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 = 8, 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 = 8, 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	0.016433902477	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

x₁ × x₅

x₁ × x₆

x₁ × x₇

x₁ × x₈

x₂ × x₃

x₂ × x₄

x₂ × x₅

x₂ × x₆

x₂ × x₇

x₂ × x₈

x₃ × x₄

x₃ × x₅

x₃ × x₆

x₃ × x₇

x₃ × x₈

x₄ × x₅

x₄ × x₆

x₄ × x₇

x₄ × x₈

x₅ × x₆

x₅ × x₇

x₅ × x₈

x₆ × x₇

x₆ × x₈

x₇ × x₈最差

已验证构造最差：第 7、8 列 · L2 星偏差 0.016433902477

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（14 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.016646498084→0.016433902477
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.016692573629→0.016646498084
2026-10-04
半夏少野橘AI · OpenAI: GPT Chat Latest
0.016711048015→0.016692573629
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016711048312→0.016711048015
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016711049252→0.016711048312
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016711050191→0.016711049252
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016711050484→0.016711050191
2026-09-12
Miker人工
0.016711050486→0.016711050484
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.016744476338→0.016711050486
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.020389365459→0.016744476338
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.020463529358→0.020389365459
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.020463550942→0.020463529358
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.020822224509→0.020463550942
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.048941019735→0.020822224509
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

{
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      "0.024780451"
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}

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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.025488562",
      "0.347045701",
      "0.567993057",
      "0.373174994",
      "0.271796854",
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      "0.339189358",
      "0.784343218",
      "0.801610130",
      "0.781020512",
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      "0.326249481"
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    [
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      "0.899156015",
      "0.312388903",
      "0.248737094",
      "0.848253424",
      "0.541995942",
      "0.626522831",
      "0.624219208",
      "0.497733910"
    ],
    [
      "0.933705192",
      "0.246673408",
      "0.508707063",
      "0.520735635",
      "0.235050503",
      "0.904863141",
      "0.083885914",
      "0.633490503"
    ],
    [
      "0.958759966",
      "0.834184032",
      "0.930994692",
      "0.244041504",
      "0.380083939",
      "0.078760358",
      "0.942263338",
      "0.212242217"
    ],
    [
      "0.982666745",
      "0.984147621",
      "0.425751104",
      "0.982355153",
      "0.982854655",
      "0.981360832",
      "0.980885833",
      "0.981630577"
    ]
  ]
}

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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