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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 20

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

Instancen = 20
ObjectiveMinimize container square side
Best known, unproven3.26857Source-precision interval: 3.26857–3.26858 (values inside match)exhibited site certificate 3.268570008Compiled by Erich Friedman; Found by Aapo Lippoen in July 2026. Source expression: s = 3.26857+.

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 = 20, 3.268570008
Current leader

3.268570008

container square side

Matches the best known
Answer sourceAapo Lippoen
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": "0.035798384",
      "x": "0.836419363",
      "y": "0.141696040"
    },
    {
      "turn": "0.028065282",
      "x": "0.484597625",
      "y": "0.161371947"
    },
    {
      "quarter": 2,
      "turn": "0.028065298",
      "x": "0.179019521",
      "y": "0.144206150"
    },
    {
      "turn": "0.028065288",
      "x": "0.809383418",
      "y": "0.161371949"
    },
    {
      "quarter": 2,
      "turn": "0.028065286",
      "x": "0.503805240",
      "y": "0.144207531"
    },
    {
      "turn": "0.028065307",
      "x": "0.189764231",
      "y": "0.451352121"
    },
    {
      "turn": "0.028065301",
      "x": "0.501288452",
      "y": "0.450608541"
    },
    {
      "turn": "0.035798328",
      "x": "0.814534522",
      "y": "0.446972478"
    },
    {
      "turn": "0.000000000",
      "x": "0.806059864",
      "y": "0.761106120"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.193940139",
      "y": "0.761106121"
    },
    {
      "quarter": 2,
      "turn": "0.028065299",
      "x": "0.505657518",
      "y": "0.446704236"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000002",
      "y": "0.761106118"
    },
    {
      "quarter": 2,
      "turn": "0.035798330",
      "x": "0.817393995",
      "y": "0.444495568"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.783582993",
      "y": "0.999999997"
    },
    {
      "quarter": 2,
      "turn": "0.028065302",
      "x": "0.194965390",
      "y": "0.446704238"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.761106123"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.216417009",
      "y": "0.999999997"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.783582993"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.783582995"
    },
    {
      "turn": "0.000000000",
      "x": "0.153029933",
      "y": "0.153029933"
    }
  ],
  "radius": "0.305944189"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.035798384",
      "x": "0.836419363",
      "y": "0.141696040"
    },
    {
      "turn": "0.028065282",
      "x": "0.484597625",
      "y": "0.161371947"
    },
    {
      "quarter": 2,
      "turn": "0.028065298",
      "x": "0.179019521",
      "y": "0.144206150"
    },
    {
      "turn": "0.028065288",
      "x": "0.809383418",
      "y": "0.161371949"
    },
    {
      "quarter": 2,
      "turn": "0.028065286",
      "x": "0.503805240",
      "y": "0.144207531"
    },
    {
      "turn": "0.028065307",
      "x": "0.189764231",
      "y": "0.451352121"
    },
    {
      "turn": "0.028065301",
      "x": "0.501288452",
      "y": "0.450608541"
    },
    {
      "turn": "0.035798328",
      "x": "0.814534522",
      "y": "0.446972478"
    },
    {
      "turn": "0.000000000",
      "x": "0.806059864",
      "y": "0.761106120"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.193940139",
      "y": "0.761106121"
    },
    {
      "quarter": 2,
      "turn": "0.028065299",
      "x": "0.505657518",
      "y": "0.446704236"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000002",
      "y": "0.761106118"
    },
    {
      "quarter": 2,
      "turn": "0.035798330",
      "x": "0.817393995",
      "y": "0.444495568"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.783582993",
      "y": "0.999999997"
    },
    {
      "quarter": 2,
      "turn": "0.028065302",
      "x": "0.194965390",
      "y": "0.446704238"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000001",
      "y": "0.761106123"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.216417009",
      "y": "0.999999997"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.783582993"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.783582995"
    },
    {
      "turn": "0.000000000",
      "x": "0.153029933",
      "y": "0.153029933"
    }
  ],
  "radius": "0.305944189"
}

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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