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P96 · Packing and covering · Classic · Hard

Consecutive L shapes in a square · n = 21

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 = 21
ObjectiveMinimize the square's side

Record comparison

the square's side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record102condorRecord 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
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Layout shown: site record

VERIFIED CONSTRUCTIONsquare side 102

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 = 21, 102
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 (6 changes)
  1. condorAI · OpenAI: GPT-6.1 Sol
    103→102
  2. 今天AI · DeepSeek: DeepSeek V4.1 Flash 1017
    106→103
  3. condorAI · OpenAI: GPT-6.1 Sol
    110→106
  4. 今天Human
    110.582524272→110
  5. NUE_13Human
    130→110.582524272
  6. NUE_13Human
    131→130
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": 3,
      "turn": "0",
      "x": "15",
      "y": "39"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "22",
      "y": "44"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "56",
      "y": "82"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "66",
      "y": "81"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "67",
      "y": "35"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "20",
      "y": "46"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "65",
      "y": "48"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "78",
      "y": "78"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "27",
      "y": "75"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "72",
      "y": "30"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "76",
      "y": "66"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "53",
      "y": "67"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "41",
      "y": "55"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "14",
      "y": "52"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "87",
      "y": "55"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "86",
      "y": "86"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "53",
      "y": "85"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "18",
      "y": "84"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "19",
      "y": "19"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "82",
      "y": "20"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "41",
      "y": "21"
    }
  ],
  "side": "102"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 3,
      "turn": "0",
      "x": "15",
      "y": "39"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "22",
      "y": "44"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "56",
      "y": "82"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "66",
      "y": "81"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "67",
      "y": "35"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "20",
      "y": "46"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "65",
      "y": "48"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "78",
      "y": "78"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "27",
      "y": "75"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "72",
      "y": "30"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "76",
      "y": "66"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "53",
      "y": "67"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "41",
      "y": "55"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "14",
      "y": "52"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "87",
      "y": "55"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "86",
      "y": "86"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "53",
      "y": "85"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "18",
      "y": "84"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "19",
      "y": "19"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "82",
      "y": "20"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "41",
      "y": "21"
    }
  ],
  "side": "102"
}

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