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P96 · Packing and covering · Classic · Hard

Consecutive L shapes in a square · n = 48

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 = 48
ObjectiveMinimize the square's side

Record comparison

the square's side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record345condorRecord 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 345

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 = 48, 345
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
    346→345
  2. condorAI · OpenAI: GPT-6.1 Sol
    352→346
  3. condorAI · OpenAI: GPT-6.1 Sol
    358→352
  4. condorAI · OpenAI: GPT-6.1 Sol
    361→358
  5. condorAI · OpenAI: GPT-6.1 Sol
    427→361
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": "186",
      "y": "213"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "336",
      "y": "265"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "185",
      "y": "212"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "96",
      "y": "219"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "334",
      "y": "263"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "182",
      "y": "209"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "221",
      "y": "112"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "221",
      "y": "104"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "21",
      "y": "235"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "11",
      "y": "235"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "329",
      "y": "258"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "176",
      "y": "203"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "120",
      "y": "104"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "106",
      "y": "104"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "32",
      "y": "326"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "16",
      "y": "326"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "83",
      "y": "326"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "65",
      "y": "326"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "212",
      "y": "159"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "212",
      "y": "139"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "22",
      "y": "289"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "22",
      "y": "267"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "318",
      "y": "247"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "164",
      "y": "191"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "210",
      "y": "320"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "210",
      "y": "294"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "72",
      "y": "281"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "72",
      "y": "253"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "218",
      "y": "239"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "218",
      "y": "209"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "245",
      "y": "65"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "245",
      "y": "33"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "311",
      "y": "68"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "311",
      "y": "34"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "308",
      "y": "307"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "272",
      "y": "307"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "308",
      "y": "139"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "270",
      "y": "139"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "173",
      "y": "80"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "173",
      "y": "40"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "142",
      "y": "299"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "142",
      "y": "257"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "88",
      "y": "44"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "44",
      "y": "44"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "46",
      "y": "180"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "46",
      "y": "134"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "295",
      "y": "224"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "140",
      "y": "167"
    }
  ],
  "side": "345"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 48
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0",
      "x": "186",
      "y": "213"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "336",
      "y": "265"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "185",
      "y": "212"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "96",
      "y": "219"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "334",
      "y": "263"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "182",
      "y": "209"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "221",
      "y": "112"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "221",
      "y": "104"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "21",
      "y": "235"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "11",
      "y": "235"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "329",
      "y": "258"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "176",
      "y": "203"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "120",
      "y": "104"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "106",
      "y": "104"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "32",
      "y": "326"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "16",
      "y": "326"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "83",
      "y": "326"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "65",
      "y": "326"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "212",
      "y": "159"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "212",
      "y": "139"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "22",
      "y": "289"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "22",
      "y": "267"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "318",
      "y": "247"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "164",
      "y": "191"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "210",
      "y": "320"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "210",
      "y": "294"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "72",
      "y": "281"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "72",
      "y": "253"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "218",
      "y": "239"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "218",
      "y": "209"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "245",
      "y": "65"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "245",
      "y": "33"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "311",
      "y": "68"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "311",
      "y": "34"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "308",
      "y": "307"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "272",
      "y": "307"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "308",
      "y": "139"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "270",
      "y": "139"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "173",
      "y": "80"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "173",
      "y": "40"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "142",
      "y": "299"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "142",
      "y": "257"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "88",
      "y": "44"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "44",
      "y": "44"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "46",
      "y": "180"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "46",
      "y": "134"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "295",
      "y": "224"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "140",
      "y": "167"
    }
  ],
  "side": "345"
}

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#310 replies