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 = 43

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 = 43

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	0.030818680280013	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

12345678910111213141516171819202122232425262728293031323334353637383940414243

0x1

已验证构造虚线框占总面积的 75.2%，按比例应含 32.3 个点，实际包含 31 个，偏差 0.0308 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（8 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.031949576924753→0.030818680280013
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.033273864745558→0.031949576924753
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.034493885849343→0.033273864745558
2026-09-04
deeppig人工
0.036915196176204→0.034493885849343
2026-09-02
deeppig人工
0.040419222630888→0.036915196176204
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.040419668280773→0.040419222630888
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.055858760694917→0.040419668280773
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.379853975119011→0.055858760694917
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": [
    [
      "0.030818680",
      "0.320585474"
    ],
    [
      "0.054074494",
      "0.532575900"
    ],
    [
      "0.077330308",
      "0.052979599"
    ],
    [
      "0.100586122",
      "0.964153017"
    ],
    [
      "0.104325890",
      "0.651946930"
    ],
    [
      "0.128446350",
      "0.855718862"
    ],
    [
      "0.144722690",
      "0.254313183"
    ],
    [
      "0.171899623",
      "0.732139755"
    ],
    [
      "0.188867205",
      "0.462034639"
    ],
    [
      "0.212629536",
      "0.141484526"
    ],
    [
      "0.249048648",
      "0.370341025"
    ],
    [
      "0.273169108",
      "0.784491512"
    ],
    [
      "0.296207370",
      "0.000000001"
    ],
    [
      "0.321410027",
      "0.586291398"
    ],
    [
      "0.345530487",
      "0.904751804"
    ],
    [
      "0.359735717",
      "0.405961677"
    ],
    [
      "0.386299274",
      "0.205113931"
    ],
    [
      "0.416492017",
      "0.698504029"
    ],
    [
      "0.439656390",
      "0.096679776"
    ],
    [
      "0.466132785",
      "0.501750985"
    ],
    [
      "0.490253245",
      "0.826474333"
    ],
    [
      "0.510233385",
      "0.294357472"
    ],
    [
      "0.538494164",
      "0.623094617"
    ],
    [
      "0.562614624",
      "0.168969213"
    ],
    [
      "0.586735083",
      "0.938588059"
    ],
    [
      "0.599552921",
      "0.429317428"
    ],
    [
      "0.625257002",
      "0.755120830"
    ],
    [
      "0.643408359",
      "0.060535057"
    ],
    [
      "0.676665165",
      "0.556962026"
    ],
    [
      "0.701826164",
      "0.224247549"
    ],
    [
      "0.726603608",
      "0.878659314"
    ],
    [
      "0.740753831",
      "0.344783613"
    ],
    [
      "0.768892411",
      "0.672553528"
    ],
    [
      "0.795922394",
      "0.107372767"
    ],
    [
      "0.821516354",
      "0.477482259"
    ],
    [
      "0.844772168",
      "0.800381842"
    ],
    [
      "0.867712347",
      "0.269144402"
    ],
    [
      "0.891283796",
      "0.598825891"
    ],
    [
      "0.914539610",
      "0.987125512"
    ],
    [
      "0.922371358",
      "0.168969213"
    ],
    [
      "0.948541978",
      "0.914319333"
    ],
    [
      "0.958262647",
      "0.380407353"
    ],
    [
      "0.985114141",
      "0.700539688"
    ]
  ]
}

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": 43
}

当前第一名的答案

{
  "points": [
    [
      "0.030818680",
      "0.320585474"
    ],
    [
      "0.054074494",
      "0.532575900"
    ],
    [
      "0.077330308",
      "0.052979599"
    ],
    [
      "0.100586122",
      "0.964153017"
    ],
    [
      "0.104325890",
      "0.651946930"
    ],
    [
      "0.128446350",
      "0.855718862"
    ],
    [
      "0.144722690",
      "0.254313183"
    ],
    [
      "0.171899623",
      "0.732139755"
    ],
    [
      "0.188867205",
      "0.462034639"
    ],
    [
      "0.212629536",
      "0.141484526"
    ],
    [
      "0.249048648",
      "0.370341025"
    ],
    [
      "0.273169108",
      "0.784491512"
    ],
    [
      "0.296207370",
      "0.000000001"
    ],
    [
      "0.321410027",
      "0.586291398"
    ],
    [
      "0.345530487",
      "0.904751804"
    ],
    [
      "0.359735717",
      "0.405961677"
    ],
    [
      "0.386299274",
      "0.205113931"
    ],
    [
      "0.416492017",
      "0.698504029"
    ],
    [
      "0.439656390",
      "0.096679776"
    ],
    [
      "0.466132785",
      "0.501750985"
    ],
    [
      "0.490253245",
      "0.826474333"
    ],
    [
      "0.510233385",
      "0.294357472"
    ],
    [
      "0.538494164",
      "0.623094617"
    ],
    [
      "0.562614624",
      "0.168969213"
    ],
    [
      "0.586735083",
      "0.938588059"
    ],
    [
      "0.599552921",
      "0.429317428"
    ],
    [
      "0.625257002",
      "0.755120830"
    ],
    [
      "0.643408359",
      "0.060535057"
    ],
    [
      "0.676665165",
      "0.556962026"
    ],
    [
      "0.701826164",
      "0.224247549"
    ],
    [
      "0.726603608",
      "0.878659314"
    ],
    [
      "0.740753831",
      "0.344783613"
    ],
    [
      "0.768892411",
      "0.672553528"
    ],
    [
      "0.795922394",
      "0.107372767"
    ],
    [
      "0.821516354",
      "0.477482259"
    ],
    [
      "0.844772168",
      "0.800381842"
    ],
    [
      "0.867712347",
      "0.269144402"
    ],
    [
      "0.891283796",
      "0.598825891"
    ],
    [
      "0.914539610",
      "0.987125512"
    ],
    [
      "0.922371358",
      "0.168969213"
    ],
    [
      "0.948541978",
      "0.914319333"
    ],
    [
      "0.958262647",
      "0.380407353"
    ],
    [
      "0.985114141",
      "0.700539688"
    ]
  ]
}

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