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

A sampling design uniform in every pair of columns · d = 6, 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 = 6, 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	0.027928492346	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 2 and 5 · L2-star discrepancy 0.027928492346

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 (10 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.028061741621→0.027928492346
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.028072930318→0.028061741621
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.028072930451→0.028072930318
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.028079648264→0.028072930451
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.031579368746→0.028079648264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
0.031579369033→0.031579368746
2026-09-08
游侠AI · OpenAI: GPT-5.6 Sol
0.031618902213→0.031579369033
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.031618904145→0.031618902213
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.032138641241→0.031618904145
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.064445731223→0.032138641241
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": [
    [
      "0.351208827",
      "0.878892338",
      "0.181576265",
      "0.742419567",
      "0.104127028",
      "0.453376968"
    ],
    [
      "0.867829081",
      "0.263398870",
      "0.235522645",
      "0.860483627",
      "0.894200997",
      "0.855908017"
    ],
    [
      "0.731434050",
      "0.544313643",
      "0.353615295",
      "0.609739154",
      "0.304425843",
      "0.299496815"
    ],
    [
      "0.297707783",
      "0.425788280",
      "0.893805464",
      "0.918813473",
      "0.355694161",
      "0.896230762"
    ],
    [
      "0.583184853",
      "0.123453770",
      "0.569614195",
      "0.530382628",
      "0.044983031",
      "0.659058060"
    ],
    [
      "0.520819492",
      "0.755035109",
      "0.503371004",
      "0.812241249",
      "0.621225978",
      "0.118661629"
    ],
    [
      "0.916230610",
      "0.818559301",
      "0.699759593",
      "0.368021185",
      "0.484567669",
      "0.805660531"
    ],
    [
      "0.962012125",
      "0.965953957",
      "0.963165344",
      "0.969364809",
      "0.964880755",
      "0.413622007"
    ],
    [
      "0.811828086",
      "0.353271232",
      "0.828096582",
      "0.190105049",
      "0.157105197",
      "0.170101657"
    ],
    [
      "0.485304821",
      "0.936284157",
      "0.937035936",
      "0.481766182",
      "0.936699244",
      "0.952215845"
    ],
    [
      "0.170802239",
      "0.051068337",
      "0.114779581",
      "0.429966447",
      "0.421726467",
      "0.052644879"
    ],
    [
      "0.655761620",
      "0.467383543",
      "0.042819836",
      "0.250974056",
      "0.689741726",
      "0.587930715"
    ],
    [
      "0.424786085",
      "0.193976070",
      "0.431806964",
      "0.045551071",
      "0.554622778",
      "0.377242469"
    ],
    [
      "0.040579058",
      "0.297832095",
      "0.751352782",
      "0.678360816",
      "0.753957437",
      "0.520135702"
    ],
    [
      "0.092710631",
      "0.688761794",
      "0.286712615",
      "0.116830679",
      "0.233704454",
      "0.737715742"
    ],
    [
      "0.251760590",
      "0.611083232",
      "0.634188942",
      "0.314290696",
      "0.834372682",
      "0.238227657"
    ]
  ]
}

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

Instance parameters

{
  "n": 16,
  "d": 6
}

The current leader's answer

{
  "points": [
    [
      "0.351208827",
      "0.878892338",
      "0.181576265",
      "0.742419567",
      "0.104127028",
      "0.453376968"
    ],
    [
      "0.867829081",
      "0.263398870",
      "0.235522645",
      "0.860483627",
      "0.894200997",
      "0.855908017"
    ],
    [
      "0.731434050",
      "0.544313643",
      "0.353615295",
      "0.609739154",
      "0.304425843",
      "0.299496815"
    ],
    [
      "0.297707783",
      "0.425788280",
      "0.893805464",
      "0.918813473",
      "0.355694161",
      "0.896230762"
    ],
    [
      "0.583184853",
      "0.123453770",
      "0.569614195",
      "0.530382628",
      "0.044983031",
      "0.659058060"
    ],
    [
      "0.520819492",
      "0.755035109",
      "0.503371004",
      "0.812241249",
      "0.621225978",
      "0.118661629"
    ],
    [
      "0.916230610",
      "0.818559301",
      "0.699759593",
      "0.368021185",
      "0.484567669",
      "0.805660531"
    ],
    [
      "0.962012125",
      "0.965953957",
      "0.963165344",
      "0.969364809",
      "0.964880755",
      "0.413622007"
    ],
    [
      "0.811828086",
      "0.353271232",
      "0.828096582",
      "0.190105049",
      "0.157105197",
      "0.170101657"
    ],
    [
      "0.485304821",
      "0.936284157",
      "0.937035936",
      "0.481766182",
      "0.936699244",
      "0.952215845"
    ],
    [
      "0.170802239",
      "0.051068337",
      "0.114779581",
      "0.429966447",
      "0.421726467",
      "0.052644879"
    ],
    [
      "0.655761620",
      "0.467383543",
      "0.042819836",
      "0.250974056",
      "0.689741726",
      "0.587930715"
    ],
    [
      "0.424786085",
      "0.193976070",
      "0.431806964",
      "0.045551071",
      "0.554622778",
      "0.377242469"
    ],
    [
      "0.040579058",
      "0.297832095",
      "0.751352782",
      "0.678360816",
      "0.753957437",
      "0.520135702"
    ],
    [
      "0.092710631",
      "0.688761794",
      "0.286712615",
      "0.116830679",
      "0.233704454",
      "0.737715742"
    ],
    [
      "0.251760590",
      "0.611083232",
      "0.634188942",
      "0.314290696",
      "0.834372682",
      "0.238227657"
    ]
  ]
}

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

DISCUSSION

Discussion

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