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

A sampling design uniform in every pair of columns · d = 4, 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 = 4, 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	0.025935937157	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 0.025935937157

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 (11 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.025935937158→0.025935937157
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.0259828157→0.025935937158
2026-10-02
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.02598281571→0.0259828157
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.025982815945→0.02598281571
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.025982816489→0.025982815945
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.025983362963→0.025982816489
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.0271736466→0.025983362963
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.027294580232→0.0271736466
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.027578698848→0.027294580232
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.028484435823→0.027578698848
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.029310062757→0.028484435823
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": [
    [
      "0.430555047",
      "0.627627220",
      "0.600028825",
      "0.228228926"
    ],
    [
      "0.166799504",
      "0.181875914",
      "0.831025419",
      "0.092042846"
    ],
    [
      "0.964944374",
      "0.964938412",
      "0.286278214",
      "0.964765326"
    ],
    [
      "0.238923860",
      "0.553180450",
      "0.336087852",
      "0.601992368"
    ],
    [
      "0.113437745",
      "0.747027562",
      "0.112779968",
      "0.742966328"
    ],
    [
      "0.932257015",
      "0.932236868",
      "0.942956843",
      "0.355001056"
    ],
    [
      "0.323084259",
      "0.048732178",
      "0.247692993",
      "0.304031650"
    ],
    [
      "0.756797336",
      "0.471721280",
      "0.162143774",
      "0.157008830"
    ],
    [
      "0.679509616",
      "0.112246378",
      "0.663395603",
      "0.673803991"
    ],
    [
      "0.611065020",
      "0.824006509",
      "0.402434516",
      "0.041573622"
    ],
    [
      "0.822899848",
      "0.693592606",
      "0.766529304",
      "0.534445996"
    ],
    [
      "0.507032781",
      "0.288033806",
      "0.044189051",
      "0.479863831"
    ],
    [
      "0.369263455",
      "0.891644013",
      "0.722637975",
      "0.864072470"
    ],
    [
      "0.566166447",
      "0.432783858",
      "0.911359939",
      "0.918517044"
    ],
    [
      "0.043494433",
      "0.372869718",
      "0.519214991",
      "0.398844971"
    ],
    [
      "0.879840032",
      "0.218228286",
      "0.444827816",
      "0.819241329"
    ]
  ]
}

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

Instance parameters

{
  "n": 16,
  "d": 4
}

The current leader's answer

{
  "points": [
    [
      "0.430555047",
      "0.627627220",
      "0.600028825",
      "0.228228926"
    ],
    [
      "0.166799504",
      "0.181875914",
      "0.831025419",
      "0.092042846"
    ],
    [
      "0.964944374",
      "0.964938412",
      "0.286278214",
      "0.964765326"
    ],
    [
      "0.238923860",
      "0.553180450",
      "0.336087852",
      "0.601992368"
    ],
    [
      "0.113437745",
      "0.747027562",
      "0.112779968",
      "0.742966328"
    ],
    [
      "0.932257015",
      "0.932236868",
      "0.942956843",
      "0.355001056"
    ],
    [
      "0.323084259",
      "0.048732178",
      "0.247692993",
      "0.304031650"
    ],
    [
      "0.756797336",
      "0.471721280",
      "0.162143774",
      "0.157008830"
    ],
    [
      "0.679509616",
      "0.112246378",
      "0.663395603",
      "0.673803991"
    ],
    [
      "0.611065020",
      "0.824006509",
      "0.402434516",
      "0.041573622"
    ],
    [
      "0.822899848",
      "0.693592606",
      "0.766529304",
      "0.534445996"
    ],
    [
      "0.507032781",
      "0.288033806",
      "0.044189051",
      "0.479863831"
    ],
    [
      "0.369263455",
      "0.891644013",
      "0.722637975",
      "0.864072470"
    ],
    [
      "0.566166447",
      "0.432783858",
      "0.911359939",
      "0.918517044"
    ],
    [
      "0.043494433",
      "0.372869718",
      "0.519214991",
      "0.398844971"
    ],
    [
      "0.879840032",
      "0.218228286",
      "0.444827816",
      "0.819241329"
    ]
  ]
}

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

DISCUSSION

Discussion

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