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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 21

Pack n tans (right isosceles triangles of leg 1) into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 21
ObjectiveMinimize container square side
Best known, unproven3.340Source-precision interval: 3.34–3.341 (values inside match)exhibited site certificate 3.340065737Compiled by Erich Friedman; Found by Maurizio Morandi in March 2008. Source expression: s = 3.340+.

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,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational 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/r (r is the piece's short side inside the unit square), smaller is better. Pages, leaderboards and literature records share this unit; nothing is converted by hand.
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 Tans in a square n = 21, 3.340065737
Current leader

3.340065737

container square side

Matches the best known
Answer sourceMaurizio Morandi
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,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational turn = tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.000000068",
      "x": "0.850302354",
      "y": "0.550906961"
    },
    {
      "quarter": 3,
      "turn": "0.414213555",
      "x": "0.233350328",
      "y": "0.577144755"
    },
    {
      "turn": "0.414213531",
      "x": "0.445054780",
      "y": "0.445054803"
    },
    {
      "quarter": 2,
      "turn": "-0.000000018",
      "x": "0.251511727",
      "y": "0.850302339"
    },
    {
      "quarter": 2,
      "turn": "-0.000000029",
      "x": "0.550907039",
      "y": "0.850302336"
    },
    {
      "quarter": 2,
      "turn": "0.414213472",
      "x": "0.656759220",
      "y": "0.233350315"
    },
    {
      "quarter": 2,
      "turn": "-0.000000077",
      "x": "0.850302323",
      "y": "0.251511641"
    },
    {
      "turn": "-0.000000019",
      "x": "0.449092972",
      "y": "0.149697661"
    },
    {
      "quarter": 2,
      "turn": "-0.000000020",
      "x": "0.550907025",
      "y": "0.550907022"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.149697659",
      "y": "0.149697659"
    },
    {
      "turn": "0.414213471",
      "x": "0.577144787",
      "y": "0.233350324"
    },
    {
      "quarter": 2,
      "turn": "-0.000000033",
      "x": "0.850302335",
      "y": "0.850302335"
    },
    {
      "turn": "-0.000000037",
      "x": "0.850302350",
      "y": "0.850302313"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.149697656"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.449092969"
    },
    {
      "turn": "-0.000000026",
      "x": "0.550907038",
      "y": "0.850302332"
    },
    {
      "quarter": 2,
      "turn": "-0.000000042",
      "x": "0.850302332",
      "y": "0.550907003"
    },
    {
      "quarter": 2,
      "turn": "0.414213529",
      "x": "0.445054784",
      "y": "0.445054806"
    },
    {
      "quarter": 2,
      "turn": "0.384561065",
      "x": "0.788572377",
      "y": "0.010822938"
    },
    {
      "quarter": 2,
      "turn": "-0.384561063",
      "x": "0.010822939",
      "y": "0.788572377"
    },
    {
      "quarter": 1,
      "turn": "0.414213548",
      "x": "0.233350335",
      "y": "0.656759269"
    }
  ],
  "radius": "0.299395305"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.000000068",
      "x": "0.850302354",
      "y": "0.550906961"
    },
    {
      "quarter": 3,
      "turn": "0.414213555",
      "x": "0.233350328",
      "y": "0.577144755"
    },
    {
      "turn": "0.414213531",
      "x": "0.445054780",
      "y": "0.445054803"
    },
    {
      "quarter": 2,
      "turn": "-0.000000018",
      "x": "0.251511727",
      "y": "0.850302339"
    },
    {
      "quarter": 2,
      "turn": "-0.000000029",
      "x": "0.550907039",
      "y": "0.850302336"
    },
    {
      "quarter": 2,
      "turn": "0.414213472",
      "x": "0.656759220",
      "y": "0.233350315"
    },
    {
      "quarter": 2,
      "turn": "-0.000000077",
      "x": "0.850302323",
      "y": "0.251511641"
    },
    {
      "turn": "-0.000000019",
      "x": "0.449092972",
      "y": "0.149697661"
    },
    {
      "quarter": 2,
      "turn": "-0.000000020",
      "x": "0.550907025",
      "y": "0.550907022"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.149697659",
      "y": "0.149697659"
    },
    {
      "turn": "0.414213471",
      "x": "0.577144787",
      "y": "0.233350324"
    },
    {
      "quarter": 2,
      "turn": "-0.000000033",
      "x": "0.850302335",
      "y": "0.850302335"
    },
    {
      "turn": "-0.000000037",
      "x": "0.850302350",
      "y": "0.850302313"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.149697656"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.449092969"
    },
    {
      "turn": "-0.000000026",
      "x": "0.550907038",
      "y": "0.850302332"
    },
    {
      "quarter": 2,
      "turn": "-0.000000042",
      "x": "0.850302332",
      "y": "0.550907003"
    },
    {
      "quarter": 2,
      "turn": "0.414213529",
      "x": "0.445054784",
      "y": "0.445054806"
    },
    {
      "quarter": 2,
      "turn": "0.384561065",
      "x": "0.788572377",
      "y": "0.010822938"
    },
    {
      "quarter": 2,
      "turn": "-0.384561063",
      "x": "0.010822939",
      "y": "0.788572377"
    },
    {
      "quarter": 1,
      "turn": "0.414213548",
      "x": "0.233350335",
      "y": "0.656759269"
    }
  ],
  "radius": "0.299395305"
}

Submit {radius, placements:[{x,y,turn,quarter},…]}, with x, y and turn as decimal strings of at most nine places. radius is the piece's short side (the leg, the domino's short side, the cell side); the container side is 1/radius. x,y is the centre of the piece's bounding box; quarter is an integer 0–3, an exact number of right angles applied about the centre before the rational turn = tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

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