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 = 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 = 8, 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	0.029755001064	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最差

x₅ × x₈

x₆ × x₇

x₆ × x₈

x₇ × x₈

已验证构造最差：第 5、7 列 · L2 星偏差 0.029755001064

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
0.029773597321→0.029755001064
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.029825507719→0.029773597321
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.029825507725→0.029825507719
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.029825507736→0.029825507725
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.034288686092→0.029825507736
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.034323725134→0.034288686092
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.034342220342→0.034323725134
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.035486094079→0.034342220342
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.070774366079→0.035486094079
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": [
    [
      "0.052209963",
      "0.115613742",
      "0.186349983",
      "0.052626274",
      "0.172403794",
      "0.229665051",
      "0.337746122",
      "0.242667182"
    ],
    [
      "0.128706636",
      "0.645231112",
      "0.693632705",
      "0.465544731",
      "0.700900766",
      "0.422579391",
      "0.254844964",
      "0.457613927"
    ],
    [
      "0.192704123",
      "0.577532718",
      "0.041871488",
      "0.771944847",
      "0.436516528",
      "0.708996352",
      "0.602119883",
      "0.387267957"
    ],
    [
      "0.252655794",
      "0.428017794",
      "0.763277488",
      "0.614409934",
      "0.318060166",
      "0.044120448",
      "0.873941132",
      "0.805212506"
    ],
    [
      "0.298216828",
      "0.932709793",
      "0.932281262",
      "0.950805361",
      "0.932735119",
      "0.932630492",
      "0.156496585",
      "0.947999278"
    ],
    [
      "0.331300146",
      "0.301021713",
      "0.564094724",
      "0.171895829",
      "0.633409530",
      "0.780872302",
      "0.801347922",
      "0.112982936"
    ],
    [
      "0.403370975",
      "0.774430431",
      "0.366933385",
      "0.306838064",
      "0.043188262",
      "0.314422411",
      "0.678583391",
      "0.583391978"
    ],
    [
      "0.473667675",
      "0.041884726",
      "0.437728339",
      "0.401056378",
      "0.566484499",
      "0.574600878",
      "0.467984441",
      "0.742455058"
    ],
    [
      "0.546218884",
      "0.501036509",
      "0.293401634",
      "0.543177819",
      "0.258335750",
      "0.501150016",
      "0.047185595",
      "0.046186992"
    ],
    [
      "0.619967502",
      "0.180799090",
      "0.828275836",
      "0.904817754",
      "0.849931860",
      "0.366695217",
      "0.742693643",
      "0.331973656"
    ],
    [
      "0.680483477",
      "0.369904096",
      "0.095548666",
      "0.235278958",
      "0.780459093",
      "0.145428174",
      "0.114169790",
      "0.663234578"
    ],
    [
      "0.749169217",
      "0.702734402",
      "0.863358461",
      "0.115055401",
      "0.375050138",
      "0.644205923",
      "0.537897079",
      "0.873074819"
    ],
    [
      "0.810882122",
      "0.838867619",
      "0.495238311",
      "0.838227174",
      "0.507018414",
      "0.093750153",
      "0.414054546",
      "0.174476680"
    ],
    [
      "0.853483431",
      "0.889359302",
      "0.235868827",
      "0.658716960",
      "0.886792297",
      "0.888967549",
      "0.923286173",
      "0.931338945"
    ],
    [
      "0.901732573",
      "0.230221121",
      "0.631903644",
      "0.702159929",
      "0.097664479",
      "0.836074044",
      "0.193458566",
      "0.525096582"
    ],
    [
      "0.958090759",
      "0.956534552",
      "0.956433478",
      "0.357962449",
      "0.960286113",
      "0.963299035",
      "0.959015035",
      "0.313445006"
    ]
  ]
}

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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.052209963",
      "0.115613742",
      "0.186349983",
      "0.052626274",
      "0.172403794",
      "0.229665051",
      "0.337746122",
      "0.242667182"
    ],
    [
      "0.128706636",
      "0.645231112",
      "0.693632705",
      "0.465544731",
      "0.700900766",
      "0.422579391",
      "0.254844964",
      "0.457613927"
    ],
    [
      "0.192704123",
      "0.577532718",
      "0.041871488",
      "0.771944847",
      "0.436516528",
      "0.708996352",
      "0.602119883",
      "0.387267957"
    ],
    [
      "0.252655794",
      "0.428017794",
      "0.763277488",
      "0.614409934",
      "0.318060166",
      "0.044120448",
      "0.873941132",
      "0.805212506"
    ],
    [
      "0.298216828",
      "0.932709793",
      "0.932281262",
      "0.950805361",
      "0.932735119",
      "0.932630492",
      "0.156496585",
      "0.947999278"
    ],
    [
      "0.331300146",
      "0.301021713",
      "0.564094724",
      "0.171895829",
      "0.633409530",
      "0.780872302",
      "0.801347922",
      "0.112982936"
    ],
    [
      "0.403370975",
      "0.774430431",
      "0.366933385",
      "0.306838064",
      "0.043188262",
      "0.314422411",
      "0.678583391",
      "0.583391978"
    ],
    [
      "0.473667675",
      "0.041884726",
      "0.437728339",
      "0.401056378",
      "0.566484499",
      "0.574600878",
      "0.467984441",
      "0.742455058"
    ],
    [
      "0.546218884",
      "0.501036509",
      "0.293401634",
      "0.543177819",
      "0.258335750",
      "0.501150016",
      "0.047185595",
      "0.046186992"
    ],
    [
      "0.619967502",
      "0.180799090",
      "0.828275836",
      "0.904817754",
      "0.849931860",
      "0.366695217",
      "0.742693643",
      "0.331973656"
    ],
    [
      "0.680483477",
      "0.369904096",
      "0.095548666",
      "0.235278958",
      "0.780459093",
      "0.145428174",
      "0.114169790",
      "0.663234578"
    ],
    [
      "0.749169217",
      "0.702734402",
      "0.863358461",
      "0.115055401",
      "0.375050138",
      "0.644205923",
      "0.537897079",
      "0.873074819"
    ],
    [
      "0.810882122",
      "0.838867619",
      "0.495238311",
      "0.838227174",
      "0.507018414",
      "0.093750153",
      "0.414054546",
      "0.174476680"
    ],
    [
      "0.853483431",
      "0.889359302",
      "0.235868827",
      "0.658716960",
      "0.886792297",
      "0.888967549",
      "0.923286173",
      "0.931338945"
    ],
    [
      "0.901732573",
      "0.230221121",
      "0.631903644",
      "0.702159929",
      "0.097664479",
      "0.836074044",
      "0.193458566",
      "0.525096582"
    ],
    [
      "0.958090759",
      "0.956534552",
      "0.956433478",
      "0.357962449",
      "0.960286113",
      "0.963299035",
      "0.959015035",
      "0.313445006"
    ]
  ]
}

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。