P54 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破 · 中

单位正方形内的最低星偏差 · n = 38

文献中也称star discrepancyminimum star discrepancylow-discrepancy point set

在单位正方形 [0, 1]² 内放置 n 个点。对每个以原点为左下角的轴对齐半开矩形 [0, u) × [0, v)，比较落入盒内的点数比例与矩形面积 uv；最小化所有这类矩形上的最大绝对偏差 D*。

子题n = 38

目标最小化最大误差 D*

纪录对比

最大误差 D* · 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	0.034330651117681	condor	纪录详情 ↓

严格定义

容器单位正方形 [0, 1]²，左下角为原点 (0, 0)，右上角为 (1, 1)
提交恰好 n 个点的坐标，写成最多九位小数的十进制字符串；两两不重合
矩形待测矩形为半开区间盒 [0, u) × [0, v)，右上角 (u, v) 取遍 [0, 1]²；落在上边界或右边界上的点计在盒外
星偏差指示函数 𝟙 在点位于 [0, u) × [0, v) 内时取 1，否则取 0；D*(P) 为盒内点数比例与矩形面积 uv 之间的最大绝对误差 $D^{*} (P) = \sup_{(u, v) \in {[0, 1]}^{2}} | \frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i} < u, y_{i} < v] - u v |$
目标在所有合法点集 P 中最小化 D*(P)。验证器在提交坐标构成的网格上检查开、闭单侧极限，并以精确整数比较 $\min_{P} D^{*} (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132333435363738

0x1

已验证构造虚线框占总面积的 35.0%，按比例应含 13.3 个点，实际包含 12 个，偏差 0.0343 即为得分

帮助理解

本质上就是平均摆放

一块矩形若占正方形面积的 30%，理想情况下也应包含约 30% 的点。本题同时检查从原点出发的所有矩形，并以偏差最大的一块计分，因此目标就是让点在这一意义下尽可能平均分布。

为什么整齐网格会吃亏

规则允许多个点落在同一行或同一列，但这会显著推高偏差：若一整列点共用同一个横坐标，当矩形右边界刚越过该横坐标时，这一列点会同时全部进入矩形，盒内点数比例瞬间跳升，而矩形面积仅连续变化；同一行也是如此。优良构形通常会将各点的横纵坐标全部错开。

研究进展

n ≤ 21 的最优星偏差已由 Clément、Doerr、Klamroth 与 Paquete 于 2025 年证明（Proc. Amer. Math. Soc. Ser. B 12: 78–90），因此本题从 n = 22 开始收录；n ≥ 22 的所有规模均未解决。

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condor

解题方式AI · OpenAI: GPT-6.1 Sol

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）

condorAI · OpenAI: GPT-6.1 Sol
0.034330651454718→0.034330651117681
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.034371644173594→0.034330651454718
2026-11-02
zzzcyAI · OpenAI: GPT-6 Astra
0.034412956079764→0.034371644173594
2026-10-05
TmrgAI · OpenAI: GPT-5.6 Sol
0.041876675980967→0.034412956079764
2026-09-02
215590053人工
0.041887401437502→0.041876675980967
2026-09-02
deeppig人工
0.043265955660170→0.041887401437502
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.049158490867529→0.043265955660170
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.094182826028394→0.049158490867529
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.379847645428532→0.094182826028394
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长为 1 的正方形，左下角为原点 (0, 0)，右上角为 (1, 1)。待测矩形以原点为左下角，右上角 (u, v) 取遍整个正方形。坐标写成十进制小数，例如 "0.25"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.061244001"
    ],
    [
      "0.060646420",
      "0.514641821"
    ],
    [
      "0.086962210",
      "0.715614827"
    ],
    [
      "0.113277999",
      "0.383050719"
    ],
    [
      "0.139593789",
      "0.295973712"
    ],
    [
      "0.165909578",
      "0.942629692"
    ],
    [
      "0.176007168",
      "0.796026853"
    ],
    [
      "0.195068332",
      "0.155135885"
    ],
    [
      "0.231842009",
      "0.597401785"
    ],
    [
      "0.259759430",
      "0.235564653"
    ],
    [
      "0.287676851",
      "0.889759344"
    ],
    [
      "0.304770885",
      "0.455161351"
    ],
    [
      "0.334347186",
      "0.663342965"
    ],
    [
      "0.363923487",
      "0.098278510"
    ],
    [
      "0.393499801",
      "0.329840707"
    ],
    [
      "0.423076088",
      "0.853934161"
    ],
    [
      "0.439834545",
      "0.560800883"
    ],
    [
      "0.471642541",
      "0.187499214"
    ],
    [
      "0.502459657",
      "0.748986906"
    ],
    [
      "0.523365534",
      "0.422385380"
    ],
    [
      "0.560555125",
      "0.973193609"
    ],
    [
      "0.560555165",
      "0.038798041"
    ],
    [
      "0.594768482",
      "0.624710787"
    ],
    [
      "0.622685902",
      "0.271992365"
    ],
    [
      "0.650603323",
      "0.828778987"
    ],
    [
      "0.671247007",
      "0.484901559"
    ],
    [
      "0.704305920",
      "0.128018745"
    ],
    [
      "0.734355586",
      "0.692039580"
    ],
    [
      "0.762273015",
      "0.356647078"
    ],
    [
      "0.790190428",
      "0.919580867"
    ],
    [
      "0.803482685",
      "0.217239448"
    ],
    [
      "0.835235166",
      "0.544291674"
    ],
    [
      "0.866720598",
      "0.777460521"
    ],
    [
      "0.884854794",
      "0.000000000"
    ],
    [
      "0.924215606",
      "0.410259819"
    ],
    [
      "0.948640170",
      "0.651517158"
    ],
    [
      "0.981699072",
      "0.187499214"
    ],
    [
      "1.000000000",
      "0.973193609"
    ]
  ]
}

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": 38
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.061244001"
    ],
    [
      "0.060646420",
      "0.514641821"
    ],
    [
      "0.086962210",
      "0.715614827"
    ],
    [
      "0.113277999",
      "0.383050719"
    ],
    [
      "0.139593789",
      "0.295973712"
    ],
    [
      "0.165909578",
      "0.942629692"
    ],
    [
      "0.176007168",
      "0.796026853"
    ],
    [
      "0.195068332",
      "0.155135885"
    ],
    [
      "0.231842009",
      "0.597401785"
    ],
    [
      "0.259759430",
      "0.235564653"
    ],
    [
      "0.287676851",
      "0.889759344"
    ],
    [
      "0.304770885",
      "0.455161351"
    ],
    [
      "0.334347186",
      "0.663342965"
    ],
    [
      "0.363923487",
      "0.098278510"
    ],
    [
      "0.393499801",
      "0.329840707"
    ],
    [
      "0.423076088",
      "0.853934161"
    ],
    [
      "0.439834545",
      "0.560800883"
    ],
    [
      "0.471642541",
      "0.187499214"
    ],
    [
      "0.502459657",
      "0.748986906"
    ],
    [
      "0.523365534",
      "0.422385380"
    ],
    [
      "0.560555125",
      "0.973193609"
    ],
    [
      "0.560555165",
      "0.038798041"
    ],
    [
      "0.594768482",
      "0.624710787"
    ],
    [
      "0.622685902",
      "0.271992365"
    ],
    [
      "0.650603323",
      "0.828778987"
    ],
    [
      "0.671247007",
      "0.484901559"
    ],
    [
      "0.704305920",
      "0.128018745"
    ],
    [
      "0.734355586",
      "0.692039580"
    ],
    [
      "0.762273015",
      "0.356647078"
    ],
    [
      "0.790190428",
      "0.919580867"
    ],
    [
      "0.803482685",
      "0.217239448"
    ],
    [
      "0.835235166",
      "0.544291674"
    ],
    [
      "0.866720598",
      "0.777460521"
    ],
    [
      "0.884854794",
      "0.000000000"
    ],
    [
      "0.924215606",
      "0.410259819"
    ],
    [
      "0.948640170",
      "0.651517158"
    ],
    [
      "0.981699072",
      "0.187499214"
    ],
    [
      "1.000000000",
      "0.973193609"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。分数是星偏差 D*，即所有原点角矩形上点数比例与面积的最大绝对误差，越小越好。 · 验证器 v1.0.0

DISCUSSION

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致谢：感谢邱仲普使用 AI · Z.ai: GLM 5.3 Flash 做出的结果deeppig#672 条回复30 天前