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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 12

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

Instancen = 12
ObjectiveMinimize container square side
Best known, unproven2.61115Source-precision interval: 2.61115–2.61116 (values inside match)exhibited site certificate 2.611150004Compiled by Erich Friedman; Found by Haowei Lin in July 2026. Source expression: s = 2.61115+.

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 = 12, 2.611150004
Current leader

2.611150004

container square side

Matches the best known
Answer sourceHaowei Lin
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": 3,
      "turn": "0.414145804",
      "x": "0.999919193",
      "y": "0.271044385"
    },
    {
      "quarter": 2,
      "turn": "-0.414213560",
      "x": "0.000000004",
      "y": "0.729084980"
    },
    {
      "quarter": 1,
      "turn": "-0.353273174",
      "x": "0.333598409",
      "y": "0.970508942"
    },
    {
      "quarter": 3,
      "turn": "0.013986251",
      "x": "0.467730757",
      "y": "0.395252860"
    },
    {
      "quarter": 1,
      "turn": "0.048583348",
      "x": "0.442975980",
      "y": "0.790619891"
    },
    {
      "turn": "-0.414058210",
      "x": "0.270915010",
      "y": "0.458098108"
    },
    {
      "quarter": 3,
      "turn": "0.048583346",
      "x": "0.827906332",
      "y": "0.790766356"
    },
    {
      "quarter": 1,
      "turn": "0.000822757",
      "x": "0.808119187",
      "y": "0.734341303"
    },
    {
      "quarter": 1,
      "turn": "0.013986251",
      "x": "0.420650829",
      "y": "0.345462700"
    },
    {
      "quarter": 3,
      "turn": "0.000822758",
      "x": "0.808749635",
      "y": "0.351210127"
    },
    {
      "turn": "0.000000000",
      "x": "0.191565848",
      "y": "0.191565848"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.729084980",
      "y": "0.000000003"
    }
  ],
  "radius": "0.382973019"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 3,
      "turn": "0.414145804",
      "x": "0.999919193",
      "y": "0.271044385"
    },
    {
      "quarter": 2,
      "turn": "-0.414213560",
      "x": "0.000000004",
      "y": "0.729084980"
    },
    {
      "quarter": 1,
      "turn": "-0.353273174",
      "x": "0.333598409",
      "y": "0.970508942"
    },
    {
      "quarter": 3,
      "turn": "0.013986251",
      "x": "0.467730757",
      "y": "0.395252860"
    },
    {
      "quarter": 1,
      "turn": "0.048583348",
      "x": "0.442975980",
      "y": "0.790619891"
    },
    {
      "turn": "-0.414058210",
      "x": "0.270915010",
      "y": "0.458098108"
    },
    {
      "quarter": 3,
      "turn": "0.048583346",
      "x": "0.827906332",
      "y": "0.790766356"
    },
    {
      "quarter": 1,
      "turn": "0.000822757",
      "x": "0.808119187",
      "y": "0.734341303"
    },
    {
      "quarter": 1,
      "turn": "0.013986251",
      "x": "0.420650829",
      "y": "0.345462700"
    },
    {
      "quarter": 3,
      "turn": "0.000822758",
      "x": "0.808749635",
      "y": "0.351210127"
    },
    {
      "turn": "0.000000000",
      "x": "0.191565848",
      "y": "0.191565848"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.729084980",
      "y": "0.000000003"
    }
  ],
  "radius": "0.382973019"
}

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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