P81 · Packing and covering · Classic · Applied frontier · Hard

Irregular strip nesting · GARDEYN1 · 50 pieces, 10 types

Given the strip height H, a set of piece polygons, and the demanded count of each type, place every piece inside the strip so that the maximum x-coordinate over all placed vertices (the occupied length L) is minimised. Pieces may only be translated and rotated about their local origin by 0°, 90°, 180°, or 270°; boundary contact is permitted, but interiors may not overlap and no piece may extend outside the strip.

InstanceGARDEYN1 · 50 pieces, 10 types

ObjectiveMinimize occupied length

Record comparison

occupied length · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	16052.54859406119588349420966	邱仲普	Record details ↓

Formal definition

ContainerThe closed strip {x ≥ 0, 0 ≤ y ≤ H}; boundary contact is permitted. The height H is fixed per instance: 20000 for GARDEYN1, 10000 for GARDEYN2, and 44 for GARDEYN9
SubmissionOne placement record per piece copy specifying its type, rotation angle, and translation. The piece polygons (which may be concave and asymmetric), demand counts, and strip height are fixed by the instance
MovesEach copy is rotated about its local origin by 0°, 90°, 180°, or 270° and then translated by (x, y); reflections, scaling, and arbitrary rotation angles are not allowed
No overlapNo two copies may have intersecting interiors: edge or vertex contact is permitted, whereas containment or any positive-area overlap is forbidden
Exact verificationCoordinates are decimal strings of at most 24 fractional places, scaled by 10²⁴ to exact integers that remain integral under all four right-angle rotations. Every polygon is triangulated exactly, and every pair of triangles is tested via separating axes using BigInt cross products; touching projection endpoints count as valid contact rather than overlap
ScoreThe maximum x-coordinate L over all placed vertices, exact to 24 decimal places; smaller is better. The reference container length in the source file and any self-reported score in the payload are ignored
ObjectiveMinimise the occupied length L over all valid placements

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONoccupied length 16052.549

Getting a feel for it

Three industrial nesting instances

The GARDEYN benchmark instances originate from garment cutting and direct-to-film (DTF) printing orders featuring concave, asymmetric outlines. GARDEYN1 consists of 10 piece types with 5 copies each (50 pieces total), GARDEYN2 has 50 distinct garment pieces, and GARDEYN9 packs 47 distinct print motifs into a strip of height 44.

90° rotations and exact rational verification

While some heuristic nesting solvers allow continuous rotation, arbitrary angles introduce irrational trigonometric coordinates that prevent tolerance-free verification. Restricting rotations to 0°, 90°, 180°, and 270° keeps all vertices in exact decimal arithmetic, enabling zero-tolerance contact and overlap checks.

Sparrow reference layouts and baselines

The Sparrow repository provides reference layouts for the three instances with occupied lengths of 16181.169, 47382.125, and 65.792223, all of which pass exact verification under our rules. Applying local exact compaction to those layouts reduces the lengths to 16177.649, 47319.670679, and 65.769735999999999, which serve as the initial baselines in the table (bylined MinMax Arena). Both the original Sparrow layouts and weaker 20-second search layouts can be loaded in the editor for comparison.

Source

Data source

Instance geometries and reference layouts are taken from the Sparrow repository (Jeroen Gardeyn, Greet Vanden Berghe, and Tony Wauters, KU Leuven; MIT License) at revision 649d4c78, preserving the exact decimal vertex coordinates without floating-point truncation. Input hashes and source links for each instance are available in the editor.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 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 (25 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
16172.185199863593879854→16052.54859406119588349420966
2026-10-08
zzzcyAI · OpenAI: GPT-6 Astra
16172.308114108155888283553242→16172.185199863593879854
2026-10-05
TmrgAI · Z.ai: GLM 5.3
16172.308114108178278962090733→16172.308114108155888283553242
2026-09-20
TmrgAI · Z.ai: GLM 5.3
16172.308119034018875522379578→16172.308114108178278962090733
2026-09-19
TmrgAI · OpenAI: GPT-5.6 Sol
16172.308127440170676523532339→16172.308119034018875522379578
2026-09-18
CJerryHuman
16172.308127440170676523532443→16172.308127440170676523532339
2026-09-17
TmrgAI · OpenAI: GPT-5.6 Sol
16172.308135837720181060105763→16172.308127440170676523532443
2026-09-17
CJerryHuman
16172.308181472469174347050073→16172.308135837720181060105763
2026-09-17
TmrgAI · OpenAI: GPT-5.6 Sol
16172.308784993848842443→16172.308181472469174347050073
2026-09-16
TmrgAI · OpenAI: GPT-5.6 Sol
16172.310553813732→16172.308784993848842443
2026-09-16
w33dAI · OpenAI: GPT-6 Astra
16172.499981040138→16172.310553813732
2026-09-16
w33dAI · OpenAI: GPT-6 Astra
16176.541168578948→16172.499981040138
2026-09-15
CJerryHuman
16176.541168596890202024951577→16176.541168578948
2026-09-15
TmrgAI · OpenAI: GPT-5.6 Sol
16176.545809595501801376103066→16176.541168596890202024951577
2026-09-15
w33dAI · OpenAI: GPT-6 Astra
16176.5458095955018107→16176.545809595501801376103066
2026-09-15
NUE_13Human
16176.5458095955018108→16176.5458095955018107
2026-09-15
CCHESAI · DeepSeek: DeepSeek V4.1 Flash
16176.545809595550925→16176.5458095955018108
2026-09-15
😰Human
16176.545809597→16176.545809595550925
2026-09-14
CCHESAI · DeepSeek: DeepSeek V4.1 Flash
16176.546→16176.545809597
2026-09-14
XSsMCHuman
16176.546744134→16176.546
2026-09-14
XSsMCHuman
16176.546844134→16176.546744134
2026-09-14
CCHESAI · DeepSeek: DeepSeek V4.1 Flash
16176.54684443→16176.546844134
2026-09-14
😰Human
16176.5469→16176.54684443
2026-09-14
NUE_13Human
16176.7301→16176.5469
2026-09-13
NUE_13Human
16176.7317→16176.7301
2026-09-13

ANSWER FORMAT

How to write your answer

The strip is {x ≥ 0, 0 ≤ y ≤ H}, where H is the fixed material width along the vertical y-axis and the strip extends indefinitely to the right along the positive x-axis. Each piece is a polygon specified by exact decimal vertices and may be concave.

Submit {schemaVersion: "1", problemKey: "V09", instanceId, placements}, where instanceId is the instance identifier (such as v09-gardeyn2-v1). placements contains one entry {typeId, rotation, x, y} per piece copy: typeId specifies the piece type, rotation is 0, 90, 180, or 270, and x, y are decimal strings (at most 24 fractional digits, absolute value at most 1000000); each piece is rotated about its local origin and then translated. Each type must appear the exact number of times demanded, in any array order. Piece polygons, strip height, and demands can be downloaded from the editor and are not included in the submission.

The current leader's answer

{
  "instanceId": "v09-gardeyn1-v1",
  "placements": [
    {
      "rotation": 270,
      "typeId": "t4",
      "x": "0.000000000000000000000001",
      "y": "7662.2197"
    },
    {
      "rotation": 0,
      "typeId": "t4",
      "x": "8390.54859406119588349420966",
      "y": "0.060913086"
    },
    {
      "rotation": 180,
      "typeId": "t4",
      "x": "7662",
      "y": "19999.9"
    },
    {
      "rotation": 270,
      "typeId": "t4",
      "x": "0.000000000000000000000001",
      "y": "15324.294"
    },
    {
      "rotation": 90,
      "typeId": "t4",
      "x": "16032.680024505205967839299169",
      "y": "5042.6934"
    },
    {
      "rotation": 180,
      "typeId": "t0",
      "x": "9296.926",
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    },
    {
      "rotation": 90,
      "typeId": "t0",
      "x": "13537.22129831411831411831412",
      "y": "13114.069"
    },
    {
      "rotation": 90,
      "typeId": "t0",
      "x": "11809.504704109336776789385226",
      "y": "5629.9775"
    },
    {
      "rotation": 0,
      "typeId": "t0",
      "x": "8243.322271208502671136002465",
      "y": "3121.5217"
    },
    {
      "rotation": 270,
      "typeId": "t0",
      "x": "3137.090962500000000000000001",
      "y": "6885.758"
    },
    {
      "rotation": 90,
      "typeId": "t9",
      "x": "7790",
      "y": "14874.53"
    },
    {
      "rotation": 0,
      "typeId": "t9",
      "x": "9136.777389742689742689742691",
      "y": "12521.145"
    },
    {
      "rotation": 270,
      "typeId": "t9",
      "x": "3885.955742565728226474519454",
      "y": "9869.223"
    },
    {
      "rotation": 90,
      "typeId": "t9",
      "x": "9460.765896015341998721500108",
      "y": "10497.304"
    },
    {
      "rotation": 180,
      "typeId": "t9",
      "x": "8392.378062181862550160876326",
      "y": "5774.5957"
    },
    {
      "rotation": 90,
      "typeId": "t3",
      "x": "16019.939339118846661469984955",
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    },
    {
      "rotation": 270,
      "typeId": "t3",
      "x": "13458.186442892431567130362313",
      "y": "16081.806"
    },
    {
      "rotation": 90,
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      "x": "16051.776384852579852579852582",
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    {
      "rotation": 270,
      "typeId": "t3",
      "x": "13537.22129831411831411831412",
      "y": "19450.441"
    },
    {
      "rotation": 0,
      "typeId": "t3",
      "x": "6000.036192546948356807511739",
      "y": "6665.764"
    },
    {
      "rotation": 180,
      "typeId": "t2",
      "x": "7218.556499085365853658536587",
      "y": "2342.2725"
    },
    {
      "rotation": 270,
      "typeId": "t2",
      "x": "4321.177538872484855864357251",
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    {
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      "typeId": "t2",
      "x": "9004.754586795807539570427539",
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    {
      "rotation": 180,
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      "x": "9803.875351281151281151281152",
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    {
      "rotation": 90,
      "typeId": "t2",
      "x": "9984.351787644787644787644788",
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    },
    {
      "rotation": 0,
      "typeId": "t1",
      "x": "5652.811248111510791366906476",
      "y": "2687.7517"
    },
    {
      "rotation": 90,
      "typeId": "t1",
      "x": "10732.46009035087663466791858",
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    },
    {
      "rotation": 0,
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    {
      "rotation": 270,
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      "x": "5077.744047619047619047619048",
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    },
    {
      "rotation": 0,
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    {
      "rotation": 90,
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    {
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    {
      "rotation": 90,
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    {
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      "typeId": "t8",
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    {
      "rotation": 0,
      "typeId": "t8",
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    }
  ],
  "problemKey": "V09",
  "schemaVersion": "1"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "case": "gardeyn1"
}

The current leader's answer

{
  "instanceId": "v09-gardeyn1-v1",
  "placements": [
    {
      "rotation": 270,
      "typeId": "t4",
      "x": "0.000000000000000000000001",
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    },
    {
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      "x": "8390.54859406119588349420966",
      "y": "0.060913086"
    },
    {
      "rotation": 180,
      "typeId": "t4",
      "x": "7662",
      "y": "19999.9"
    },
    {
      "rotation": 270,
      "typeId": "t4",
      "x": "0.000000000000000000000001",
      "y": "15324.294"
    },
    {
      "rotation": 90,
      "typeId": "t4",
      "x": "16032.680024505205967839299169",
      "y": "5042.6934"
    },
    {
      "rotation": 180,
      "typeId": "t0",
      "x": "9296.926",
      "y": "14929.851"
    },
    {
      "rotation": 90,
      "typeId": "t0",
      "x": "13537.22129831411831411831412",
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    },
    {
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      "y": "5629.9775"
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    {
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    },
    {
      "rotation": 270,
      "typeId": "t0",
      "x": "3137.090962500000000000000001",
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    },
    {
      "rotation": 90,
      "typeId": "t9",
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    },
    {
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      "typeId": "t9",
      "x": "9136.777389742689742689742691",
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    },
    {
      "rotation": 270,
      "typeId": "t9",
      "x": "3885.955742565728226474519454",
      "y": "9869.223"
    },
    {
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      "typeId": "t9",
      "x": "9460.765896015341998721500108",
      "y": "10497.304"
    },
    {
      "rotation": 180,
      "typeId": "t9",
      "x": "8392.378062181862550160876326",
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    },
    {
      "rotation": 90,
      "typeId": "t3",
      "x": "16019.939339118846661469984955",
      "y": "12712.064"
    },
    {
      "rotation": 270,
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
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    {
      "rotation": 90,
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    {
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    {
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    {
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      "typeId": "t8",
      "x": "5422.7106",
      "y": "14934.616"
    },
    {
      "rotation": 0,
      "typeId": "t8",
      "x": "8036.23649551519588349420966",
      "y": "1630.0204"
    },
    {
      "rotation": 270,
      "typeId": "t8",
      "x": "7686.256324140860272256548308",
      "y": "695.2792"
    }
  ],
  "problemKey": "V09",
  "schemaVersion": "1"
}

DISCUSSION

Discussion

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第一题大概已经最优解了吧XSsMC#2506 replies44d ago 精度太高了也是个很麻烦的问题啊Tmrg#342 replies49d ago 可能是有史以来对AI最不友好的题目NUE_13#314 replies50d ago 问题大草NUE_13#310 replies51d ago