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 = 41

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 = 41

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	0.031856645499737	邱仲普	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

1234567891011121314151617181920212223242526272829303132333435363738394041

0x1

已验证构造虚线框占总面积的 44.6%，按比例应含 18.3 个点，实际包含 17 个，偏差 0.0319 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。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 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（11 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.031856645886445→0.031856645499737
2026-10-08
zzzcyAI · OpenAI: GPT-6 Astra
0.032150776090367→0.031856645886445
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.033285509696449→0.032150776090367
2026-10-03
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.033285509853659→0.033285509696449
2026-09-19
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.033285510000569→0.033285509853659
2026-09-19
游侠AI · OpenAI: GPT-5.6 Sol
0.034311929894017→0.033285510000569
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.035694477987672→0.034311929894017
2026-09-04
deeppig人工
0.041242700152757→0.035694477987672
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.041243323436334→0.041242700152757
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.082386574080784→0.041243323436334
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.379851278998186→0.082386574080784
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": [
    [
      "0.031856645",
      "0.307358702"
    ],
    [
      "0.056246889",
      "0.761099366"
    ],
    [
      "0.073902162",
      "0.031856644"
    ],
    [
      "0.105027377",
      "0.556866095"
    ],
    [
      "0.129417621",
      "0.894142941"
    ],
    [
      "0.137994303",
      "0.419303319"
    ],
    [
      "0.170040374",
      "0.158493870"
    ],
    [
      "0.199294878",
      "0.681262012"
    ],
    [
      "0.225769032",
      "0.481153152"
    ],
    [
      "0.250480670",
      "0.266204492"
    ],
    [
      "0.275759084",
      "0.819632327"
    ],
    [
      "0.297372155",
      "0.097088875"
    ],
    [
      "0.324539572",
      "0.612133732"
    ],
    [
      "0.348929816",
      "0.944589193"
    ],
    [
      "0.362316797",
      "0.373029772"
    ],
    [
      "0.390239412",
      "0.204964961"
    ],
    [
      "0.417517203",
      "0.728560915"
    ],
    [
      "0.440578401",
      "0.525434401"
    ],
    [
      "0.470881035",
      "0.035940803"
    ],
    [
      "0.495271279",
      "0.860485855"
    ],
    [
      "0.512181525",
      "0.335336223"
    ],
    [
      "0.544051767",
      "0.652388472"
    ],
    [
      "0.568442011",
      "0.123372603"
    ],
    [
      "0.592832255",
      "0.972623639"
    ],
    [
      "0.601787544",
      "0.449923084"
    ],
    [
      "0.627608551",
      "0.787568328"
    ],
    [
      "0.646316630",
      "0.235584726"
    ],
    [
      "0.679250564",
      "0.581042382"
    ],
    [
      "0.705071571",
      "0.175940060"
    ],
    [
      "0.730892578",
      "0.916554747"
    ],
    [
      "0.744850690",
      "0.388683553"
    ],
    [
      "0.772128481",
      "0.703170101"
    ],
    [
      "0.796552277",
      "0.051866133"
    ],
    [
      "0.826684062",
      "0.500043587"
    ],
    [
      "0.853961851",
      "0.833749548"
    ],
    [
      "0.870009839",
      "0.281074974"
    ],
    [
      "0.901835730",
      "0.626997658"
    ],
    [
      "0.927959819",
      "0.070805146"
    ],
    [
      "0.958685913",
      "0.982469057"
    ],
    [
      "0.960593216",
      "0.423871144"
    ],
    [
      "0.985669970",
      "0.734770195"
    ]
  ]
}

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": 41
}

当前第一名的答案

{
  "points": [
    [
      "0.031856645",
      "0.307358702"
    ],
    [
      "0.056246889",
      "0.761099366"
    ],
    [
      "0.073902162",
      "0.031856644"
    ],
    [
      "0.105027377",
      "0.556866095"
    ],
    [
      "0.129417621",
      "0.894142941"
    ],
    [
      "0.137994303",
      "0.419303319"
    ],
    [
      "0.170040374",
      "0.158493870"
    ],
    [
      "0.199294878",
      "0.681262012"
    ],
    [
      "0.225769032",
      "0.481153152"
    ],
    [
      "0.250480670",
      "0.266204492"
    ],
    [
      "0.275759084",
      "0.819632327"
    ],
    [
      "0.297372155",
      "0.097088875"
    ],
    [
      "0.324539572",
      "0.612133732"
    ],
    [
      "0.348929816",
      "0.944589193"
    ],
    [
      "0.362316797",
      "0.373029772"
    ],
    [
      "0.390239412",
      "0.204964961"
    ],
    [
      "0.417517203",
      "0.728560915"
    ],
    [
      "0.440578401",
      "0.525434401"
    ],
    [
      "0.470881035",
      "0.035940803"
    ],
    [
      "0.495271279",
      "0.860485855"
    ],
    [
      "0.512181525",
      "0.335336223"
    ],
    [
      "0.544051767",
      "0.652388472"
    ],
    [
      "0.568442011",
      "0.123372603"
    ],
    [
      "0.592832255",
      "0.972623639"
    ],
    [
      "0.601787544",
      "0.449923084"
    ],
    [
      "0.627608551",
      "0.787568328"
    ],
    [
      "0.646316630",
      "0.235584726"
    ],
    [
      "0.679250564",
      "0.581042382"
    ],
    [
      "0.705071571",
      "0.175940060"
    ],
    [
      "0.730892578",
      "0.916554747"
    ],
    [
      "0.744850690",
      "0.388683553"
    ],
    [
      "0.772128481",
      "0.703170101"
    ],
    [
      "0.796552277",
      "0.051866133"
    ],
    [
      "0.826684062",
      "0.500043587"
    ],
    [
      "0.853961851",
      "0.833749548"
    ],
    [
      "0.870009839",
      "0.281074974"
    ],
    [
      "0.901835730",
      "0.626997658"
    ],
    [
      "0.927959819",
      "0.070805146"
    ],
    [
      "0.958685913",
      "0.982469057"
    ],
    [
      "0.960593216",
      "0.423871144"
    ],
    [
      "0.985669970",
      "0.734770195"
    ]
  ]
}

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