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

A sampling design uniform in every pair of columns · d = 4, n = 32

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 = 4, n = 32

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	0.013890754713	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

x₁ × x₂

x₁ × x₃

x₁ × x₄WORST

x₂ × x₃

x₂ × x₄

x₃ × x₄

VERIFIED CONSTRUCTIONworst: columns 1 and 4 · L2-star discrepancy 0.013890754713

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
0.013891317352→0.013890754713
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.013908200123→0.013891317352
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.013908200394→0.013908200123
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.013909495977→0.013908200394
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016563782223→0.013909495977
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.016592155407→0.016563782223
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.016592196638→0.016592155407
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.016821591669→0.016592196638
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.282597082631→0.016821591669
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": [
    [
      "0.802435814",
      "0.272149198",
      "0.394049743",
      "0.490195791"
    ],
    [
      "0.603767614",
      "0.082422713",
      "0.083181276",
      "0.553280762"
    ],
    [
      "0.053268495",
      "0.213925726",
      "0.160060235",
      "0.227024735"
    ],
    [
      "0.762233519",
      "0.776407985",
      "0.053216561",
      "0.745513505"
    ],
    [
      "0.623711516",
      "0.986646053",
      "0.860614959",
      "0.984841594"
    ],
    [
      "0.175565532",
      "0.538791925",
      "0.362471476",
      "0.811844696"
    ],
    [
      "0.019805214",
      "0.664852233",
      "0.538170129",
      "0.590847168"
    ],
    [
      "0.105974884",
      "0.897455520",
      "0.906892631",
      "0.417184243"
    ],
    [
      "0.635971865",
      "0.698407213",
      "0.973154906",
      "0.098135997"
    ],
    [
      "0.534894682",
      "0.842780906",
      "0.424345817",
      "0.189321559"
    ],
    [
      "0.738139444",
      "0.049076330",
      "0.742264724",
      "0.263156166"
    ],
    [
      "0.949719463",
      "0.706566788",
      "0.278223162",
      "0.960964516"
    ],
    [
      "0.694463929",
      "0.383944242",
      "0.326534843",
      "0.326268999"
    ],
    [
      "0.838519936",
      "0.564644908",
      "0.841363579",
      "0.164437719"
    ],
    [
      "0.412531692",
      "0.460025602",
      "0.498810109",
      "0.021568493"
    ],
    [
      "0.930791419",
      "0.432039526",
      "0.885356122",
      "0.617480463"
    ],
    [
      "0.861411576",
      "0.879944772",
      "0.183770128",
      "0.878836521"
    ],
    [
      "0.495896095",
      "0.499513275",
      "0.210334516",
      "0.452499118"
    ],
    [
      "0.238418885",
      "0.340066173",
      "0.025307402",
      "0.134829687"
    ],
    [
      "0.893440212",
      "0.172631574",
      "0.565829770",
      "0.400028274"
    ],
    [
      "0.563240644",
      "0.307061491",
      "0.659828133",
      "0.671272045"
    ],
    [
      "0.464439110",
      "0.155233296",
      "0.938497412",
      "0.894903233"
    ],
    [
      "0.277018124",
      "0.021880704",
      "0.456756158",
      "0.711117509"
    ],
    [
      "0.138621329",
      "0.122318930",
      "0.254514624",
      "0.062249472"
    ],
    [
      "0.310035526",
      "0.744715479",
      "0.625349754",
      "0.294121683"
    ],
    [
      "0.349860728",
      "0.621576429",
      "0.116877506",
      "0.364844360"
    ],
    [
      "0.090587757",
      "0.412469843",
      "0.703063218",
      "0.932370118"
    ],
    [
      "0.661884168",
      "0.591038517",
      "0.592877047",
      "0.847238233"
    ],
    [
      "0.204636700",
      "0.805833835",
      "0.774415658",
      "0.515333367"
    ],
    [
      "0.972269849",
      "0.964801569",
      "0.690762856",
      "0.089559214"
    ],
    [
      "0.443932732",
      "0.935995430",
      "0.296934243",
      "0.638402925"
    ],
    [
      "0.375561480",
      "0.246900290",
      "0.810833060",
      "0.775034355"
    ]
  ]
}

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

Instance parameters

{
  "n": 32,
  "d": 4
}

The current leader's answer

{
  "points": [
    [
      "0.802435814",
      "0.272149198",
      "0.394049743",
      "0.490195791"
    ],
    [
      "0.603767614",
      "0.082422713",
      "0.083181276",
      "0.553280762"
    ],
    [
      "0.053268495",
      "0.213925726",
      "0.160060235",
      "0.227024735"
    ],
    [
      "0.762233519",
      "0.776407985",
      "0.053216561",
      "0.745513505"
    ],
    [
      "0.623711516",
      "0.986646053",
      "0.860614959",
      "0.984841594"
    ],
    [
      "0.175565532",
      "0.538791925",
      "0.362471476",
      "0.811844696"
    ],
    [
      "0.019805214",
      "0.664852233",
      "0.538170129",
      "0.590847168"
    ],
    [
      "0.105974884",
      "0.897455520",
      "0.906892631",
      "0.417184243"
    ],
    [
      "0.635971865",
      "0.698407213",
      "0.973154906",
      "0.098135997"
    ],
    [
      "0.534894682",
      "0.842780906",
      "0.424345817",
      "0.189321559"
    ],
    [
      "0.738139444",
      "0.049076330",
      "0.742264724",
      "0.263156166"
    ],
    [
      "0.949719463",
      "0.706566788",
      "0.278223162",
      "0.960964516"
    ],
    [
      "0.694463929",
      "0.383944242",
      "0.326534843",
      "0.326268999"
    ],
    [
      "0.838519936",
      "0.564644908",
      "0.841363579",
      "0.164437719"
    ],
    [
      "0.412531692",
      "0.460025602",
      "0.498810109",
      "0.021568493"
    ],
    [
      "0.930791419",
      "0.432039526",
      "0.885356122",
      "0.617480463"
    ],
    [
      "0.861411576",
      "0.879944772",
      "0.183770128",
      "0.878836521"
    ],
    [
      "0.495896095",
      "0.499513275",
      "0.210334516",
      "0.452499118"
    ],
    [
      "0.238418885",
      "0.340066173",
      "0.025307402",
      "0.134829687"
    ],
    [
      "0.893440212",
      "0.172631574",
      "0.565829770",
      "0.400028274"
    ],
    [
      "0.563240644",
      "0.307061491",
      "0.659828133",
      "0.671272045"
    ],
    [
      "0.464439110",
      "0.155233296",
      "0.938497412",
      "0.894903233"
    ],
    [
      "0.277018124",
      "0.021880704",
      "0.456756158",
      "0.711117509"
    ],
    [
      "0.138621329",
      "0.122318930",
      "0.254514624",
      "0.062249472"
    ],
    [
      "0.310035526",
      "0.744715479",
      "0.625349754",
      "0.294121683"
    ],
    [
      "0.349860728",
      "0.621576429",
      "0.116877506",
      "0.364844360"
    ],
    [
      "0.090587757",
      "0.412469843",
      "0.703063218",
      "0.932370118"
    ],
    [
      "0.661884168",
      "0.591038517",
      "0.592877047",
      "0.847238233"
    ],
    [
      "0.204636700",
      "0.805833835",
      "0.774415658",
      "0.515333367"
    ],
    [
      "0.972269849",
      "0.964801569",
      "0.690762856",
      "0.089559214"
    ],
    [
      "0.443932732",
      "0.935995430",
      "0.296934243",
      "0.638402925"
    ],
    [
      "0.375561480",
      "0.246900290",
      "0.810833060",
      "0.775034355"
    ]
  ]
}

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

DISCUSSION

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