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P91 · Packing and covering · Classic · Hard

L's in a square · n = 23

Pack n L-trominoes (three unit cells) into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 23
ObjectiveMinimize container square side
Best known, unproven8.914Source-precision interval: 8.914–8.915 (values inside match)exhibited site certificate 8.914213814Compiled by Erich Friedman; Found by Maurizio Morandi in June 2012. Source expression: s = 15/2 + √2 = 8.914+.

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 L's in a square n = 23, 8.914213814
Current leader

8.914213814

container square side

Matches the best known
Answer sourceMaurizio Morandi
Solution methodPublished reference construction
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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": 1,
      "turn": "0.414213551",
      "x": "0.388810090",
      "y": "0.667280109"
    },
    {
      "quarter": 2,
      "turn": "-0.414213557",
      "x": "0.634713372",
      "y": "0.421376822"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.570797819",
      "y": "0.112180395"
    },
    {
      "quarter": 3,
      "turn": "0.414213552",
      "x": "0.308758507",
      "y": "0.747331691"
    },
    {
      "quarter": 1,
      "turn": "-0.410087471",
      "x": "0.158804200",
      "y": "0.916322717"
    },
    {
      "quarter": 3,
      "turn": "0.414208431",
      "x": "0.555257537",
      "y": "0.500832654"
    },
    {
      "turn": "-0.413819943",
      "x": "0.802537265",
      "y": "0.271721892"
    },
    {
      "quarter": 2,
      "turn": "-0.000000008",
      "x": "0.775639208",
      "y": "0.775639207"
    },
    {
      "turn": "-0.000000011",
      "x": "0.112180398",
      "y": "0.673082374"
    },
    {
      "turn": "-0.000000005",
      "x": "0.112180397",
      "y": "0.448721582"
    },
    {
      "quarter": 3,
      "turn": "-0.000000003",
      "x": "0.663458813",
      "y": "0.887819604"
    },
    {
      "quarter": 1,
      "turn": "-0.000000012",
      "x": "0.887819602",
      "y": "0.663458809"
    },
    {
      "quarter": 2,
      "turn": "-0.000000007",
      "x": "0.224360792",
      "y": "0.448721584"
    },
    {
      "quarter": 2,
      "turn": "-0.000000017",
      "x": "0.887819601",
      "y": "0.439098020"
    },
    {
      "turn": "0.000000000",
      "x": "0.112180396",
      "y": "0.112180396"
    },
    {
      "turn": "-0.000000001",
      "x": "0.336541186",
      "y": "0.112180396"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.448721582",
      "y": "0.224360791"
    },
    {
      "quarter": 2,
      "turn": "-0.000000001",
      "x": "0.439098025",
      "y": "0.887819604"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.887819605",
      "y": "0.887819602"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.663458811",
      "y": "0.663458813"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.336541187",
      "y": "0.336541187"
    },
    {
      "quarter": 2,
      "turn": "0.000000001",
      "x": "0.112180398",
      "y": "0.224360792"
    },
    {
      "quarter": 1,
      "turn": "0.015538692",
      "x": "0.884309660",
      "y": "0.115612152"
    }
  ],
  "radius": "0.112180392"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 1,
      "turn": "0.414213551",
      "x": "0.388810090",
      "y": "0.667280109"
    },
    {
      "quarter": 2,
      "turn": "-0.414213557",
      "x": "0.634713372",
      "y": "0.421376822"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.570797819",
      "y": "0.112180395"
    },
    {
      "quarter": 3,
      "turn": "0.414213552",
      "x": "0.308758507",
      "y": "0.747331691"
    },
    {
      "quarter": 1,
      "turn": "-0.410087471",
      "x": "0.158804200",
      "y": "0.916322717"
    },
    {
      "quarter": 3,
      "turn": "0.414208431",
      "x": "0.555257537",
      "y": "0.500832654"
    },
    {
      "turn": "-0.413819943",
      "x": "0.802537265",
      "y": "0.271721892"
    },
    {
      "quarter": 2,
      "turn": "-0.000000008",
      "x": "0.775639208",
      "y": "0.775639207"
    },
    {
      "turn": "-0.000000011",
      "x": "0.112180398",
      "y": "0.673082374"
    },
    {
      "turn": "-0.000000005",
      "x": "0.112180397",
      "y": "0.448721582"
    },
    {
      "quarter": 3,
      "turn": "-0.000000003",
      "x": "0.663458813",
      "y": "0.887819604"
    },
    {
      "quarter": 1,
      "turn": "-0.000000012",
      "x": "0.887819602",
      "y": "0.663458809"
    },
    {
      "quarter": 2,
      "turn": "-0.000000007",
      "x": "0.224360792",
      "y": "0.448721584"
    },
    {
      "quarter": 2,
      "turn": "-0.000000017",
      "x": "0.887819601",
      "y": "0.439098020"
    },
    {
      "turn": "0.000000000",
      "x": "0.112180396",
      "y": "0.112180396"
    },
    {
      "turn": "-0.000000001",
      "x": "0.336541186",
      "y": "0.112180396"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.448721582",
      "y": "0.224360791"
    },
    {
      "quarter": 2,
      "turn": "-0.000000001",
      "x": "0.439098025",
      "y": "0.887819604"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.887819605",
      "y": "0.887819602"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.663458811",
      "y": "0.663458813"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.336541187",
      "y": "0.336541187"
    },
    {
      "quarter": 2,
      "turn": "0.000000001",
      "x": "0.112180398",
      "y": "0.224360792"
    },
    {
      "quarter": 1,
      "turn": "0.015538692",
      "x": "0.884309660",
      "y": "0.115612152"
    }
  ],
  "radius": "0.112180392"
}

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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