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 = 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 = 6, 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	0.015173875807	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

x₂ × x₆

x₃ × x₄

x₃ × x₅最差

x₃ × x₆

x₄ × x₅

x₄ × x₆

x₅ × x₆

已验证构造最差：第 3、5 列 · L2 星偏差 0.015173875807

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（11 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.015283955259→0.015173875807
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.015297720781→0.015283955259
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.015297720785→0.015297720781
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.015297720801→0.015297720785
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.015297720851→0.015297720801
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.01529858852→0.015297720851
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.01857214541→0.01529858852
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.018650389162→0.01857214541
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.018656632636→0.018650389162
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.019098818459→0.018656632636
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.028722229844→0.019098818459
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": [
    [
      "0.963208881",
      "0.115531135",
      "0.963851281",
      "0.969301493",
      "0.965242247",
      "0.844618432"
    ],
    [
      "0.460029898",
      "0.195578511",
      "0.279728987",
      "0.458907182",
      "0.211309164",
      "0.338171339"
    ],
    [
      "0.763639736",
      "0.747227791",
      "0.434109851",
      "0.498098665",
      "0.724632682",
      "0.238401763"
    ],
    [
      "0.570542361",
      "0.559668242",
      "0.528952522",
      "0.611176536",
      "0.307151901",
      "0.020812721"
    ],
    [
      "0.947164426",
      "0.956519060",
      "0.365356952",
      "0.948838991",
      "0.246202443",
      "0.587148303"
    ],
    [
      "0.156876526",
      "0.389128131",
      "0.394890588",
      "0.023538988",
      "0.626288972",
      "0.169857076"
    ],
    [
      "0.215064444",
      "0.654511503",
      "0.461245246",
      "0.875721922",
      "0.943262484",
      "0.537960081"
    ],
    [
      "0.427059483",
      "0.881656092",
      "0.563060921",
      "0.108203792",
      "0.463081288",
      "0.757382461"
    ],
    [
      "0.246752622",
      "0.485756793",
      "0.315341468",
      "0.301651298",
      "0.405612254",
      "0.872085858"
    ],
    [
      "0.665134142",
      "0.427000014",
      "0.849794175",
      "0.425949690",
      "0.662540466",
      "0.616860466"
    ],
    [
      "0.118039568",
      "0.024168786",
      "0.216424459",
      "0.649705978",
      "0.497256744",
      "0.481678019"
    ],
    [
      "0.867079751",
      "0.682989002",
      "0.938200052",
      "0.330052303",
      "0.369776987",
      "0.513516462"
    ],
    [
      "0.494901218",
      "0.622448083",
      "0.755802549",
      "0.686842961",
      "0.533443384",
      "0.301393557"
    ],
    [
      "0.188162551",
      "0.775593343",
      "0.719723461",
      "0.394713690",
      "0.177697030",
      "0.821300652"
    ],
    [
      "0.346930790",
      "0.351073087",
      "0.616207525",
      "0.570769793",
      "0.021229179",
      "0.441263692"
    ],
    [
      "0.370156548",
      "0.940305104",
      "0.924932348",
      "0.934454812",
      "0.887087586",
      "0.936178364"
    ],
    [
      "0.891717165",
      "0.134176305",
      "0.585716706",
      "0.776024129",
      "0.594919060",
      "0.956107134"
    ],
    [
      "0.828826351",
      "0.454842093",
      "0.085361688",
      "0.841218335",
      "0.146803715",
      "0.270713401"
    ],
    [
      "0.390503643",
      "0.096700260",
      "0.119923196",
      "0.364618563",
      "0.764989466",
      "0.055300915"
    ],
    [
      "0.312134999",
      "0.712050402",
      "0.154696223",
      "0.803585814",
      "0.691984707",
      "0.644286289"
    ],
    [
      "0.602642079",
      "0.321361578",
      "0.346985960",
      "0.716272413",
      "0.836786972",
      "0.791439918"
    ],
    [
      "0.977072108",
      "0.976793823",
      "0.976267740",
      "0.553495711",
      "0.973973235",
      "0.977683357"
    ],
    [
      "0.798620641",
      "0.228521322",
      "0.783879814",
      "0.084408355",
      "0.869526440",
      "0.567234586"
    ],
    [
      "0.730118518",
      "0.598197264",
      "0.180608389",
      "0.057895088",
      "0.054562816",
      "0.912121303"
    ],
    [
      "0.915241623",
      "0.914330486",
      "0.245451779",
      "0.275308772",
      "0.915865885",
      "0.115254680"
    ],
    [
      "0.021731834",
      "0.523361061",
      "0.653194474",
      "0.210506757",
      "0.803388117",
      "0.372221108"
    ],
    [
      "0.053043906",
      "0.260320264",
      "0.050467958",
      "0.529503284",
      "0.341040399",
      "0.723165687"
    ],
    [
      "0.534508711",
      "0.057645645",
      "0.488761412",
      "0.242140307",
      "0.114330469",
      "0.677311866"
    ],
    [
      "0.085277090",
      "0.848001193",
      "0.822899230",
      "0.750738403",
      "0.081150438",
      "0.088901373"
    ],
    [
      "0.280531304",
      "0.152463088",
      "0.890436763",
      "0.142173485",
      "0.265508915",
      "0.212082852"
    ],
    [
      "0.631651525",
      "0.812197014",
      "0.022023998",
      "0.175277494",
      "0.564511383",
      "0.410741710"
    ],
    [
      "0.699751617",
      "0.285942637",
      "0.692121629",
      "0.904001918",
      "0.431684170",
      "0.132979539"
    ]
  ]
}

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": 6
}

当前第一名的答案

{
  "points": [
    [
      "0.963208881",
      "0.115531135",
      "0.963851281",
      "0.969301493",
      "0.965242247",
      "0.844618432"
    ],
    [
      "0.460029898",
      "0.195578511",
      "0.279728987",
      "0.458907182",
      "0.211309164",
      "0.338171339"
    ],
    [
      "0.763639736",
      "0.747227791",
      "0.434109851",
      "0.498098665",
      "0.724632682",
      "0.238401763"
    ],
    [
      "0.570542361",
      "0.559668242",
      "0.528952522",
      "0.611176536",
      "0.307151901",
      "0.020812721"
    ],
    [
      "0.947164426",
      "0.956519060",
      "0.365356952",
      "0.948838991",
      "0.246202443",
      "0.587148303"
    ],
    [
      "0.156876526",
      "0.389128131",
      "0.394890588",
      "0.023538988",
      "0.626288972",
      "0.169857076"
    ],
    [
      "0.215064444",
      "0.654511503",
      "0.461245246",
      "0.875721922",
      "0.943262484",
      "0.537960081"
    ],
    [
      "0.427059483",
      "0.881656092",
      "0.563060921",
      "0.108203792",
      "0.463081288",
      "0.757382461"
    ],
    [
      "0.246752622",
      "0.485756793",
      "0.315341468",
      "0.301651298",
      "0.405612254",
      "0.872085858"
    ],
    [
      "0.665134142",
      "0.427000014",
      "0.849794175",
      "0.425949690",
      "0.662540466",
      "0.616860466"
    ],
    [
      "0.118039568",
      "0.024168786",
      "0.216424459",
      "0.649705978",
      "0.497256744",
      "0.481678019"
    ],
    [
      "0.867079751",
      "0.682989002",
      "0.938200052",
      "0.330052303",
      "0.369776987",
      "0.513516462"
    ],
    [
      "0.494901218",
      "0.622448083",
      "0.755802549",
      "0.686842961",
      "0.533443384",
      "0.301393557"
    ],
    [
      "0.188162551",
      "0.775593343",
      "0.719723461",
      "0.394713690",
      "0.177697030",
      "0.821300652"
    ],
    [
      "0.346930790",
      "0.351073087",
      "0.616207525",
      "0.570769793",
      "0.021229179",
      "0.441263692"
    ],
    [
      "0.370156548",
      "0.940305104",
      "0.924932348",
      "0.934454812",
      "0.887087586",
      "0.936178364"
    ],
    [
      "0.891717165",
      "0.134176305",
      "0.585716706",
      "0.776024129",
      "0.594919060",
      "0.956107134"
    ],
    [
      "0.828826351",
      "0.454842093",
      "0.085361688",
      "0.841218335",
      "0.146803715",
      "0.270713401"
    ],
    [
      "0.390503643",
      "0.096700260",
      "0.119923196",
      "0.364618563",
      "0.764989466",
      "0.055300915"
    ],
    [
      "0.312134999",
      "0.712050402",
      "0.154696223",
      "0.803585814",
      "0.691984707",
      "0.644286289"
    ],
    [
      "0.602642079",
      "0.321361578",
      "0.346985960",
      "0.716272413",
      "0.836786972",
      "0.791439918"
    ],
    [
      "0.977072108",
      "0.976793823",
      "0.976267740",
      "0.553495711",
      "0.973973235",
      "0.977683357"
    ],
    [
      "0.798620641",
      "0.228521322",
      "0.783879814",
      "0.084408355",
      "0.869526440",
      "0.567234586"
    ],
    [
      "0.730118518",
      "0.598197264",
      "0.180608389",
      "0.057895088",
      "0.054562816",
      "0.912121303"
    ],
    [
      "0.915241623",
      "0.914330486",
      "0.245451779",
      "0.275308772",
      "0.915865885",
      "0.115254680"
    ],
    [
      "0.021731834",
      "0.523361061",
      "0.653194474",
      "0.210506757",
      "0.803388117",
      "0.372221108"
    ],
    [
      "0.053043906",
      "0.260320264",
      "0.050467958",
      "0.529503284",
      "0.341040399",
      "0.723165687"
    ],
    [
      "0.534508711",
      "0.057645645",
      "0.488761412",
      "0.242140307",
      "0.114330469",
      "0.677311866"
    ],
    [
      "0.085277090",
      "0.848001193",
      "0.822899230",
      "0.750738403",
      "0.081150438",
      "0.088901373"
    ],
    [
      "0.280531304",
      "0.152463088",
      "0.890436763",
      "0.142173485",
      "0.265508915",
      "0.212082852"
    ],
    [
      "0.631651525",
      "0.812197014",
      "0.022023998",
      "0.175277494",
      "0.564511383",
      "0.410741710"
    ],
    [
      "0.699751617",
      "0.285942637",
      "0.692121629",
      "0.904001918",
      "0.431684170",
      "0.132979539"
    ]
  ]
}

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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