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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 14

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

Instancen = 14
ObjectiveMinimize container square side
Best known, unproven2.76345Source-precision interval: 2.76345–2.76346 (values inside match)exhibited site certificate 2.763450004Compiled by Erich Friedman; Found by Haowei Lin in July 2026. Source expression: s = 2.76345+.

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 = 14, 2.763450004
Current leader

2.763450004

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": "0.000000000",
      "x": "0.180985610",
      "y": "0.819014390"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.743987130",
      "y": "0.000046262"
    },
    {
      "quarter": 1,
      "turn": "0.388386939",
      "x": "0.451792235",
      "y": "0.744300941"
    },
    {
      "turn": "0.388386565",
      "x": "0.721020783",
      "y": "0.500007594"
    },
    {
      "quarter": 1,
      "turn": "-0.064088731",
      "x": "0.245973243",
      "y": "0.515777301"
    },
    {
      "quarter": 1,
      "turn": "0.388197584",
      "x": "0.453854735",
      "y": "0.255695299"
    },
    {
      "quarter": 1,
      "turn": "-0.064086212",
      "x": "0.251775641",
      "y": "0.843597422"
    },
    {
      "quarter": 3,
      "turn": "0.387808321",
      "x": "0.988359826",
      "y": "0.744312328"
    },
    {
      "quarter": 2,
      "turn": "0.388386461",
      "x": "0.721020788",
      "y": "0.500007643"
    },
    {
      "quarter": 1,
      "turn": "-0.022414808",
      "x": "0.269691295",
      "y": "0.188913279"
    },
    {
      "quarter": 3,
      "turn": "-0.064088602",
      "x": "0.202608285",
      "y": "0.156401614"
    },
    {
      "turn": "0.405652766",
      "x": "0.719253061",
      "y": "0.996230878"
    },
    {
      "quarter": 3,
      "turn": "-0.064088731",
      "x": "0.205567858",
      "y": "0.484586754"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.255952302"
    }
  ],
  "radius": "0.361866507"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 14,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.180985610",
      "y": "0.819014390"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.743987130",
      "y": "0.000046262"
    },
    {
      "quarter": 1,
      "turn": "0.388386939",
      "x": "0.451792235",
      "y": "0.744300941"
    },
    {
      "turn": "0.388386565",
      "x": "0.721020783",
      "y": "0.500007594"
    },
    {
      "quarter": 1,
      "turn": "-0.064088731",
      "x": "0.245973243",
      "y": "0.515777301"
    },
    {
      "quarter": 1,
      "turn": "0.388197584",
      "x": "0.453854735",
      "y": "0.255695299"
    },
    {
      "quarter": 1,
      "turn": "-0.064086212",
      "x": "0.251775641",
      "y": "0.843597422"
    },
    {
      "quarter": 3,
      "turn": "0.387808321",
      "x": "0.988359826",
      "y": "0.744312328"
    },
    {
      "quarter": 2,
      "turn": "0.388386461",
      "x": "0.721020788",
      "y": "0.500007643"
    },
    {
      "quarter": 1,
      "turn": "-0.022414808",
      "x": "0.269691295",
      "y": "0.188913279"
    },
    {
      "quarter": 3,
      "turn": "-0.064088602",
      "x": "0.202608285",
      "y": "0.156401614"
    },
    {
      "turn": "0.405652766",
      "x": "0.719253061",
      "y": "0.996230878"
    },
    {
      "quarter": 3,
      "turn": "-0.064088731",
      "x": "0.205567858",
      "y": "0.484586754"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.255952302"
    }
  ],
  "radius": "0.361866507"
}

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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