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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 17

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

Instancen = 17
ObjectiveMinimize container square side
Best known, unproven2.970Source-precision interval: 2.97–2.971 (values inside match)exhibited site certificate 2.970710991Compiled by Erich Friedman; Found by David W. Cantrell in August 2007. Source expression: s = 2.970+.

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 = 17, 2.970710991
Current leader

2.970710991

container square side

Matches the best known
Answer sourceDavid W. Cantrell
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": [
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.831690116",
      "y": "0.495070354"
    },
    {
      "quarter": 2,
      "turn": "-0.412509052",
      "x": "0.000694056",
      "y": "0.716995177"
    },
    {
      "quarter": 2,
      "turn": "0.414213543",
      "x": "0.238026113",
      "y": "0.476052243"
    },
    {
      "quarter": 2,
      "turn": "0.414213558",
      "x": "0.238026113",
      "y": "0.000000005"
    },
    {
      "quarter": 2,
      "turn": "0.411976182",
      "x": "0.718335617",
      "y": "0.000911991"
    },
    {
      "quarter": 2,
      "turn": "-0.000000002",
      "x": "0.831690120",
      "y": "0.831690120"
    },
    {
      "turn": "-0.000000011",
      "x": "0.821975243",
      "y": "0.504785224"
    },
    {
      "turn": "0.414213542",
      "x": "0.238026115",
      "y": "0.476052242"
    },
    {
      "quarter": 3,
      "turn": "0.414213554",
      "x": "0.476052224",
      "y": "0.238026121"
    },
    {
      "quarter": 1,
      "turn": "0.414213558",
      "x": "0.476052229",
      "y": "0.238026113"
    },
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.485355486",
      "y": "0.466748977"
    },
    {
      "quarter": 2,
      "turn": "0.014647703",
      "x": "0.826832680",
      "y": "0.163308023"
    },
    {
      "turn": "-0.000000008",
      "x": "0.831690120",
      "y": "0.831690111"
    },
    {
      "turn": "-0.028271264",
      "x": "0.495133388",
      "y": "0.812608962"
    },
    {
      "quarter": 2,
      "turn": "-0.028274536",
      "x": "0.158530820",
      "y": "0.822448851"
    },
    {
      "quarter": 1,
      "turn": "0.414213559",
      "x": "0.000000004",
      "y": "0.238026120"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.495070364",
      "y": "0.831690118"
    }
  ],
  "radius": "0.336619753"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.831690116",
      "y": "0.495070354"
    },
    {
      "quarter": 2,
      "turn": "-0.412509052",
      "x": "0.000694056",
      "y": "0.716995177"
    },
    {
      "quarter": 2,
      "turn": "0.414213543",
      "x": "0.238026113",
      "y": "0.476052243"
    },
    {
      "quarter": 2,
      "turn": "0.414213558",
      "x": "0.238026113",
      "y": "0.000000005"
    },
    {
      "quarter": 2,
      "turn": "0.411976182",
      "x": "0.718335617",
      "y": "0.000911991"
    },
    {
      "quarter": 2,
      "turn": "-0.000000002",
      "x": "0.831690120",
      "y": "0.831690120"
    },
    {
      "turn": "-0.000000011",
      "x": "0.821975243",
      "y": "0.504785224"
    },
    {
      "turn": "0.414213542",
      "x": "0.238026115",
      "y": "0.476052242"
    },
    {
      "quarter": 3,
      "turn": "0.414213554",
      "x": "0.476052224",
      "y": "0.238026121"
    },
    {
      "quarter": 1,
      "turn": "0.414213558",
      "x": "0.476052229",
      "y": "0.238026113"
    },
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.485355486",
      "y": "0.466748977"
    },
    {
      "quarter": 2,
      "turn": "0.014647703",
      "x": "0.826832680",
      "y": "0.163308023"
    },
    {
      "turn": "-0.000000008",
      "x": "0.831690120",
      "y": "0.831690111"
    },
    {
      "turn": "-0.028271264",
      "x": "0.495133388",
      "y": "0.812608962"
    },
    {
      "quarter": 2,
      "turn": "-0.028274536",
      "x": "0.158530820",
      "y": "0.822448851"
    },
    {
      "quarter": 1,
      "turn": "0.414213559",
      "x": "0.000000004",
      "y": "0.238026120"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.495070364",
      "y": "0.831690118"
    }
  ],
  "radius": "0.336619753"
}

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

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