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

Tans in a square · n = 19

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

Instancen = 19
ObjectiveMinimize container square side
Best known, unproven3.201Source-precision interval: 3.201–3.202 (values inside match)exhibited site certificate 3.201496211Compiled by Erich Friedman; Found by Maurizio Morandi in February 2008. Source expression: s = 3.201+.

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 = 19, 3.201496211
Current leader

3.201496211

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": "0.219115100",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.531469060"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.531469058"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.532960235",
      "y": "0.185175259"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.249604552",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.561958510",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.156176980",
      "y": "0.465025926"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.219115100"
    },
    {
      "quarter": 3,
      "turn": "-0.075206786",
      "x": "0.177779241",
      "y": "0.131069708"
    },
    {
      "turn": "0.000000000",
      "x": "0.531469058",
      "y": "0.843823017"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.843823017"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.531469060"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.279850674",
      "y": "0.562219841"
    },
    {
      "turn": "-0.414213562",
      "x": "0.279850671",
      "y": "0.562219839"
    },
    {
      "turn": "-0.048915272",
      "x": "0.859811014",
      "y": "0.170674536"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500718276",
      "y": "0.341352240"
    },
    {
      "quarter": 2,
      "turn": "-0.337772885",
      "x": "0.029493573",
      "y": "0.781109919"
    }
  ],
  "radius": "0.312353954"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.219115100",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.531469060"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.531469058"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.532960235",
      "y": "0.185175259"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.249604552",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.561958510",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.156176980",
      "y": "0.465025926"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.219115100"
    },
    {
      "quarter": 3,
      "turn": "-0.075206786",
      "x": "0.177779241",
      "y": "0.131069708"
    },
    {
      "turn": "0.000000000",
      "x": "0.531469058",
      "y": "0.843823017"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.843823017"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.531469060"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.279850674",
      "y": "0.562219841"
    },
    {
      "turn": "-0.414213562",
      "x": "0.279850671",
      "y": "0.562219839"
    },
    {
      "turn": "-0.048915272",
      "x": "0.859811014",
      "y": "0.170674536"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500718276",
      "y": "0.341352240"
    },
    {
      "quarter": 2,
      "turn": "-0.337772885",
      "x": "0.029493573",
      "y": "0.781109919"
    }
  ],
  "radius": "0.312353954"
}

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?