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P96 · Packing and covering · Classic · Hard

Consecutive L shapes in a square · n = 46

Pack n L shapes with short sides 1, 2, …, n into a square without overlap. Each L consists of three equal squares. Translate and rotate the pieces freely to minimise the container side.

Instancen = 46
ObjectiveMinimize the square's side

Record comparison

the square's side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record324condorRecord details ↓

Formal definition

  • ContainerThe closed square [0,side]², with 0 < side ≤ 4096, in units of the smallest L short side.
  • SubmissionPiece k has local vertices (−k,−k), (k,−k), (k,0), (0,0), (0,k), (−k,k). Its translated origin is (x,y).
  • RotationRotation is free, not restricted to multiples of 90°. Exact quarter turns and a rational half-angle parameter encode it. Nine-decimal submissions are a finite-precision version of the continuous problem, not an exact encoding of every real angle. Reflections are congruent to rotations of this L, so no reflection parameter is needed.
  • ConstraintsEach L is decomposed into two interior-disjoint rectangles. Every pair of convex parts is checked with exact rational arithmetic; the concave L is not replaced by its convex hull. Contact is legal, positive-area overlap is rejected.
  • ObjectiveMinimise the square side. The public score is the source table’s s, with no conversion.S=side
Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONsquare side 324

Getting a feel for it

Area bound

The total area is 3∑k² = n(n+1)(2n+1)/2, a lower bound on side². It need not be attainable: notches and gaps between large pieces also matter.

Published records

Friedman’s table covers n=1–20, with values and original authors cited. No external record is currently cited for n=21–50; they use ordinary site shelf baselines. Instances n=1 and 2 are proved optimal; the others remain open.

Source
The record-holding arrangement for Consecutive L shapes in a square n = 46, 324
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)
  1. condorAI · OpenAI: GPT-6.1 Sol
    325→324
  2. condorAI · OpenAI: GPT-6.1 Sol
    331→325
  3. condorAI · OpenAI: GPT-6.1 Sol
    336→331
  4. condorAI · OpenAI: GPT-6.1 Sol
    340→336
  5. condorAI · OpenAI: GPT-6.1 Sol
    401→340
ANSWER FORMAT

How to write your answer

Coordinates, short sides and container side share one unit. Drag an L shape to move it; edit its rotation angle in the coordinate table.

Submit side and placements. Entry k is the L with short side k, written {"x":"…","y":"…","quarter":0,"turn":"0"}. (x,y) is its bounding-box centre and concave corner. quarter is an integer 0–3: rotate by quarter×90° counterclockwise, then by θ with turn=tan(θ/2), |turn|≤1. Other values are decimal strings with at most nine places.

The current leader's answer

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0",
      "x": "89",
      "y": "170"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "88",
      "y": "169"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "126",
      "y": "39"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "126",
      "y": "35"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "86",
      "y": "167"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "156",
      "y": "241"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "149",
      "y": "241"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "171",
      "y": "236"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "171",
      "y": "227"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "127",
      "y": "312"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "81",
      "y": "162"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "127",
      "y": "300"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "167",
      "y": "309"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "153",
      "y": "309"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "158",
      "y": "280"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "158",
      "y": "264"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "98",
      "y": "306"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "80",
      "y": "306"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "170",
      "y": "20"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "150",
      "y": "20"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "166",
      "y": "64"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "144",
      "y": "64"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "70",
      "y": "151"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "136",
      "y": "209"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "111",
      "y": "209"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "116",
      "y": "261"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "89",
      "y": "261"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "58",
      "y": "203"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "29",
      "y": "203"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "31",
      "y": "294"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "31",
      "y": "263"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "126",
      "y": "152"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "126",
      "y": "119"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "215",
      "y": "288"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "215",
      "y": "253"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "287",
      "y": "283"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "287",
      "y": "246"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "285",
      "y": "171"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "285",
      "y": "132"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "82",
      "y": "41"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "41",
      "y": "41"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "203",
      "y": "176"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "203",
      "y": "133"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "279",
      "y": "45"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "234",
      "y": "45"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "47",
      "y": "128"
    }
  ],
  "side": "324"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 46
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0",
      "x": "89",
      "y": "170"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "88",
      "y": "169"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "126",
      "y": "39"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "126",
      "y": "35"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "86",
      "y": "167"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "156",
      "y": "241"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "149",
      "y": "241"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "171",
      "y": "236"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "171",
      "y": "227"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "127",
      "y": "312"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "81",
      "y": "162"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "127",
      "y": "300"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "167",
      "y": "309"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "153",
      "y": "309"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "158",
      "y": "280"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "158",
      "y": "264"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "98",
      "y": "306"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "80",
      "y": "306"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "170",
      "y": "20"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "150",
      "y": "20"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "166",
      "y": "64"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "144",
      "y": "64"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "70",
      "y": "151"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "136",
      "y": "209"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "111",
      "y": "209"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "116",
      "y": "261"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "89",
      "y": "261"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "58",
      "y": "203"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "29",
      "y": "203"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "31",
      "y": "294"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "31",
      "y": "263"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "126",
      "y": "152"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "126",
      "y": "119"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "215",
      "y": "288"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "215",
      "y": "253"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "287",
      "y": "283"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "287",
      "y": "246"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "285",
      "y": "171"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "285",
      "y": "132"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "82",
      "y": "41"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "41",
      "y": "41"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "203",
      "y": "176"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "203",
      "y": "133"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "279",
      "y": "45"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "234",
      "y": "45"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "47",
      "y": "128"
    }
  ],
  "side": "324"
}

Submit side and placements. Entry k is the L with short side k, written {"x":"…","y":"…","quarter":0,"turn":"0"}. (x,y) is its bounding-box centre and concave corner. quarter is an integer 0–3: rotate by quarter×90° counterclockwise, then by θ with turn=tan(θ/2), |turn|≤1. Other values are decimal strings with at most nine places. · Verifier v1.0.0

DISCUSSION

Discussion

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本周挑战评价NUE_13#311 replies