P62 · Extremal configurations · Formed here · Applied frontier · Hard

A sampling design uniform in every pair of columns · d = 6, n = 24

Submit an n-row, d-column sampling table. Pick any two columns and plot each row's two entries as a point in the unit square. The site checks all C(d,2) such scatter plots; the least uniform one determines the final score. Make that worst score as small as possible.

Instanced = 6, n = 24

ObjectiveMinimize the least-uniform pair's discrepancy

Record comparison

the least-uniform pair's discrepancy · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.019512950797	zzzcy	Record details ↓

Formal definition

ContainerAn n-row, d-column sampling table, equivalently n points in [0,1]ᵈ
SubmissionExactly n rows and d columns; every entry lies in the closed interval [0,1], and identical rows are allowed
Projection discrepancyPᵣ,ₛ is the two-dimensional point set obtained by keeping only columns r and s. The indicator 𝟙 is 1 when a point lies in the origin-anchored rectangle [0,u] × [0,v], and 0 otherwise ${{D_{2}}^{*} (P_{r, s})}^{2} = \int_{0}^{1} \int_{0}^{1} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u, x_{i s} \leq v] - u v)}^{2} d u d v$
ScoreCompute the L2-star discrepancy of all C(d,2) column pairs; the largest one is the submission's score $score (P) = \max_{1 \leq r < s \leq d} {D_{2}}^{*} (P_{r, s})$
ObjectiveMake this worst-pair score as small as possible. The verifier computes the squared discrepancy exactly and displays it rounded up at the twelfth decimal $\min_{P \in {({[0, 1]}^{d})}^{n}} score (P)$

Challenge this record ↗

Layout shown: site record

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VERIFIED CONSTRUCTIONworst: columns 4 and 6 · L2-star discrepancy 0.019512950797

Getting a feel for it

Think of it as an experiment plan

Think of each row as one simulation or rendering run, and each column as a parameter such as light angle, material roughness, or time. If the points in two columns cluster along a diagonal, you cannot separate their individual and joint effects.

Every plot shares the same table

Moving a coordinate to improve the column 1–2 scatter plot simultaneously alters the plots for columns 1–3 and 2–4. Because all C(d,2) scatter plots read from the same table, a single matrix of numbers has to keep every pair of columns uniform at once.

Where the frontier is

Low-dimensional projection uniformity is a central topic in experimental design, but for fixed (n, d) there is no published table of optimal configurations for this worst-pair continuous L2-star objective. Every sub-problem is open.

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 (12 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.019594903781→0.019512950797
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.019612987086→0.019594903781
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612987199→0.019612987086
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612987849→0.019612987199
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612988794→0.019612987849
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612990278→0.019612988794
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612990977→0.019612990278
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019613048175→0.019612990977
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.023007114764→0.019613048175
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.023032156459→0.023007114764
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.02345890633→0.023032156459
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.052217573059→0.02345890633
2026-09-01

ANSWER FORMAT

How to write your answer

Read the answer as an n-row, d-column table: each row is one sample and each column is a parameter in [0,1]. Every number has at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1].

The current leader's answer

{
  "points": [
    [
      "0.085164731",
      "0.749595743",
      "0.801984807",
      "0.279713678",
      "0.073673114",
      "0.062639194"
    ],
    [
      "0.450319708",
      "0.551277400",
      "0.305394962",
      "0.032163954",
      "0.288802009",
      "0.166366049"
    ],
    [
      "0.336054057",
      "0.327506942",
      "0.881555085",
      "0.899657499",
      "0.949705905",
      "0.292919839"
    ],
    [
      "0.852225655",
      "0.620317641",
      "0.944631814",
      "0.930015758",
      "0.451284082",
      "0.962127186"
    ],
    [
      "0.496119311",
      "0.597618519",
      "0.703959828",
      "0.501627374",
      "0.855617694",
      "0.694835392"
    ],
    [
      "0.921798501",
      "0.155398101",
      "0.847590603",
      "0.587994014",
      "0.325364387",
      "0.782028647"
    ],
    [
      "0.305531007",
      "0.840609445",
      "0.128828217",
      "0.117202500",
      "0.427602592",
      "0.740313265"
    ],
    [
      "0.206165159",
      "0.656639065",
      "0.075586468",
      "0.624869746",
      "0.203816162",
      "0.563242913"
    ],
    [
      "0.540212823",
      "0.033372423",
      "0.032160652",
      "0.330405844",
      "0.542018124",
      "0.219602020"
    ],
    [
      "0.673748501",
      "0.888355868",
      "0.672322492",
      "0.807064281",
      "0.245631557",
      "0.261726506"
    ],
    [
      "0.368250115",
      "0.925679900",
      "0.915985002",
      "0.372403114",
      "0.719688563",
      "0.918926032"
    ],
    [
      "0.395971205",
      "0.078328992",
      "0.443220803",
      "0.858398379",
      "0.109926038",
      "0.652956047"
    ],
    [
      "0.627940606",
      "0.292366051",
      "0.528931272",
      "0.204052971",
      "0.160660993",
      "0.871317204"
    ],
    [
      "0.715119871",
      "0.247967700",
      "0.162655336",
      "0.677675320",
      "0.758516911",
      "0.028087210"
    ],
    [
      "0.584807228",
      "0.792922608",
      "0.485244550",
      "0.722910276",
      "0.587683274",
      "0.514800420"
    ],
    [
      "0.947878599",
      "0.946457164",
      "0.235836352",
      "0.945199070",
      "0.935750165",
      "0.943092634"
    ],
    [
      "0.164318415",
      "0.120291222",
      "0.630083280",
      "0.157105055",
      "0.805049303",
      "0.478174664"
    ],
    [
      "0.974643287",
      "0.973257534",
      "0.983912117",
      "0.973534038",
      "0.690316962",
      "0.342985984"
    ],
    [
      "0.031016787",
      "0.201722391",
      "0.363366027",
      "0.460521679",
      "0.373275323",
      "0.374796897"
    ],
    [
      "0.114837435",
      "0.403299854",
      "0.260906664",
      "0.774184287",
      "0.671570821",
      "0.826997813"
    ],
    [
      "0.257995849",
      "0.493662740",
      "0.577537871",
      "0.547310914",
      "0.488458008",
      "0.117387069"
    ],
    [
      "0.760756135",
      "0.363522859",
      "0.751725489",
      "0.074588463",
      "0.626130641",
      "0.606980661"
    ],
    [
      "0.808047079",
      "0.446792819",
      "0.211382302",
      "0.404880272",
      "0.030243331",
      "0.430169549"
    ],
    [
      "0.879283183",
      "0.702228654",
      "0.410053481",
      "0.240360584",
      "0.896533770",
      "0.320795956"
    ]
  ]
}

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

Instance parameters

{
  "n": 24,
  "d": 6
}

The current leader's answer

{
  "points": [
    [
      "0.085164731",
      "0.749595743",
      "0.801984807",
      "0.279713678",
      "0.073673114",
      "0.062639194"
    ],
    [
      "0.450319708",
      "0.551277400",
      "0.305394962",
      "0.032163954",
      "0.288802009",
      "0.166366049"
    ],
    [
      "0.336054057",
      "0.327506942",
      "0.881555085",
      "0.899657499",
      "0.949705905",
      "0.292919839"
    ],
    [
      "0.852225655",
      "0.620317641",
      "0.944631814",
      "0.930015758",
      "0.451284082",
      "0.962127186"
    ],
    [
      "0.496119311",
      "0.597618519",
      "0.703959828",
      "0.501627374",
      "0.855617694",
      "0.694835392"
    ],
    [
      "0.921798501",
      "0.155398101",
      "0.847590603",
      "0.587994014",
      "0.325364387",
      "0.782028647"
    ],
    [
      "0.305531007",
      "0.840609445",
      "0.128828217",
      "0.117202500",
      "0.427602592",
      "0.740313265"
    ],
    [
      "0.206165159",
      "0.656639065",
      "0.075586468",
      "0.624869746",
      "0.203816162",
      "0.563242913"
    ],
    [
      "0.540212823",
      "0.033372423",
      "0.032160652",
      "0.330405844",
      "0.542018124",
      "0.219602020"
    ],
    [
      "0.673748501",
      "0.888355868",
      "0.672322492",
      "0.807064281",
      "0.245631557",
      "0.261726506"
    ],
    [
      "0.368250115",
      "0.925679900",
      "0.915985002",
      "0.372403114",
      "0.719688563",
      "0.918926032"
    ],
    [
      "0.395971205",
      "0.078328992",
      "0.443220803",
      "0.858398379",
      "0.109926038",
      "0.652956047"
    ],
    [
      "0.627940606",
      "0.292366051",
      "0.528931272",
      "0.204052971",
      "0.160660993",
      "0.871317204"
    ],
    [
      "0.715119871",
      "0.247967700",
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    [
      "0.584807228",
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      "0.722910276",
      "0.587683274",
      "0.514800420"
    ],
    [
      "0.947878599",
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      "0.235836352",
      "0.945199070",
      "0.935750165",
      "0.943092634"
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    [
      "0.164318415",
      "0.120291222",
      "0.630083280",
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      "0.478174664"
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    [
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      "0.983912117",
      "0.973534038",
      "0.690316962",
      "0.342985984"
    ],
    [
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      "0.201722391",
      "0.363366027",
      "0.460521679",
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      "0.403299854",
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      "0.671570821",
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    [
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    [
      "0.760756135",
      "0.363522859",
      "0.751725489",
      "0.074588463",
      "0.626130641",
      "0.606980661"
    ],
    [
      "0.808047079",
      "0.446792819",
      "0.211382302",
      "0.404880272",
      "0.030243331",
      "0.430169549"
    ],
    [
      "0.879283183",
      "0.702228654",
      "0.410053481",
      "0.240360584",
      "0.896533770",
      "0.320795956"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1]. · Verifier v1.0.0

DISCUSSION

Discussion

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