P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 11

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

Instancen = 11

ObjectiveMinimize container square side

Best known, unproven2.490Source-precision interval: 2.49–2.491 (values inside match)exhibited site certificate 2.490959084Compiled by Erich Friedman; Found by Maurizio Morandi in June 2008. Source expression: s = 2.490+.

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.

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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

2.490959084

container square side

Matches the best known

Answer sourceMaurizio Morandi

Solution methodPublished reference construction

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Record holder's solver noteNo solver note yet (expand)＋

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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": "0.515787640",
      "y": "0.884110848"
    },
    {
      "turn": "0.490962183",
      "x": "0.858256770",
      "y": "0.816615052"
    },
    {
      "turn": "-0.484525357",
      "x": "0.229352203",
      "y": "0.784651311"
    },
    {
      "turn": "-0.067308898",
      "x": "0.716513540",
      "y": "0.633230104"
    },
    {
      "turn": "0.072534148",
      "x": "0.143642667",
      "y": "0.569302622"
    },
    {
      "turn": "-0.285417374",
      "x": "0.427425173",
      "y": "0.470011513"
    },
    {
      "turn": "-0.577350269",
      "x": "0.884110848",
      "y": "0.432504204"
    },
    {
      "turn": "0.267949192",
      "x": "0.200725900",
      "y": "0.238064781"
    },
    {
      "turn": "0.267949192",
      "x": "0.598548200",
      "y": "0.231778305"
    },
    {
      "turn": "-0.267949192",
      "x": "0.397822301",
      "y": "0.115889152"
    },
    {
      "turn": "-0.267949192",
      "x": "0.799274100",
      "y": "0.115889152"
    }
  ],
  "radius": "0.231778303"
}

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

Instance parameters

{
  "n": 11,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.515787640",
      "y": "0.884110848"
    },
    {
      "turn": "0.490962183",
      "x": "0.858256770",
      "y": "0.816615052"
    },
    {
      "turn": "-0.484525357",
      "x": "0.229352203",
      "y": "0.784651311"
    },
    {
      "turn": "-0.067308898",
      "x": "0.716513540",
      "y": "0.633230104"
    },
    {
      "turn": "0.072534148",
      "x": "0.143642667",
      "y": "0.569302622"
    },
    {
      "turn": "-0.285417374",
      "x": "0.427425173",
      "y": "0.470011513"
    },
    {
      "turn": "-0.577350269",
      "x": "0.884110848",
      "y": "0.432504204"
    },
    {
      "turn": "0.267949192",
      "x": "0.200725900",
      "y": "0.238064781"
    },
    {
      "turn": "0.267949192",
      "x": "0.598548200",
      "y": "0.231778305"
    },
    {
      "turn": "-0.267949192",
      "x": "0.397822301",
      "y": "0.115889152"
    },
    {
      "turn": "-0.267949192",
      "x": "0.799274100",
      "y": "0.115889152"
    }
  ],
  "radius": "0.231778303"
}

DISCUSSION

Discussion

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