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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 23

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

Instancen = 23
ObjectiveMinimize container square side
Best known, unproven3.464Source-precision interval: 3.464–3.465 (values inside match)exhibited site certificate 3.46400001Compiled by Erich Friedman; Found by Maurizio Morandi in December 2007. Source expression: s = 3.464+.

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 = 23, 3.46400001
Current leader

3.46400001

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": "-0.414213562",
      "x": "0.999999997",
      "y": "0.204134726"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.781514924",
      "y": "0.000000003"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.855654945",
      "y": "0.552614563"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345053",
      "y": "0.433035151"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.433035150"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345051",
      "y": "0.144345053"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.144345053"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.855654950",
      "y": "0.841304660"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.204134728",
      "y": "0.999999997"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690100"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.204134726",
      "y": "0.577380201"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.433035150"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.144345050"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035148",
      "y": "0.144345050"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.795865274",
      "y": "0.408269471"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.577380199",
      "y": "0.204134728"
    },
    {
      "turn": "-0.414213562",
      "x": "0.795865271",
      "y": "0.393919107"
    },
    {
      "quarter": 3,
      "turn": "0.012428988",
      "x": "0.859287111",
      "y": "0.852111969"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.795865272"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690097"
    },
    {
      "turn": "-0.414213562",
      "x": "0.552614501",
      "y": "0.517590526"
    },
    {
      "quarter": 1,
      "turn": "0.335237318",
      "x": "0.580791168",
      "y": "0.613337157"
    },
    {
      "quarter": 2,
      "turn": "0.012428983",
      "x": "0.570507806",
      "y": "0.852111970"
    }
  ],
  "radius": "0.288683602"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.204134726"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.781514924",
      "y": "0.000000003"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.855654945",
      "y": "0.552614563"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345053",
      "y": "0.433035151"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.433035150"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345051",
      "y": "0.144345053"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.144345053"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.855654950",
      "y": "0.841304660"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.204134728",
      "y": "0.999999997"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690100"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.204134726",
      "y": "0.577380201"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.433035150"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.144345050"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035148",
      "y": "0.144345050"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.795865274",
      "y": "0.408269471"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.577380199",
      "y": "0.204134728"
    },
    {
      "turn": "-0.414213562",
      "x": "0.795865271",
      "y": "0.393919107"
    },
    {
      "quarter": 3,
      "turn": "0.012428988",
      "x": "0.859287111",
      "y": "0.852111969"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.795865272"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690097"
    },
    {
      "turn": "-0.414213562",
      "x": "0.552614501",
      "y": "0.517590526"
    },
    {
      "quarter": 1,
      "turn": "0.335237318",
      "x": "0.580791168",
      "y": "0.613337157"
    },
    {
      "quarter": 2,
      "turn": "0.012428983",
      "x": "0.570507806",
      "y": "0.852111970"
    }
  ],
  "radius": "0.288683602"
}

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

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