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

A sampling design uniform in every pair of columns · d = 4, 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 = 4, 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	0.017943197477	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 0.017943197477

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 (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.017966653631→0.017943197477
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.017988750894→0.017966653631
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.017988752052→0.017988750894
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.017988764514→0.017988752052
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.020881772942→0.017988764514
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.020938056423→0.020881772942
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.021197647336→0.020938056423
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": [
    [
      "0.111773255",
      "0.458628084",
      "0.834391859",
      "0.734229758"
    ],
    [
      "0.738576191",
      "0.742541198",
      "0.734176699",
      "0.071882901"
    ],
    [
      "0.413306640",
      "0.704000723",
      "0.542212656",
      "0.904882533"
    ],
    [
      "0.541717107",
      "0.837881418",
      "0.409603709",
      "0.769324989"
    ],
    [
      "0.694518441",
      "0.201162891",
      "0.201910282",
      "0.683246099"
    ],
    [
      "0.375647084",
      "0.301489070",
      "0.908855342",
      "0.623412219"
    ],
    [
      "0.205701963",
      "0.585005690",
      "0.700583711",
      "0.856817302"
    ],
    [
      "0.589187028",
      "0.546158942",
      "0.246833360",
      "0.235371008"
    ],
    [
      "0.782021246",
      "0.496279230",
      "0.445802051",
      "0.532942235"
    ],
    [
      "0.336783324",
      "0.656782283",
      "0.156640390",
      "0.339226538"
    ],
    [
      "0.865236045",
      "0.869946651",
      "0.865154144",
      "0.200843767"
    ],
    [
      "0.642623101",
      "0.359756221",
      "0.652618005",
      "0.384909449"
    ],
    [
      "0.032481894",
      "0.157537912",
      "0.598790472",
      "0.288529854"
    ],
    [
      "0.614673462",
      "0.960780109",
      "0.970024612",
      "0.971227952"
    ],
    [
      "0.233237398",
      "0.913672749",
      "0.289573021",
      "0.444206381"
    ],
    [
      "0.283832305",
      "0.032361444",
      "0.492386769",
      "0.171102682"
    ],
    [
      "0.941655746",
      "0.616878306",
      "0.955200819",
      "0.428296437"
    ],
    [
      "0.509929850",
      "0.105440458",
      "0.782411193",
      "0.490642330"
    ],
    [
      "0.073517312",
      "0.786370717",
      "0.080140119",
      "0.574540599"
    ],
    [
      "0.157787186",
      "0.254678169",
      "0.366006180",
      "0.033392080"
    ],
    [
      "0.457193027",
      "0.414694588",
      "0.031522442",
      "0.114388691"
    ],
    [
      "0.831194789",
      "0.075317096",
      "0.108743330",
      "0.815237628"
    ],
    [
      "0.969311338",
      "0.969970218",
      "0.559760440",
      "0.640019718"
    ],
    [
      "0.907654629",
      "0.328585272",
      "0.316603308",
      "0.945009581"
    ]
  ]
}

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

Instance parameters

{
  "n": 24,
  "d": 4
}

The current leader's answer

{
  "points": [
    [
      "0.111773255",
      "0.458628084",
      "0.834391859",
      "0.734229758"
    ],
    [
      "0.738576191",
      "0.742541198",
      "0.734176699",
      "0.071882901"
    ],
    [
      "0.413306640",
      "0.704000723",
      "0.542212656",
      "0.904882533"
    ],
    [
      "0.541717107",
      "0.837881418",
      "0.409603709",
      "0.769324989"
    ],
    [
      "0.694518441",
      "0.201162891",
      "0.201910282",
      "0.683246099"
    ],
    [
      "0.375647084",
      "0.301489070",
      "0.908855342",
      "0.623412219"
    ],
    [
      "0.205701963",
      "0.585005690",
      "0.700583711",
      "0.856817302"
    ],
    [
      "0.589187028",
      "0.546158942",
      "0.246833360",
      "0.235371008"
    ],
    [
      "0.782021246",
      "0.496279230",
      "0.445802051",
      "0.532942235"
    ],
    [
      "0.336783324",
      "0.656782283",
      "0.156640390",
      "0.339226538"
    ],
    [
      "0.865236045",
      "0.869946651",
      "0.865154144",
      "0.200843767"
    ],
    [
      "0.642623101",
      "0.359756221",
      "0.652618005",
      "0.384909449"
    ],
    [
      "0.032481894",
      "0.157537912",
      "0.598790472",
      "0.288529854"
    ],
    [
      "0.614673462",
      "0.960780109",
      "0.970024612",
      "0.971227952"
    ],
    [
      "0.233237398",
      "0.913672749",
      "0.289573021",
      "0.444206381"
    ],
    [
      "0.283832305",
      "0.032361444",
      "0.492386769",
      "0.171102682"
    ],
    [
      "0.941655746",
      "0.616878306",
      "0.955200819",
      "0.428296437"
    ],
    [
      "0.509929850",
      "0.105440458",
      "0.782411193",
      "0.490642330"
    ],
    [
      "0.073517312",
      "0.786370717",
      "0.080140119",
      "0.574540599"
    ],
    [
      "0.157787186",
      "0.254678169",
      "0.366006180",
      "0.033392080"
    ],
    [
      "0.457193027",
      "0.414694588",
      "0.031522442",
      "0.114388691"
    ],
    [
      "0.831194789",
      "0.075317096",
      "0.108743330",
      "0.815237628"
    ],
    [
      "0.969311338",
      "0.969970218",
      "0.559760440",
      "0.640019718"
    ],
    [
      "0.907654629",
      "0.328585272",
      "0.316603308",
      "0.945009581"
    ]
  ]
}

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

DISCUSSION

Discussion

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