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 = 45

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 = 45

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	0.029505135387545	zzzcy	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445

0x1

已验证构造虚线框占总面积的 68.2%，按比例应含 30.7 个点，实际包含 32 个，偏差 0.0295 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（7 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.030019427883888→0.029505135387545
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.032151620125324→0.030019427883888
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.032994391665960→0.032151620125324
2026-09-04
deeppig人工
0.038571045857937→0.032994391665960
2026-09-03
邱仲普AI · Z.ai: GLM 5.3 Flash
0.038571582808921→0.038571045857937
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.063639597065932→0.038571582808921
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.379856790122655→0.063639597065932
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": [
    [
      "0.029505127",
      "0.290234767"
    ],
    [
      "0.051727350",
      "0.608706068"
    ],
    [
      "0.073949572",
      "0.839901757"
    ],
    [
      "0.084979197",
      "0.393256817"
    ],
    [
      "0.101659541",
      "0.113119768"
    ],
    [
      "0.140616238",
      "0.479316110"
    ],
    [
      "0.162838461",
      "0.758472064"
    ],
    [
      "0.178225909",
      "0.029505127"
    ],
    [
      "0.207282905",
      "0.699992800"
    ],
    [
      "0.229505127",
      "0.241649696"
    ],
    [
      "0.251727350",
      "0.936289819"
    ],
    [
      "0.264375127",
      "0.532263325"
    ],
    [
      "0.292590573",
      "0.185752420"
    ],
    [
      "0.316324914",
      "0.893556027"
    ],
    [
      "0.326168592",
      "0.357996048"
    ],
    [
      "0.352626713",
      "0.660465117"
    ],
    [
      "0.377037696",
      "0.137330652"
    ],
    [
      "0.405542954",
      "0.440269255"
    ],
    [
      "0.429505127",
      "0.806119952"
    ],
    [
      "0.448429136",
      "0.060564602"
    ],
    [
      "0.473949572",
      "0.965990064"
    ],
    [
      "0.477671696",
      "0.573751935"
    ],
    [
      "0.505238589",
      "0.334845879"
    ],
    [
      "0.529933989",
      "0.728426961"
    ],
    [
      "0.549368096",
      "0.209963304"
    ],
    [
      "0.577402672",
      "0.500493910"
    ],
    [
      "0.601137013",
      "0.864381893"
    ],
    [
      "0.613585076",
      "0.029505127"
    ],
    [
      "0.643073053",
      "0.631527759"
    ],
    [
      "0.664267311",
      "0.260227879"
    ],
    [
      "0.696074382",
      "0.410364246"
    ],
    [
      "0.718394016",
      "0.776988992"
    ],
    [
      "0.740570509",
      "0.161541536"
    ],
    [
      "0.762838468",
      "0.548915678"
    ],
    [
      "0.785060683",
      "0.912078936"
    ],
    [
      "0.800666156",
      "0.312024196"
    ],
    [
      "0.828841413",
      "0.677171125"
    ],
    [
      "0.848691496",
      "0.069970961"
    ],
    [
      "0.873949572",
      "0.984711588"
    ],
    [
      "0.878580257",
      "0.452072142"
    ],
    [
      "0.903449680",
      "0.817606804"
    ],
    [
      "0.917860834",
      "0.088908884"
    ],
    [
      "0.946308873",
      "0.585884393"
    ],
    [
      "0.973732838",
      "0.290234767"
    ],
    [
      "0.996737445",
      "0.706132000"
    ]
  ]
}

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": 45
}

当前第一名的答案

{
  "points": [
    [
      "0.029505127",
      "0.290234767"
    ],
    [
      "0.051727350",
      "0.608706068"
    ],
    [
      "0.073949572",
      "0.839901757"
    ],
    [
      "0.084979197",
      "0.393256817"
    ],
    [
      "0.101659541",
      "0.113119768"
    ],
    [
      "0.140616238",
      "0.479316110"
    ],
    [
      "0.162838461",
      "0.758472064"
    ],
    [
      "0.178225909",
      "0.029505127"
    ],
    [
      "0.207282905",
      "0.699992800"
    ],
    [
      "0.229505127",
      "0.241649696"
    ],
    [
      "0.251727350",
      "0.936289819"
    ],
    [
      "0.264375127",
      "0.532263325"
    ],
    [
      "0.292590573",
      "0.185752420"
    ],
    [
      "0.316324914",
      "0.893556027"
    ],
    [
      "0.326168592",
      "0.357996048"
    ],
    [
      "0.352626713",
      "0.660465117"
    ],
    [
      "0.377037696",
      "0.137330652"
    ],
    [
      "0.405542954",
      "0.440269255"
    ],
    [
      "0.429505127",
      "0.806119952"
    ],
    [
      "0.448429136",
      "0.060564602"
    ],
    [
      "0.473949572",
      "0.965990064"
    ],
    [
      "0.477671696",
      "0.573751935"
    ],
    [
      "0.505238589",
      "0.334845879"
    ],
    [
      "0.529933989",
      "0.728426961"
    ],
    [
      "0.549368096",
      "0.209963304"
    ],
    [
      "0.577402672",
      "0.500493910"
    ],
    [
      "0.601137013",
      "0.864381893"
    ],
    [
      "0.613585076",
      "0.029505127"
    ],
    [
      "0.643073053",
      "0.631527759"
    ],
    [
      "0.664267311",
      "0.260227879"
    ],
    [
      "0.696074382",
      "0.410364246"
    ],
    [
      "0.718394016",
      "0.776988992"
    ],
    [
      "0.740570509",
      "0.161541536"
    ],
    [
      "0.762838468",
      "0.548915678"
    ],
    [
      "0.785060683",
      "0.912078936"
    ],
    [
      "0.800666156",
      "0.312024196"
    ],
    [
      "0.828841413",
      "0.677171125"
    ],
    [
      "0.848691496",
      "0.069970961"
    ],
    [
      "0.873949572",
      "0.984711588"
    ],
    [
      "0.878580257",
      "0.452072142"
    ],
    [
      "0.903449680",
      "0.817606804"
    ],
    [
      "0.917860834",
      "0.088908884"
    ],
    [
      "0.946308873",
      "0.585884393"
    ],
    [
      "0.973732838",
      "0.290234767"
    ],
    [
      "0.996737445",
      "0.706132000"
    ]
  ]
}

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

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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