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 = 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 = 8, 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	0.020993502457	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最差

x₃ × x₈

x₄ × x₅

x₄ × x₆

x₄ × x₇

x₄ × x₈

x₅ × x₆

x₅ × x₇

x₅ × x₈

x₆ × x₇

x₆ × x₈

x₇ × x₈

已验证构造最差：第 3、7 列 · L2 星偏差 0.020993502457

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
0.021024290694→0.020993502457
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.021110282136→0.021024290694
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.021110282228→0.021110282136
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.021110344552→0.021110282228
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.025309864367→0.021110344552
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.025349563662→0.025309864367
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.02534958784→0.025349563662
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.025653739028→0.02534958784
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.049498863502→0.025653739028
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": [
    [
      "0.652639744",
      "0.976596496",
      "0.238759761",
      "0.977597143",
      "0.977889893",
      "0.973885482",
      "0.972801059",
      "0.974081393"
    ],
    [
      "0.388751462",
      "0.936734235",
      "0.906955090",
      "0.922338349",
      "0.594895963",
      "0.270548061",
      "0.149992887",
      "0.852207832"
    ],
    [
      "0.435894198",
      "0.033938680",
      "0.205208416",
      "0.162610733",
      "0.081101106",
      "0.120807622",
      "0.711600877",
      "0.079778947"
    ],
    [
      "0.327789483",
      "0.872973838",
      "0.965031313",
      "0.194941630",
      "0.926106481",
      "0.873035088",
      "0.925998601",
      "0.943724539"
    ],
    [
      "0.215659093",
      "0.798922074",
      "0.342260778",
      "0.712756684",
      "0.377836393",
      "0.196449380",
      "0.887224412",
      "0.574266873"
    ],
    [
      "0.852627933",
      "0.203051678",
      "0.936716048",
      "0.943575240",
      "0.944392704",
      "0.622625469",
      "0.943759143",
      "0.328749983"
    ],
    [
      "0.822565497",
      "0.759786682",
      "0.160334570",
      "0.116754622",
      "0.629372104",
      "0.594799669",
      "0.413787565",
      "0.726042853"
    ],
    [
      "0.080748326",
      "0.614969949",
      "0.070140316",
      "0.622942643",
      "0.670152820",
      "0.488165191",
      "0.767465847",
      "0.205202095"
    ],
    [
      "0.167209971",
      "0.564055880",
      "0.293632040",
      "0.492623267",
      "0.030914583",
      "0.755001567",
      "0.272056661",
      "0.491122204"
    ],
    [
      "0.780208868",
      "0.845447953",
      "0.840361265",
      "0.753810670",
      "0.127123631",
      "0.406687173",
      "0.537748020",
      "0.170096865"
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    [
      "0.123808136",
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      "0.795215031",
      "0.323538399",
      "0.763258394",
      "0.071945275",
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      "0.677645059"
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    [
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      "0.871736236",
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      "0.886678894"
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      "0.844225223",
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    [
      "0.731205057",
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      "0.030629461",
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      "0.198257725",
      "0.312046355",
      "0.222461289",
      "0.624308104"
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    [
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      "0.410734526",
      "0.403935676",
      "0.465558117",
      "0.443994551",
      "0.584315703",
      "0.812053918"
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    [
      "0.583971806",
      "0.711861582",
      "0.516580317",
      "0.547390921",
      "0.329206688",
      "0.029608091",
      "0.369687174",
      "0.295460333"
    ],
    [
      "0.490682133",
      "0.661442670",
      "0.747488252",
      "0.230309000",
      "0.418770838",
      "0.706680689",
      "0.030611251",
      "0.127901961"
    ],
    [
      "0.681659851",
      "0.515369016",
      "0.696891474",
      "0.080007870",
      "0.558163139",
      "0.237328862",
      "0.811661514",
      "0.447245054"
    ],
    [
      "0.033179997",
      "0.167120544",
      "0.462074244",
      "0.034256041",
      "0.256130663",
      "0.542297799",
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      "0.393744834"
    ],
    [
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    [
      "0.952917383",
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      "0.783699837",
      "0.897604392",
      "0.161125638",
      "0.061950848",
      "0.918601640"
    ],
    [
      "0.256886627",
      "0.398182492",
      "0.569487078",
      "0.446679630",
      "0.849247079",
      "0.354742731",
      "0.320129065",
      "0.030137364"
    ],
    [
      "0.356156693",
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      "0.121798739",
      "0.870876265",
      "0.514850846",
      "0.910170942",
      "0.456813229",
      "0.251871530"
    ]
  ]
}

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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.652639744",
      "0.976596496",
      "0.238759761",
      "0.977597143",
      "0.977889893",
      "0.973885482",
      "0.972801059",
      "0.974081393"
    ],
    [
      "0.388751462",
      "0.936734235",
      "0.906955090",
      "0.922338349",
      "0.594895963",
      "0.270548061",
      "0.149992887",
      "0.852207832"
    ],
    [
      "0.435894198",
      "0.033938680",
      "0.205208416",
      "0.162610733",
      "0.081101106",
      "0.120807622",
      "0.711600877",
      "0.079778947"
    ],
    [
      "0.327789483",
      "0.872973838",
      "0.965031313",
      "0.194941630",
      "0.926106481",
      "0.873035088",
      "0.925998601",
      "0.943724539"
    ],
    [
      "0.215659093",
      "0.798922074",
      "0.342260778",
      "0.712756684",
      "0.377836393",
      "0.196449380",
      "0.887224412",
      "0.574266873"
    ],
    [
      "0.852627933",
      "0.203051678",
      "0.936716048",
      "0.943575240",
      "0.944392704",
      "0.622625469",
      "0.943759143",
      "0.328749983"
    ],
    [
      "0.822565497",
      "0.759786682",
      "0.160334570",
      "0.116754622",
      "0.629372104",
      "0.594799669",
      "0.413787565",
      "0.726042853"
    ],
    [
      "0.080748326",
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      "0.070140316",
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      "0.488165191",
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      "0.205202095"
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    [
      "0.167209971",
      "0.564055880",
      "0.293632040",
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      "0.755001567",
      "0.272056661",
      "0.491122204"
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    [
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      "0.753810670",
      "0.127123631",
      "0.406687173",
      "0.537748020",
      "0.170096865"
    ],
    [
      "0.123808136",
      "0.238034383",
      "0.795215031",
      "0.323538399",
      "0.763258394",
      "0.071945275",
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      "0.677645059"
    ],
    [
      "0.891098271",
      "0.363526595",
      "0.871736236",
      "0.584175143",
      "0.300459885",
      "0.949326934",
      "0.852971776",
      "0.886678894"
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      "0.918823174",
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      "0.359948858",
      "0.803499550",
      "0.807098934",
      "0.628818764",
      "0.349401730"
    ],
    [
      "0.629347736",
      "0.070799497",
      "0.658229419",
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      "0.715795399",
      "0.844225223",
      "0.177956001",
      "0.539874912"
    ],
    [
      "0.731205057",
      "0.325059586",
      "0.030629461",
      "0.278661287",
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      "0.624308104"
    ],
    [
      "0.533659004",
      "0.117648742",
      "0.410734526",
      "0.403935676",
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      "0.443994551",
      "0.584315703",
      "0.812053918"
    ],
    [
      "0.583971806",
      "0.711861582",
      "0.516580317",
      "0.547390921",
      "0.329206688",
      "0.029608091",
      "0.369687174",
      "0.295460333"
    ],
    [
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      "0.661442670",
      "0.747488252",
      "0.230309000",
      "0.418770838",
      "0.706680689",
      "0.030611251",
      "0.127901961"
    ],
    [
      "0.681659851",
      "0.515369016",
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      "0.080007870",
      "0.558163139",
      "0.237328862",
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      "0.447245054"
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    [
      "0.033179997",
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      "0.462074244",
      "0.034256041",
      "0.256130663",
      "0.542297799",
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    [
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      "0.661778601",
      "0.661714638",
      "0.762762110"
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    [
      "0.952917383",
      "0.482925320",
      "0.252434450",
      "0.783699837",
      "0.897604392",
      "0.161125638",
      "0.061950848",
      "0.918601640"
    ],
    [
      "0.256886627",
      "0.398182492",
      "0.569487078",
      "0.446679630",
      "0.849247079",
      "0.354742731",
      "0.320129065",
      "0.030137364"
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    [
      "0.356156693",
      "0.279745911",
      "0.121798739",
      "0.870876265",
      "0.514850846",
      "0.910170942",
      "0.456813229",
      "0.251871530"
    ]
  ]
}

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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