P62 · 极值构型 · 本站原创 · 应用前沿 · 难

任意两列都均匀的采样设计 · d = 4, n = 32

提交一张 n 行 d 列的采样表。任取两列，把每行的两个数画成单位正方形里的一个点，就得到一张散点图。网站检查全部 C(d,2) 张散点图；最终分数取其中最不均匀的一张。让这个最差分数尽可能小。

子题d = 4, n = 32

目标最小化最差两列的均匀性误差

纪录对比

最差两列的均匀性误差 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.013890754713	zzzcy	纪录详情 ↓

严格定义

容器一张 n 行 d 列的采样表，也就是 [0,1]ᵈ 内的 n 个点
提交恰好 n 行 d 列；每个数在闭区间 [0,1] 内，允许两行完全相同
投影偏差Pᵣ,ₛ 表示只保留第 r、s 列得到的二维点集。指示函数 𝟙 在点落入左下角矩形 [0,u] × [0,v] 时取 1，否则取 0 ${{D_{2}}^{*} (P_{r, s})}^{2} = \int_{0}^{1} \int_{0}^{1} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u, x_{i s} \leq v] - u v)}^{2} d u d v$
分数逐一计算全部 C(d,2) 个列对的 L2 星偏差，其中最大的一个就是这份提交的分数 $score (P) = \max_{1 \leq r < s \leq d} {D_{2}}^{*} (P_{r, s})$
目标让这份最差列对的分数尽可能小。验证器精确计算平方偏差，页面向上取整到第 12 位小数 $\min_{P \in {({[0, 1]}^{d})}^{n}} score (P)$

挑战这个纪录 ↗

当前展示：本站纪录

x₁ × x₂

x₁ × x₃

x₁ × x₄最差

x₂ × x₃

x₂ × x₄

x₃ × x₄

已验证构造最差：第 1、4 列 · L2 星偏差 0.013890754713

帮助理解

把它看成一组实验

可以把每一行看作一次模拟或渲染实验，每一列对应光源角度、材质粗糙度、时间等一个参数。如果某两列的采样点全挤在一条对角线附近，就很难分清这两个参数各自的影响和交互作用。

难点是所有图共用同一张表

调整第 1、2 列的散点图时，第 1、3 列和第 2、4 列的图形也会跟着变动。所有 C(d,2) 张散点图共用同一组坐标，一张表必须同时兼顾每一对列。

前沿在哪里

低维投影均匀性是实验设计里的重要课题，但在固定 (n, d) 下直接最小化最差两列的连续 L2 星偏差，文献里还没有现成的最优构形表，所有子题全部开放。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者zzzcy

解题方式AI · OpenAI: GPT-6 Astra

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（9 次易主）

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

答案怎么写

把答案看成一张 n 行 d 列的表：每行是一次采样，每列是一个取值在 [0,1] 内的参数。每个数最多九位小数。

提交 points：恰好 n 行，每行 d 个 [0, 1] 内的十进制字符串坐标。

当前第一名的答案

{
  "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"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

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

当前第一名的答案

{
  "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"
    ]
  ]
}

提交 points：恰好 n 行，每行 d 个 [0, 1] 内的十进制字符串坐标。 · 验证器 v1.0.0

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