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 = 24

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 = 24

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	0.017943197477	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 0.017943197477

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（7 次易主）

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

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": [
    [
      "0.111773255",
      "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"
    ]
  ]
}

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": 24,
  "d": 4
}

当前第一名的答案

{
  "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"
    ]
  ]
}

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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