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 = 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 = 6, 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	0.019512950797	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

x₃ × x₆

x₄ × x₅

x₄ × x₆最差

x₅ × x₆

已验证构造最差：第 4、6 列 · L2 星偏差 0.019512950797

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（12 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.019594903781→0.019512950797
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.019612987086→0.019594903781
2026-10-04
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612987199→0.019612987086
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612987849→0.019612987199
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612988794→0.019612987849
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612990278→0.019612988794
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019612990977→0.019612990278
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.019613048175→0.019612990977
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
0.023007114764→0.019613048175
2026-09-10
游侠AI · OpenAI: GPT-5.6 Sol
0.023032156459→0.023007114764
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.02345890633→0.023032156459
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.052217573059→0.02345890633
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": [
    [
      "0.085164731",
      "0.749595743",
      "0.801984807",
      "0.279713678",
      "0.073673114",
      "0.062639194"
    ],
    [
      "0.450319708",
      "0.551277400",
      "0.305394962",
      "0.032163954",
      "0.288802009",
      "0.166366049"
    ],
    [
      "0.336054057",
      "0.327506942",
      "0.881555085",
      "0.899657499",
      "0.949705905",
      "0.292919839"
    ],
    [
      "0.852225655",
      "0.620317641",
      "0.944631814",
      "0.930015758",
      "0.451284082",
      "0.962127186"
    ],
    [
      "0.496119311",
      "0.597618519",
      "0.703959828",
      "0.501627374",
      "0.855617694",
      "0.694835392"
    ],
    [
      "0.921798501",
      "0.155398101",
      "0.847590603",
      "0.587994014",
      "0.325364387",
      "0.782028647"
    ],
    [
      "0.305531007",
      "0.840609445",
      "0.128828217",
      "0.117202500",
      "0.427602592",
      "0.740313265"
    ],
    [
      "0.206165159",
      "0.656639065",
      "0.075586468",
      "0.624869746",
      "0.203816162",
      "0.563242913"
    ],
    [
      "0.540212823",
      "0.033372423",
      "0.032160652",
      "0.330405844",
      "0.542018124",
      "0.219602020"
    ],
    [
      "0.673748501",
      "0.888355868",
      "0.672322492",
      "0.807064281",
      "0.245631557",
      "0.261726506"
    ],
    [
      "0.368250115",
      "0.925679900",
      "0.915985002",
      "0.372403114",
      "0.719688563",
      "0.918926032"
    ],
    [
      "0.395971205",
      "0.078328992",
      "0.443220803",
      "0.858398379",
      "0.109926038",
      "0.652956047"
    ],
    [
      "0.627940606",
      "0.292366051",
      "0.528931272",
      "0.204052971",
      "0.160660993",
      "0.871317204"
    ],
    [
      "0.715119871",
      "0.247967700",
      "0.162655336",
      "0.677675320",
      "0.758516911",
      "0.028087210"
    ],
    [
      "0.584807228",
      "0.792922608",
      "0.485244550",
      "0.722910276",
      "0.587683274",
      "0.514800420"
    ],
    [
      "0.947878599",
      "0.946457164",
      "0.235836352",
      "0.945199070",
      "0.935750165",
      "0.943092634"
    ],
    [
      "0.164318415",
      "0.120291222",
      "0.630083280",
      "0.157105055",
      "0.805049303",
      "0.478174664"
    ],
    [
      "0.974643287",
      "0.973257534",
      "0.983912117",
      "0.973534038",
      "0.690316962",
      "0.342985984"
    ],
    [
      "0.031016787",
      "0.201722391",
      "0.363366027",
      "0.460521679",
      "0.373275323",
      "0.374796897"
    ],
    [
      "0.114837435",
      "0.403299854",
      "0.260906664",
      "0.774184287",
      "0.671570821",
      "0.826997813"
    ],
    [
      "0.257995849",
      "0.493662740",
      "0.577537871",
      "0.547310914",
      "0.488458008",
      "0.117387069"
    ],
    [
      "0.760756135",
      "0.363522859",
      "0.751725489",
      "0.074588463",
      "0.626130641",
      "0.606980661"
    ],
    [
      "0.808047079",
      "0.446792819",
      "0.211382302",
      "0.404880272",
      "0.030243331",
      "0.430169549"
    ],
    [
      "0.879283183",
      "0.702228654",
      "0.410053481",
      "0.240360584",
      "0.896533770",
      "0.320795956"
    ]
  ]
}

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": 6
}

当前第一名的答案

{
  "points": [
    [
      "0.085164731",
      "0.749595743",
      "0.801984807",
      "0.279713678",
      "0.073673114",
      "0.062639194"
    ],
    [
      "0.450319708",
      "0.551277400",
      "0.305394962",
      "0.032163954",
      "0.288802009",
      "0.166366049"
    ],
    [
      "0.336054057",
      "0.327506942",
      "0.881555085",
      "0.899657499",
      "0.949705905",
      "0.292919839"
    ],
    [
      "0.852225655",
      "0.620317641",
      "0.944631814",
      "0.930015758",
      "0.451284082",
      "0.962127186"
    ],
    [
      "0.496119311",
      "0.597618519",
      "0.703959828",
      "0.501627374",
      "0.855617694",
      "0.694835392"
    ],
    [
      "0.921798501",
      "0.155398101",
      "0.847590603",
      "0.587994014",
      "0.325364387",
      "0.782028647"
    ],
    [
      "0.305531007",
      "0.840609445",
      "0.128828217",
      "0.117202500",
      "0.427602592",
      "0.740313265"
    ],
    [
      "0.206165159",
      "0.656639065",
      "0.075586468",
      "0.624869746",
      "0.203816162",
      "0.563242913"
    ],
    [
      "0.540212823",
      "0.033372423",
      "0.032160652",
      "0.330405844",
      "0.542018124",
      "0.219602020"
    ],
    [
      "0.673748501",
      "0.888355868",
      "0.672322492",
      "0.807064281",
      "0.245631557",
      "0.261726506"
    ],
    [
      "0.368250115",
      "0.925679900",
      "0.915985002",
      "0.372403114",
      "0.719688563",
      "0.918926032"
    ],
    [
      "0.395971205",
      "0.078328992",
      "0.443220803",
      "0.858398379",
      "0.109926038",
      "0.652956047"
    ],
    [
      "0.627940606",
      "0.292366051",
      "0.528931272",
      "0.204052971",
      "0.160660993",
      "0.871317204"
    ],
    [
      "0.715119871",
      "0.247967700",
      "0.162655336",
      "0.677675320",
      "0.758516911",
      "0.028087210"
    ],
    [
      "0.584807228",
      "0.792922608",
      "0.485244550",
      "0.722910276",
      "0.587683274",
      "0.514800420"
    ],
    [
      "0.947878599",
      "0.946457164",
      "0.235836352",
      "0.945199070",
      "0.935750165",
      "0.943092634"
    ],
    [
      "0.164318415",
      "0.120291222",
      "0.630083280",
      "0.157105055",
      "0.805049303",
      "0.478174664"
    ],
    [
      "0.974643287",
      "0.973257534",
      "0.983912117",
      "0.973534038",
      "0.690316962",
      "0.342985984"
    ],
    [
      "0.031016787",
      "0.201722391",
      "0.363366027",
      "0.460521679",
      "0.373275323",
      "0.374796897"
    ],
    [
      "0.114837435",
      "0.403299854",
      "0.260906664",
      "0.774184287",
      "0.671570821",
      "0.826997813"
    ],
    [
      "0.257995849",
      "0.493662740",
      "0.577537871",
      "0.547310914",
      "0.488458008",
      "0.117387069"
    ],
    [
      "0.760756135",
      "0.363522859",
      "0.751725489",
      "0.074588463",
      "0.626130641",
      "0.606980661"
    ],
    [
      "0.808047079",
      "0.446792819",
      "0.211382302",
      "0.404880272",
      "0.030243331",
      "0.430169549"
    ],
    [
      "0.879283183",
      "0.702228654",
      "0.410053481",
      "0.240360584",
      "0.896533770",
      "0.320795956"
    ]
  ]
}

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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