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P96 · Packing and covering · Classic · Hard

Consecutive L shapes in a square · n = 43

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 = 43
ObjectiveMinimize the square's side

Record comparison

the square's side · Lower is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record294condorRecord 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 294

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 = 43, 294
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
    295→294
  2. condorAI · OpenAI: GPT-6.1 Terra
    296→295
  3. condorAI · OpenAI: GPT-6.1 Sol
    301→296
  4. condorAI · OpenAI: GPT-6.1 Sol
    304→301
  5. condorAI · OpenAI: GPT-6.1 Sol
    307→304
  6. condorAI · OpenAI: GPT-6.1 Sol
    371→307
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": "171",
      "y": "82"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "170",
      "y": "81"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "72",
      "y": "187"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "242",
      "y": "86"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "168",
      "y": "79"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "236",
      "y": "86"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "79",
      "y": "76"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "79",
      "y": "68"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "283",
      "y": "209"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "163",
      "y": "74"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "283",
      "y": "198"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "101",
      "y": "197"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "88",
      "y": "197"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "258",
      "y": "202"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "243",
      "y": "202"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "107",
      "y": "103"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "90",
      "y": "103"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "209",
      "y": "196"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "209",
      "y": "177"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "91",
      "y": "164"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "153",
      "y": "64"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "91",
      "y": "142"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "48",
      "y": "84"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "24",
      "y": "84"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "165",
      "y": "184"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "139",
      "y": "184"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "153",
      "y": "266"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "153",
      "y": "238"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "60",
      "y": "30"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "30",
      "y": "30"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "262",
      "y": "156"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "262",
      "y": "124"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "34",
      "y": "176"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "34",
      "y": "142"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "195",
      "y": "122"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "159",
      "y": "122"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "257",
      "y": "256"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "219",
      "y": "256"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "255",
      "y": "40"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "215",
      "y": "40"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "84",
      "y": "252"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "42",
      "y": "252"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "132",
      "y": "43"
    }
  ],
  "side": "294"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 43
}

The current leader's answer

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0",
      "x": "171",
      "y": "82"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "170",
      "y": "81"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "72",
      "y": "187"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "242",
      "y": "86"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "168",
      "y": "79"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "236",
      "y": "86"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "79",
      "y": "76"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "79",
      "y": "68"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "283",
      "y": "209"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "163",
      "y": "74"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "283",
      "y": "198"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "101",
      "y": "197"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "88",
      "y": "197"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "258",
      "y": "202"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "243",
      "y": "202"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "107",
      "y": "103"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "90",
      "y": "103"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "209",
      "y": "196"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "209",
      "y": "177"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "91",
      "y": "164"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "153",
      "y": "64"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "91",
      "y": "142"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "48",
      "y": "84"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "24",
      "y": "84"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "165",
      "y": "184"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "139",
      "y": "184"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "153",
      "y": "266"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "153",
      "y": "238"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "60",
      "y": "30"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "30",
      "y": "30"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "262",
      "y": "156"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "262",
      "y": "124"
    },
    {
      "quarter": 3,
      "turn": "0",
      "x": "34",
      "y": "176"
    },
    {
      "quarter": 1,
      "turn": "0",
      "x": "34",
      "y": "142"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "195",
      "y": "122"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "159",
      "y": "122"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "257",
      "y": "256"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "219",
      "y": "256"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "255",
      "y": "40"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "215",
      "y": "40"
    },
    {
      "quarter": 2,
      "turn": "0",
      "x": "84",
      "y": "252"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "42",
      "y": "252"
    },
    {
      "quarter": 0,
      "turn": "0",
      "x": "132",
      "y": "43"
    }
  ],
  "side": "294"
}

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