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

A sampling design uniform in every pair of columns · d = 8, n = 16

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 = 8, n = 16

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	0.029755001064	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 ↗

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VERIFIED CONSTRUCTIONworst: columns 5 and 7 · L2-star discrepancy 0.029755001064

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 (9 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.029773597321→0.029755001064
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.029825507719→0.029773597321
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.029825507725→0.029825507719
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.029825507736→0.029825507725
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.034288686092→0.029825507736
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.034323725134→0.034288686092
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.034342220342→0.034323725134
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.035486094079→0.034342220342
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.070774366079→0.035486094079
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

{
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    ],
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      "0.700900766",
      "0.422579391",
      "0.254844964",
      "0.457613927"
    ],
    [
      "0.192704123",
      "0.577532718",
      "0.041871488",
      "0.771944847",
      "0.436516528",
      "0.708996352",
      "0.602119883",
      "0.387267957"
    ],
    [
      "0.252655794",
      "0.428017794",
      "0.763277488",
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      "0.318060166",
      "0.044120448",
      "0.873941132",
      "0.805212506"
    ],
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      "0.298216828",
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      "0.950805361",
      "0.932735119",
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      "0.156496585",
      "0.947999278"
    ],
    [
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      "0.633409530",
      "0.780872302",
      "0.801347922",
      "0.112982936"
    ],
    [
      "0.403370975",
      "0.774430431",
      "0.366933385",
      "0.306838064",
      "0.043188262",
      "0.314422411",
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      "0.583391978"
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    [
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    [
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      "0.543177819",
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      "0.501150016",
      "0.047185595",
      "0.046186992"
    ],
    [
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      "0.828275836",
      "0.904817754",
      "0.849931860",
      "0.366695217",
      "0.742693643",
      "0.331973656"
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    [
      "0.680483477",
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      "0.235278958",
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      "0.663234578"
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    [
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      "0.863358461",
      "0.115055401",
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      "0.537897079",
      "0.873074819"
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      "0.810882122",
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    [
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      "0.886792297",
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      "0.525096582"
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      "0.960286113",
      "0.963299035",
      "0.959015035",
      "0.313445006"
    ]
  ]
}

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

Instance parameters

{
  "n": 16,
  "d": 8
}

The current leader's answer

{
  "points": [
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      "0.115613742",
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      "0.229665051",
      "0.337746122",
      "0.242667182"
    ],
    [
      "0.128706636",
      "0.645231112",
      "0.693632705",
      "0.465544731",
      "0.700900766",
      "0.422579391",
      "0.254844964",
      "0.457613927"
    ],
    [
      "0.192704123",
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      "0.387267957"
    ],
    [
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      "0.763277488",
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      "0.805212506"
    ],
    [
      "0.298216828",
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      "0.932281262",
      "0.950805361",
      "0.932735119",
      "0.932630492",
      "0.156496585",
      "0.947999278"
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    [
      "0.331300146",
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      "0.780872302",
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    ],
    [
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      "0.366933385",
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    ],
    [
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  ]
}

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

DISCUSSION

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