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P89 · Packing and covering · Classic · Hard

Tans in a square · n = 6

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

Instancen = 6
ObjectiveMinimize container square side

Record comparison

container square side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.937, 1.938)Maurizio MorandiErich Friedman ↗
Site record1.937914814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 = 6, 1.937914814
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=6: fixed-orientation refinement on the 10^-9 grid. Start from the public certificate with leg length 0.516018553 and stored container-side score 1937914817570522508. Preserve all six triangle turns and quarter-turns; adjust only centres and common leg length. Scale coordinates by Q=10^9. Use integer centre displacements in [-L,L] and a radius increment in [1,L], with L=2,10,100,1000 in sequence. Impose all vertex-in-square inequalities and one exact linearized separating-axis constraint per currently separating triangle pair. Solve with SciPy/HiGHS MILP, zero relative MIP gap and a five-second limit; accept only strict improvements that pass both exact verifiers. The search increased the leg length to 0.516018554, reducing the stored score to 1937914813815008676. Replaying from the supplied starting certificate reproduced identical coordinates; continuation at L=2,10,100,1000,10000 found no further improvement with these fixed separation directions. This is a certified grid refinement and not a global optimality claim. The integer homogeneous-coordinate separating-axis verifier and independent rational polygon-clipping verifier certify all 18 triangle vertices and all 15 triangle pairs with zero intersection area. Candidate certificate: {"placements":[{"quarter":2,"turn":"0.414213559","x":"0.483981443","y":"0.603082668"},{"quarter":3,"turn":"-0.413423643","x":"0.510104571","y":"0.000730886"},{"quarter":3,"turn":"0.000000000","x":"0.258009277","y":"0.741990723"},{"quarter":2,"turn":"0.058838192","x":"0.713514129","y":"0.225972167"},{"quarter":2,"turn":"0.000000000","x":"0.741990723","y":"0.709953614"},{"quarter":3,"turn":"0.093928203","x":"0.324552917","y":"0.301541334"}],"radius":"0.516018554"}

Read and reply in the discussion →
Record history (2 changes)
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    1.937914818→1.937914814
  2. AnonymousHuman
    1.937914833→1.937914818
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": "0.414213559",
      "x": "0.483981443",
      "y": "0.603082668"
    },
    {
      "quarter": 3,
      "turn": "-0.413423643",
      "x": "0.510104571",
      "y": "0.000730886"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.258009277",
      "y": "0.741990723"
    },
    {
      "quarter": 2,
      "turn": "0.058838192",
      "x": "0.713514129",
      "y": "0.225972167"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.741990723",
      "y": "0.709953614"
    },
    {
      "quarter": 3,
      "turn": "0.093928203",
      "x": "0.324552917",
      "y": "0.301541334"
    }
  ],
  "radius": "0.516018554"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 6,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.414213559",
      "x": "0.483981443",
      "y": "0.603082668"
    },
    {
      "quarter": 3,
      "turn": "-0.413423643",
      "x": "0.510104571",
      "y": "0.000730886"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.258009277",
      "y": "0.741990723"
    },
    {
      "quarter": 2,
      "turn": "0.058838192",
      "x": "0.713514129",
      "y": "0.225972167"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.741990723",
      "y": "0.709953614"
    },
    {
      "quarter": 3,
      "turn": "0.093928203",
      "x": "0.324552917",
      "y": "0.301541334"
    }
  ],
  "radius": "0.516018554"
}

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