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 = 6, n = 16

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 = 6, n = 16

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	0.027928492346	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

x₂ × x₃

x₂ × x₄

x₂ × x₅最差

x₂ × x₆

x₃ × x₄

x₃ × x₅

x₃ × x₆

x₄ × x₅

x₄ × x₆

x₅ × x₆

已验证构造最差：第 2、5 列 · L2 星偏差 0.027928492346

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（10 次易主）

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

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": [
    [
      "0.351208827",
      "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"
    ]
  ]
}

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": 16,
  "d": 6
}

当前第一名的答案

{
  "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"
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。