P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 9

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

Instancen = 9

ObjectiveMinimize container square side

Best known, unproven2.287Source-precision interval: 2.287–2.288 (values inside match)exhibited site certificate 2.287509502Compiled by Erich Friedman; Found by Maurizio Morandi in July 2008. Source expression: s = 2.287+.

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

2.287509502

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)＋

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": "0.472515919",
      "y": "0.873803743"
    },
    {
      "turn": "0.001769001",
      "x": "0.126968795",
      "y": "0.781869520"
    },
    {
      "turn": "0.490641802",
      "x": "0.845547124",
      "y": "0.800384645"
    },
    {
      "turn": "-0.067568259",
      "x": "0.691094248",
      "y": "0.600769289"
    },
    {
      "turn": "-0.340435332",
      "x": "0.345975815",
      "y": "0.532587553"
    },
    {
      "turn": "0.490641802",
      "x": "0.845547124",
      "y": "0.391282984"
    },
    {
      "turn": "0.267949192",
      "x": "0.437156657",
      "y": "0.252392514"
    },
    {
      "turn": "-0.267949192",
      "x": "0.218578329",
      "y": "0.126196257"
    },
    {
      "turn": "-0.267949192",
      "x": "0.655734986",
      "y": "0.126196257"
    }
  ],
  "radius": "0.252392512"
}

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

Instance parameters

{
  "n": 9,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.472515919",
      "y": "0.873803743"
    },
    {
      "turn": "0.001769001",
      "x": "0.126968795",
      "y": "0.781869520"
    },
    {
      "turn": "0.490641802",
      "x": "0.845547124",
      "y": "0.800384645"
    },
    {
      "turn": "-0.067568259",
      "x": "0.691094248",
      "y": "0.600769289"
    },
    {
      "turn": "-0.340435332",
      "x": "0.345975815",
      "y": "0.532587553"
    },
    {
      "turn": "0.490641802",
      "x": "0.845547124",
      "y": "0.391282984"
    },
    {
      "turn": "0.267949192",
      "x": "0.437156657",
      "y": "0.252392514"
    },
    {
      "turn": "-0.267949192",
      "x": "0.218578329",
      "y": "0.126196257"
    },
    {
      "turn": "-0.267949192",
      "x": "0.655734986",
      "y": "0.126196257"
    }
  ],
  "radius": "0.252392512"
}

DISCUSSION

Discussion

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