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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 17

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

Instancen = 17
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[2.97, 2.971)David W. CantrellErich Friedman ↗
Site record2.970710956邱仲普Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

  • ContainerThe editor and certificate normalise the container to the unit square [0,1]², while public scores are reported in the literature's unit-leg scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn,quarter},…]}, with x, y, and turn written as decimal strings of at most nine places. Here radius is the leg length of each piece inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the centre of the piece's bounding box; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
  • ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Pieces may rotate independently by arbitrary angles.
  • PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
  • ObjectiveContainer square side = 1/r (where r is the piece's leg length inside the unit square), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.
Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source
The record-holding arrangement for Tans in a square n = 17, 2.970710956
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普
Solution methodAI · OpenAI: GPT-5.6 Sol
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver notePublic method and reproducibility notes (expand)＋

P89 n=17: exact-grid refinement of the previous public certificate. The normalized triangle leg grows from 0.336619755 to 0.336619757; the minimized, upward-rounded container side falls from 2.970710973276063374 to 2.970710955625816105. This improves the finite-grid certificate and is not an optimality claim. Method: keep all turns and quarter rotations fixed. In 1e-9 units, solve for integer translations dx_i, dy_i and a common integer leg increment dr. Require every transformed vertex to stay in the unit square. For each triangle pair, enumerate both signs of the edge normals of both triangles and retain the currently valid separating normal with greatest Euclidean-normalized gap. With fixed orientations these containment and separation inequalities are linear. Maximize dr, with |dx_i|,|dy_i| <= L and 1 <= dr <= L, using scipy.optimize.milp/HiGHS (5 seconds, relative MIP gap 0). Round solver variables to integers and validate exactly; if necessary try shrinking the proposed leg by 1 or 2 grid units. Apply L=2,10,100,1000 in order. The accepted leg was 0.336619757 at L=2; later attempts did not improve it. A further sweep L=2,10,100,1000,10000,100000 also gave no exactly valid improvement. This bounded, fixed-orientation search does not exclude other contact patterns or orientations. Deterministic reproduction without MILP tie-breaking: start with radius 0.336619755 and these placement rows [x,y,turn,quarter], in order: ["0.831690116","0.495070354","-0.000000011",2] ["0.000694056","0.716995177","-0.412509052",2] ["0.238026113","0.476052243","0.414213543",2] ["0.238026113","0.000000005","0.414213558",2] ["0.718335617","0.000911991","0.411976182",2] ["0.831690120","0.831690120","-0.000000002",2] ["0.821975243","0.504785224","-0.000000011",0] ["0.238026115","0.476052242","0.414213542",0] ["0.476052224","0.238026121","0.414213554",3] ["0.476052229","0.238026113","0.414213558",1] ["0.485355486","0.466748977","-0.000000011",2] ["0.826832680","0.163308023","0.014647703",2] ["0.831690120","0.831690111","-0.000000008",0] ["0.495133388","0.812608962","-0.028271264",0] ["0.158530820","0.822448851","-0.028274536",2] ["0.000000004","0.238026120","0.414213559",1] ["0.495070364","0.831690118","-0.000000006",2] Add the following integer [dx,dy] rows times 1e-9 to the corresponding centers, preserve turn and quarter, and set radius=0.336619757: [[1,0],[-2,-2],[0,-2],[0,-1],[-2,-2],[0,0],[2,0],[0,-2],[2,-1],[1,0],[-1,1],[0,-1],[1,2],[1,2],[0,1],[-2,0],[-1,1]] Validation: parse all decimals on the 1e-9 grid. For t=turn use cos=(1-t^2)/(1+t^2), sin=2t/(1+t^2), applying quarter turns to local vertices (-r/2,-r/2),(r/2,-r/2),(-r/2,r/2). Check all vertices in [0,1]^2 and all 136 pairs with exact integer separating axes after clearing denominators. Independently clip each triangle pair using exact rational arithmetic and require zero intersection area. Both checks pass. With R=336619757, stored score=ceil(10^27/R)=2970710955625816105.

Read and reply in the discussion →
Record history (2 changes)
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.970710974→2.970710956
  2. 😰Human
    2.970710991→2.970710974
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 written as decimal strings of at most nine places. Here radius is the leg length of each piece inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the centre of the piece's bounding box; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.831690117",
      "y": "0.495070354"
    },
    {
      "quarter": 2,
      "turn": "-0.412509052",
      "x": "0.000694054",
      "y": "0.716995175"
    },
    {
      "quarter": 2,
      "turn": "0.414213543",
      "x": "0.238026113",
      "y": "0.476052241"
    },
    {
      "quarter": 2,
      "turn": "0.414213558",
      "x": "0.238026113",
      "y": "0.000000004"
    },
    {
      "quarter": 2,
      "turn": "0.411976182",
      "x": "0.718335615",
      "y": "0.000911989"
    },
    {
      "quarter": 2,
      "turn": "-0.000000002",
      "x": "0.831690120",
      "y": "0.831690120"
    },
    {
      "turn": "-0.000000011",
      "x": "0.821975245",
      "y": "0.504785224"
    },
    {
      "turn": "0.414213542",
      "x": "0.238026115",
      "y": "0.476052240"
    },
    {
      "quarter": 3,
      "turn": "0.414213554",
      "x": "0.476052226",
      "y": "0.238026120"
    },
    {
      "quarter": 1,
      "turn": "0.414213558",
      "x": "0.476052230",
      "y": "0.238026113"
    },
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.485355485",
      "y": "0.466748978"
    },
    {
      "quarter": 2,
      "turn": "0.014647703",
      "x": "0.826832680",
      "y": "0.163308022"
    },
    {
      "turn": "-0.000000008",
      "x": "0.831690121",
      "y": "0.831690113"
    },
    {
      "turn": "-0.028271264",
      "x": "0.495133389",
      "y": "0.812608964"
    },
    {
      "quarter": 2,
      "turn": "-0.028274536",
      "x": "0.158530820",
      "y": "0.822448852"
    },
    {
      "quarter": 1,
      "turn": "0.414213559",
      "x": "0.000000002",
      "y": "0.238026120"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.495070363",
      "y": "0.831690119"
    }
  ],
  "radius": "0.336619757"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.831690117",
      "y": "0.495070354"
    },
    {
      "quarter": 2,
      "turn": "-0.412509052",
      "x": "0.000694054",
      "y": "0.716995175"
    },
    {
      "quarter": 2,
      "turn": "0.414213543",
      "x": "0.238026113",
      "y": "0.476052241"
    },
    {
      "quarter": 2,
      "turn": "0.414213558",
      "x": "0.238026113",
      "y": "0.000000004"
    },
    {
      "quarter": 2,
      "turn": "0.411976182",
      "x": "0.718335615",
      "y": "0.000911989"
    },
    {
      "quarter": 2,
      "turn": "-0.000000002",
      "x": "0.831690120",
      "y": "0.831690120"
    },
    {
      "turn": "-0.000000011",
      "x": "0.821975245",
      "y": "0.504785224"
    },
    {
      "turn": "0.414213542",
      "x": "0.238026115",
      "y": "0.476052240"
    },
    {
      "quarter": 3,
      "turn": "0.414213554",
      "x": "0.476052226",
      "y": "0.238026120"
    },
    {
      "quarter": 1,
      "turn": "0.414213558",
      "x": "0.476052230",
      "y": "0.238026113"
    },
    {
      "quarter": 2,
      "turn": "-0.000000011",
      "x": "0.485355485",
      "y": "0.466748978"
    },
    {
      "quarter": 2,
      "turn": "0.014647703",
      "x": "0.826832680",
      "y": "0.163308022"
    },
    {
      "turn": "-0.000000008",
      "x": "0.831690121",
      "y": "0.831690113"
    },
    {
      "turn": "-0.028271264",
      "x": "0.495133389",
      "y": "0.812608964"
    },
    {
      "quarter": 2,
      "turn": "-0.028274536",
      "x": "0.158530820",
      "y": "0.822448852"
    },
    {
      "quarter": 1,
      "turn": "0.414213559",
      "x": "0.000000002",
      "y": "0.238026120"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.495070363",
      "y": "0.831690119"
    }
  ],
  "radius": "0.336619757"
}

Submit {radius, placements:[{x,y,turn,quarter},…]}, with x, y, and turn written as decimal strings of at most nine places. Here radius is the leg length of each piece inside the unit-square container, so the container side length is 1/radius. The coordinates x, y specify the centre of the piece's bounding box; quarter is an integer in 0–3 specifying an exact multiple of 90° applied before the rational rotation turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

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n=4,8,16,18时的最优解证明😰#1064 replies本题n=4、8、16、18时的最优解NUE_13#310 replies外部记录更新（9.23更新）NUE_13#310 repliesbest value at n=1😰#1061 replies