      "x": "0.570272228",
      "y": "0.628284930"
    },
    {
      "turn": "-0.131652498",
      "x": "0.628284930",
      "y": "0.527803983"
    },
    {
      "turn": "-0.131652498",
      "x": "0.503284930",
      "y": "0.512259526"
    },
    {
      "turn": "-0.131652498",
      "x": "0.699759526",
      "y": "0.404006351"
    },
    {
      "turn": "-0.131652498",
      "x": "0.561297632",
      "y": "0.411778579"
    },
    {
      "turn": "-0.131652498",
      "x": "0.378284930",
      "y": "0.496715070"
    },
    {
      "turn": "-0.131652498",
      "x": "0.757772228",
      "y": "0.303525404"
    },
    {
      "turn": "-0.131652498",
      "x": "0.619310334",
      "y": "0.311297632"
    },
    {
      "turn": "-0.131652498",
      "x": "0.436297632",
      "y": "0.396234123"
    },
    {
      "turn": "0.131652498",
      "x": "0.286778579",
      "y": "0.380689666"
    },
    {
      "turn": "-0.131652498",
      "x": "0.815784930",
      "y": "0.203044457"
    },
    {
      "turn": "-0.131652498",
      "x": "0.677323036",
      "y": "0.210816685"
    },
    {
      "turn": "0.131652498",
      "x": "0.362740474",
      "y": "0.280208719"
    },
    {
      "turn": "0.131652498",
      "x": "0.228765877",
      "y": "0.280208719"
    },
    {
      "turn": "-0.414213562",
      "x": "0.875000000",
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    },
    {
      "turn": "-0.414213562",
      "x": "0.758974596",
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    },
    {
      "turn": "-0.414213562",
      "x": "0.642949192",
      "y": "0.058012702"
    },
    {
      "turn": "-0.414213562",
      "x": "0.526923789",
      "y": "0.058012702"
    },
    {
      "turn": "0.131652498",
      "x": "0.380689666",
      "y": "0.079246825"
    },
    {
      "turn": "0.131652498",
      "x": "0.246715070",
      "y": "0.079246825"
    },
    {
      "turn": "0.131652498",
      "x": "0.170753175",
      "y": "0.179727772"
    }
  ],
  "radius": "0.082042348"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.438702368",
      "y": "0.179727772"
    },
    {
      "turn": "0.131652498",
      "x": "0.304727772",
      "y": "0.179727772"
    },
    {
      "turn": "0.131652498",
      "x": "0.112740474",
      "y": "0.079246825"
    },
    {
      "turn": "-0.131652498",
      "x": "0.445272228",
      "y": "0.612740474"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512259526",
      "y": "0.728765877"
    },
    {
      "turn": "-0.131652498",
      "x": "0.570272228",
      "y": "0.628284930"
    },
    {
      "turn": "-0.131652498",
      "x": "0.628284930",
      "y": "0.527803983"
    },
    {
      "turn": "-0.131652498",
      "x": "0.503284930",
      "y": "0.512259526"
    },
    {
      "turn": "-0.131652498",
      "x": "0.699759526",
      "y": "0.404006351"
    },
    {
      "turn": "-0.131652498",
      "x": "0.561297632",
      "y": "0.411778579"
    },
    {
      "turn": "-0.131652498",
      "x": "0.378284930",
      "y": "0.496715070"
    },
    {
      "turn": "-0.131652498",
      "x": "0.757772228",
      "y": "0.303525404"
    },
    {
      "turn": "-0.131652498",
      "x": "0.619310334",
      "y": "0.311297632"
    },
    {
      "turn": "-0.131652498",
      "x": "0.436297632",
      "y": "0.396234123"
    },
    {
      "turn": "0.131652498",
      "x": "0.286778579",
      "y": "0.380689666"
    },
    {
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      "x": "0.815784930",
      "y": "0.203044457"
    },
    {
      "turn": "-0.131652498",
      "x": "0.677323036",
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    },
    {
      "turn": "0.131652498",
      "x": "0.362740474",
      "y": "0.280208719"
    },
    {
      "turn": "0.131652498",
      "x": "0.228765877",
      "y": "0.280208719"
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    {
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      "x": "0.875000000",
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    },
    {
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      "x": "0.758974596",
      "y": "0.058012702"
    },
    {
      "turn": "-0.414213562",
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    {
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    },
    {
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    },
    {
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      "x": "0.246715070",
      "y": "0.079246825"
    },
    {
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      "x": "0.170753175",
      "y": "0.179727772"
    }
  ],
  "radius": "0.082042348"
}

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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