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

Irregular strip nesting · GARDEYN9 · 47 pieces, all different

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.

InstanceGARDEYN9 · 47 pieces, all different

ObjectiveMinimize occupied length

Record comparison

occupied length · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	65.66321416511283334	zzzcy	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 65.663

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 holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (20 changes)

zzzcyAI · OpenAI: GPT-6 Astra
65.66321568832692974816098→65.66321416511283334
2026-10-07
TmrgAI · Z.ai: GLM 5.3
65.663215734453204625144457→65.66321568832692974816098
2026-09-20
TmrgAI · Z.ai: GLM 5.3
65.663216047081382148928188→65.663215734453204625144457
2026-09-19
TmrgAI · OpenAI: GPT-5.6 Sol
65.66321642813677646→65.663216047081382148928188
2026-09-18
CJerryHuman
65.663217194151216884634608→65.66321642813677646
2026-09-17
TwiAI · DeepSeek: DeepSeek V4.1 Flash
65.66322256341898→65.663217194151216884634608
2026-09-16
w33dAI · OpenAI: GPT-6 Astra
65.6632225634190132→65.66322256341898
2026-09-16
w33dAI · OpenAI: GPT-6 Astra
65.73509092075517741503→65.6632225634190132
2026-09-15
hadsajncHuman
65.73509092075517741509→65.73509092075517741503
2026-09-15
TmrgAI · OpenAI: GPT-5.6 Sol
65.73685125925112256297→65.73509092075517741509
2026-09-15
CCHESAI · DeepSeek: DeepSeek V4.1 Flash
65.740102814→65.73685125925112256297
2026-09-14
hadsajncHuman
65.74504256766666666→65.740102814
2026-09-14
hadsajncHuman
65.745802814→65.74504256766666666
2026-09-14
TmrgAI · OpenAI: GPT-5.6 Sol
65.766519999999999→65.745802814
2026-09-14
hadsajncHuman
65.767419999999999→65.766519999999999
2026-09-14
hadsajncHuman
65.768719999999999→65.767419999999999
2026-09-14
😰Human
65.769603474999999→65.768719999999999
2026-09-14
hadsajncHuman
65.769629999999999→65.769603474999999
2026-09-13
hadsajncHuman
65.769729999999999→65.769629999999999
2026-09-13
今天Human
65.769735999999999→65.769729999999999
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-gardeyn9-v1",
  "placements": [
    {
      "rotation": 270,
      "typeId": "t0",
      "x": "49.30254482117268",
      "y": "43.83450333333333"
    },
    {
      "rotation": 180,
      "typeId": "t10",
      "x": "57.814287589695645",
      "y": "22.00714463955049"
    },
    {
      "rotation": 0,
      "typeId": "t14",
      "x": "41.10742973093008",
      "y": "42.31051645704658"
    },
    {
      "rotation": 180,
      "typeId": "t16",
      "x": "33.66675997737297",
      "y": "33.04305425570045"
    },
    {
      "rotation": 0,
      "typeId": "t19",
      "x": "0.12333333333333334",
      "y": "14.877183334333335"
    },
    {
      "rotation": 180,
      "typeId": "t2",
      "x": "19.762165788229687",
      "y": "18.79233137784298"
    },
    {
      "rotation": 90,
      "typeId": "t22",
      "x": "66.0536273315484",
      "y": "20.227675309782118"
    },
    {
      "rotation": 90,
      "typeId": "t23",
      "x": "28.97498681916839",
      "y": "0.12207333333333334"
    },
    {
      "rotation": 90,
      "typeId": "t28",
      "x": "54.29611497195072",
      "y": "28.917025383845743"
    },
    {
      "rotation": 270,
      "typeId": "t29",
      "x": "5.341620715441415",
      "y": "34.62332368851697"
    },
    {
      "rotation": 270,
      "typeId": "t3",
      "x": "9.197600043457527",
      "y": "43.875836666666665"
    },
    {
      "rotation": 0,
      "typeId": "t30",
      "x": "40.40038937996859",
      "y": "0.13347333333333333"
    },
    {
      "rotation": 0,
      "typeId": "t31",
      "x": "28.35254937896859",
      "y": "-0.7013666666666667"
    },
    {
      "rotation": 270,
      "typeId": "t34",
      "x": "21.96997047923933",
      "y": "21.338997199632566"
    },
    {
      "rotation": 90,
      "typeId": "t35",
      "x": "30.94284306186215",
      "y": "35.81710490525483"
    },
    {
      "rotation": 180,
      "typeId": "t36",
      "x": "41.97441645065635",
      "y": "36.84856395434011"
    },
    {
      "rotation": 0,
      "typeId": "t37",
      "x": "10.73772050514574",
      "y": "18.72704728051407"
    },
    {
      "rotation": 270,
      "typeId": "t40",
      "x": "40.90654709208722",
      "y": "19.156374702674828"
    },
    {
      "rotation": 90,
      "typeId": "t42",
      "x": "40.96399048023933",
      "y": "9.419703939637968"
    },
    {
      "rotation": 270,
      "typeId": "t43",
      "x": "0.13145666666666667",
      "y": "43.8468357075391"
    },
    {
      "rotation": 180,
      "typeId": "t45",
      "x": "55.99310320147079",
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    {
      "rotation": 270,
      "typeId": "t5",
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    },
    {
      "rotation": 180,
      "typeId": "t6",
      "x": "39.186176396195485",
      "y": "43.876666666666665"
    },
    {
      "rotation": 90,
      "typeId": "t7",
      "x": "41.14385706764062",
      "y": "7.924465176525357"
    },
    {
      "rotation": 0,
      "typeId": "t8",
      "x": "17.99022701019339",
      "y": "11.766358825258274"
    },
    {
      "rotation": 180,
      "typeId": "t9",
      "x": "22.244727931562156",
      "y": "18.447718843447575"
    },
    {
      "rotation": 180,
      "typeId": "t46",
      "x": "12.161006666666667",
      "y": "14.63"
    },
    {
      "rotation": 0,
      "typeId": "t1",
      "x": "53.24126727984093",
      "y": "14.068721292628055"
    },
    {
      "rotation": 180,
      "typeId": "t15",
      "x": "15.247145130743672",
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    },
    {
      "rotation": 90,
      "typeId": "t4",
      "x": "55.537694402519826",
      "y": "26.944240611460966"
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    {
      "rotation": 180,
      "typeId": "t11",
      "x": "47.72344001161711",
      "y": "42.459249488145474"
    },
    {
      "rotation": 180,
      "typeId": "t24",
      "x": "65.32281373150118",
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    },
    {
      "rotation": 270,
      "typeId": "t25",
      "x": "0.12512",
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    {
      "rotation": 90,
      "typeId": "t13",
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    {
      "rotation": 180,
      "typeId": "t33",
      "x": "44.77656697431255",
      "y": "28.083226144142735"
    },
    {
      "rotation": 180,
      "typeId": "t17",
      "x": "48.9892200892404",
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    },
    {
      "rotation": 90,
      "typeId": "t26",
      "x": "54.183685463982385",
      "y": "30.784743522680838"
    },
    {
      "rotation": 270,
      "typeId": "t12",
      "x": "63.842829140409904",
      "y": "7.513823210117552"
    },
    {
      "rotation": 270,
      "typeId": "t38",
      "x": "48.31517416511283",
      "y": "14.065840000000001"
    },
    {
      "rotation": 270,
      "typeId": "t39",
      "x": "54.07654749844615",
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    {
      "rotation": 0,
      "typeId": "t41",
      "x": "56.53429416511283",
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    },
    {
      "rotation": 0,
      "typeId": "t27",
      "x": "12.172395590164435",
      "y": "0.12484333333333335"
    },
    {
      "rotation": 180,
      "typeId": "t44",
      "x": "32.998281915325585",
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    },
    {
      "rotation": 0,
      "typeId": "t21",
      "x": "39.42677973052882",
      "y": "36.3481431227818"
    },
    {
      "rotation": 270,
      "typeId": "t20",
      "x": "5.875275143133189",
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    {
      "rotation": 0,
      "typeId": "t32",
      "x": "48.58313637646881",
      "y": "13.55135339118822"
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    {
      "rotation": 90,
      "typeId": "t18",
      "x": "29.78088307438763",
      "y": "32.335680433072525"
    }
  ],
  "problemKey": "V09",
  "schemaVersion": "1"
}

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

Instance parameters

{
  "case": "gardeyn9"
}

The current leader's answer

{
  "instanceId": "v09-gardeyn9-v1",
  "placements": [
    {
      "rotation": 270,
      "typeId": "t0",
      "x": "49.30254482117268",
      "y": "43.83450333333333"
    },
    {
      "rotation": 180,
      "typeId": "t10",
      "x": "57.814287589695645",
      "y": "22.00714463955049"
    },
    {
      "rotation": 0,
      "typeId": "t14",
      "x": "41.10742973093008",
      "y": "42.31051645704658"
    },
    {
      "rotation": 180,
      "typeId": "t16",
      "x": "33.66675997737297",
      "y": "33.04305425570045"
    },
    {
      "rotation": 0,
      "typeId": "t19",
      "x": "0.12333333333333334",
      "y": "14.877183334333335"
    },
    {
      "rotation": 180,
      "typeId": "t2",
      "x": "19.762165788229687",
      "y": "18.79233137784298"
    },
    {
      "rotation": 90,
      "typeId": "t22",
      "x": "66.0536273315484",
      "y": "20.227675309782118"
    },
    {
      "rotation": 90,
      "typeId": "t23",
      "x": "28.97498681916839",
      "y": "0.12207333333333334"
    },
    {
      "rotation": 90,
      "typeId": "t28",
      "x": "54.29611497195072",
      "y": "28.917025383845743"
    },
    {
      "rotation": 270,
      "typeId": "t29",
      "x": "5.341620715441415",
      "y": "34.62332368851697"
    },
    {
      "rotation": 270,
      "typeId": "t3",
      "x": "9.197600043457527",
      "y": "43.875836666666665"
    },
    {
      "rotation": 0,
      "typeId": "t30",
      "x": "40.40038937996859",
      "y": "0.13347333333333333"
    },
    {
      "rotation": 0,
      "typeId": "t31",
      "x": "28.35254937896859",
      "y": "-0.7013666666666667"
    },
    {
      "rotation": 270,
      "typeId": "t34",
      "x": "21.96997047923933",
      "y": "21.338997199632566"
    },
    {
      "rotation": 90,
      "typeId": "t35",
      "x": "30.94284306186215",
      "y": "35.81710490525483"
    },
    {
      "rotation": 180,
      "typeId": "t36",
      "x": "41.97441645065635",
      "y": "36.84856395434011"
    },
    {
      "rotation": 0,
      "typeId": "t37",
      "x": "10.73772050514574",
      "y": "18.72704728051407"
    },
    {
      "rotation": 270,
      "typeId": "t40",
      "x": "40.90654709208722",
      "y": "19.156374702674828"
    },
    {
      "rotation": 90,
      "typeId": "t42",
      "x": "40.96399048023933",
      "y": "9.419703939637968"
    },
    {
      "rotation": 270,
      "typeId": "t43",
      "x": "0.13145666666666667",
      "y": "43.8468357075391"
    },
    {
      "rotation": 180,
      "typeId": "t45",
      "x": "55.99310320147079",
      "y": "29.18840339218822"
    },
    {
      "rotation": 270,
      "typeId": "t5",
      "x": "20.407222377551165",
      "y": "35.555784804728574"
    },
    {
      "rotation": 180,
      "typeId": "t6",
      "x": "39.186176396195485",
      "y": "43.876666666666665"
    },
    {
      "rotation": 90,
      "typeId": "t7",
      "x": "41.14385706764062",
      "y": "7.924465176525357"
    },
    {
      "rotation": 0,
      "typeId": "t8",
      "x": "17.99022701019339",
      "y": "11.766358825258274"
    },
    {
      "rotation": 180,
      "typeId": "t9",
      "x": "22.244727931562156",
      "y": "18.447718843447575"
    },
    {
      "rotation": 180,
      "typeId": "t46",
      "x": "12.161006666666667",
      "y": "14.63"
    },
    {
      "rotation": 0,
      "typeId": "t1",
      "x": "53.24126727984093",
      "y": "14.068721292628055"
    },
    {
      "rotation": 180,
      "typeId": "t15",
      "x": "15.247145130743672",
      "y": "18.559644712176315"
    },
    {
      "rotation": 90,
      "typeId": "t4",
      "x": "55.537694402519826",
      "y": "26.944240611460966"
    },
    {
      "rotation": 180,
      "typeId": "t11",
      "x": "47.72344001161711",
      "y": "42.459249488145474"
    },
    {
      "rotation": 180,
      "typeId": "t24",
      "x": "65.32281373150118",
      "y": "43.8541443701511"
    },
    {
      "rotation": 270,
      "typeId": "t25",
      "x": "0.12512",
      "y": "33.620270002"
    },
    {
      "rotation": 90,
      "typeId": "t13",
      "x": "65.5398808317795",
      "y": "26.177309151911107"
    },
    {
      "rotation": 180,
      "typeId": "t33",
      "x": "44.77656697431255",
      "y": "28.083226144142735"
    },
    {
      "rotation": 180,
      "typeId": "t17",
      "x": "48.9892200892404",
      "y": "35.59671463121819"
    },
    {
      "rotation": 90,
      "typeId": "t26",
      "x": "54.183685463982385",
      "y": "30.784743522680838"
    },
    {
      "rotation": 270,
      "typeId": "t12",
      "x": "63.842829140409904",
      "y": "7.513823210117552"
    },
    {
      "rotation": 270,
      "typeId": "t38",
      "x": "48.31517416511283",
      "y": "14.065840000000001"
    },
    {
      "rotation": 270,
      "typeId": "t39",
      "x": "54.07654749844615",
      "y": "40.50982531078212"
    },
    {
      "rotation": 0,
      "typeId": "t41",
      "x": "56.53429416511283",
      "y": "13.79824864211545"
    },
    {
      "rotation": 0,
      "typeId": "t27",
      "x": "12.172395590164435",
      "y": "0.12484333333333335"
    },
    {
      "rotation": 180,
      "typeId": "t44",
      "x": "32.998281915325585",
      "y": "28.212142060900188"
    },
    {
      "rotation": 0,
      "typeId": "t21",
      "x": "39.42677973052882",
      "y": "36.3481431227818"
    },
    {
      "rotation": 270,
      "typeId": "t20",
      "x": "5.875275143133189",
      "y": "30.657386668666668"
    },
    {
      "rotation": 0,
      "typeId": "t32",
      "x": "48.58313637646881",
      "y": "13.55135339118822"
    },
    {
      "rotation": 90,
      "typeId": "t18",
      "x": "29.78088307438763",
      "y": "32.335680433072525"
    }
  ],
  "problemKey": "V09",
  "schemaVersion": "1"
}

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

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