MinMax Arena
中/EN
P89 · Packing and covering · Classic · Hard

Tans in a square · n = 13

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

Instancen = 13
ObjectiveMinimize container square side
Best known, unproven2.688Source-precision interval: 2.688–2.689 (values inside match)exhibited site certificate 2.688240167Compiled by Erich Friedman; Found by Maurizio Morandi in March 2008. Source expression: s = 2.688+.

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 = 13, 2.688240167
Current leader

2.688240167

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": [
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.660552731",
      "y": "0.441382568"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.312824115",
      "y": "0.526074123"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.736962938",
      "y": "0.999999997"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.575861177",
      "y": "0.263037060"
    },
    {
      "turn": "-0.414213562",
      "x": "0.312824112",
      "y": "0.476287074"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.288078700",
      "y": "0.813856599"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.263037060"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.575861174",
      "y": "0.263037063"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.263037065",
      "y": "0.000000003"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.736962938"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.813632779",
      "y": "0.186227244"
    },
    {
      "turn": "0.000000000",
      "x": "0.660306922",
      "y": "0.813618951"
    },
    {
      "quarter": 2,
      "turn": "-0.359721205",
      "x": "0.024893534",
      "y": "0.738143536"
    }
  ],
  "radius": "0.371990573"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.660552731",
      "y": "0.441382568"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.312824115",
      "y": "0.526074123"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.736962938",
      "y": "0.999999997"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.575861177",
      "y": "0.263037060"
    },
    {
      "turn": "-0.414213562",
      "x": "0.312824112",
      "y": "0.476287074"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.288078700",
      "y": "0.813856599"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.263037060"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.575861174",
      "y": "0.263037063"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.263037065",
      "y": "0.000000003"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.736962938"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.813632779",
      "y": "0.186227244"
    },
    {
      "turn": "0.000000000",
      "x": "0.660306922",
      "y": "0.813618951"
    },
    {
      "quarter": 2,
      "turn": "-0.359721205",
      "x": "0.024893534",
      "y": "0.738143536"
    }
  ],
  "radius": "0.371990573"
}

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

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?