P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 13

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

Instancen = 13

ObjectiveMinimize container triangle side

Best known, unproven6.309Source-precision interval: 6.309–6.31 (values inside match)exhibited site certificate 6.309401135Compiled by Erich Friedman; Found by Erich Friedman in 1997. Source expression: s = 4 + 4 / √3 = 6.309+.

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.

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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

Current leader

6.309401135

container triangle side

Matches the best known

Answer sourceErich Friedman

Solution methodPublished reference construction

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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": "0.154006351",
      "y": "0.108253175"
    },
    {
      "turn": "-0.131652498",
      "x": "0.516746825",
      "y": "0.678525404"
    },
    {
      "turn": "-0.131652498",
      "x": "0.595993649",
      "y": "0.541265877"
    },
    {
      "turn": "-0.131652498",
      "x": "0.675240474",
      "y": "0.404006351"
    },
    {
      "turn": "0.131652498",
      "x": "0.391746825",
      "y": "0.520031755"
    },
    {
      "turn": "-0.131652498",
      "x": "0.754487298",
      "y": "0.266746825"
    },
    {
      "turn": "0.131652498",
      "x": "0.455889290",
      "y": "0.314142465"
    },
    {
      "turn": "0.131652498",
      "x": "0.312500000",
      "y": "0.382772228"
    },
    {
      "turn": "-0.414213562",
      "x": "0.829246825",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.670753175",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.512259526",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.353765877",
      "y": "0.079246825"
    },
    {
      "turn": "0.131652498",
      "x": "0.233253175",
      "y": "0.245512702"
    }
  ],
  "radius": "0.112071933"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.154006351",
      "y": "0.108253175"
    },
    {
      "turn": "-0.131652498",
      "x": "0.516746825",
      "y": "0.678525404"
    },
    {
      "turn": "-0.131652498",
      "x": "0.595993649",
      "y": "0.541265877"
    },
    {
      "turn": "-0.131652498",
      "x": "0.675240474",
      "y": "0.404006351"
    },
    {
      "turn": "0.131652498",
      "x": "0.391746825",
      "y": "0.520031755"
    },
    {
      "turn": "-0.131652498",
      "x": "0.754487298",
      "y": "0.266746825"
    },
    {
      "turn": "0.131652498",
      "x": "0.455889290",
      "y": "0.314142465"
    },
    {
      "turn": "0.131652498",
      "x": "0.312500000",
      "y": "0.382772228"
    },
    {
      "turn": "-0.414213562",
      "x": "0.829246825",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.670753175",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.512259526",
      "y": "0.079246825"
    },
    {
      "turn": "-0.414213562",
      "x": "0.353765877",
      "y": "0.079246825"
    },
    {
      "turn": "0.131652498",
      "x": "0.233253175",
      "y": "0.245512702"
    }
  ],
  "radius": "0.112071933"
}

DISCUSSION

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