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 = 44

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 = 44

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	0.030161427358826	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344

0x1

已验证构造虚线框占总面积的 62.9%，按比例应含 27.7 个点，实际包含 29 个，偏差 0.0302 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
0.031017898646262→0.030161427358826
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.031976959018267→0.031017898646262
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.032806263812625→0.031976959018267
2026-09-04
deeppig人工
0.039228399297181→0.032806263812625
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.039229004513825→0.039228399297181
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.121900826974174→0.039229004513825
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.379855371901447→0.121900826974174
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": [
    [
      "0.021811877",
      "0.051602900"
    ],
    [
      "0.052351542",
      "0.717310099"
    ],
    [
      "0.073459051",
      "0.400603193"
    ],
    [
      "0.098178203",
      "0.490688853"
    ],
    [
      "0.120825631",
      "0.825533527"
    ],
    [
      "0.131983530",
      "0.154916808"
    ],
    [
      "0.166404882",
      "0.623497202"
    ],
    [
      "0.188696793",
      "0.297497193"
    ],
    [
      "0.211973361",
      "0.545233816"
    ],
    [
      "0.234478829",
      "0.915715020"
    ],
    [
      "0.245461059",
      "0.115897561"
    ],
    [
      "0.279833120",
      "0.959023047"
    ],
    [
      "0.281112424",
      "0.343018655"
    ],
    [
      "0.305883613",
      "0.776786855"
    ],
    [
      "0.327167336",
      "0.239897374"
    ],
    [
      "0.355510230",
      "0.676045374"
    ],
    [
      "0.376434626",
      "0.451244180"
    ],
    [
      "0.405180514",
      "0.193522471"
    ],
    [
      "0.429914974",
      "0.859843751"
    ],
    [
      "0.448022767",
      "0.020940658"
    ],
    [
      "0.476902238",
      "0.587802507"
    ],
    [
      "0.504431675",
      "0.366929814"
    ],
    [
      "0.529002460",
      "0.743338660"
    ],
    [
      "0.548828955",
      "0.091275485"
    ],
    [
      "0.575613360",
      "0.512839225"
    ],
    [
      "0.598220779",
      "0.884647396"
    ],
    [
      "0.612360452",
      "0.266839623"
    ],
    [
      "0.641801241",
      "0.985885017"
    ],
    [
      "0.642141008",
      "0.646927809"
    ],
    [
      "0.663585867",
      "0.215157797"
    ],
    [
      "0.694988445",
      "0.420639492"
    ],
    [
      "0.718621691",
      "0.796067476"
    ],
    [
      "0.739024217",
      "0.000000000"
    ],
    [
      "0.766025236",
      "0.562288444"
    ],
    [
      "0.789570541",
      "0.934224640"
    ],
    [
      "0.799489013",
      "0.319795312"
    ],
    [
      "0.827053492",
      "0.693709955"
    ],
    [
      "0.847212987",
      "0.165640392"
    ],
    [
      "0.876788829",
      "0.463163590"
    ],
    [
      "0.901505678",
      "0.835152885"
    ],
    [
      "0.916540981",
      "0.066585778"
    ],
    [
      "0.944995890",
      "0.600279118"
    ],
    [
      "0.972280908",
      "0.297799853"
    ],
    [
      "0.998634269",
      "0.959393167"
    ]
  ]
}

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": 44
}

当前第一名的答案

{
  "points": [
    [
      "0.021811877",
      "0.051602900"
    ],
    [
      "0.052351542",
      "0.717310099"
    ],
    [
      "0.073459051",
      "0.400603193"
    ],
    [
      "0.098178203",
      "0.490688853"
    ],
    [
      "0.120825631",
      "0.825533527"
    ],
    [
      "0.131983530",
      "0.154916808"
    ],
    [
      "0.166404882",
      "0.623497202"
    ],
    [
      "0.188696793",
      "0.297497193"
    ],
    [
      "0.211973361",
      "0.545233816"
    ],
    [
      "0.234478829",
      "0.915715020"
    ],
    [
      "0.245461059",
      "0.115897561"
    ],
    [
      "0.279833120",
      "0.959023047"
    ],
    [
      "0.281112424",
      "0.343018655"
    ],
    [
      "0.305883613",
      "0.776786855"
    ],
    [
      "0.327167336",
      "0.239897374"
    ],
    [
      "0.355510230",
      "0.676045374"
    ],
    [
      "0.376434626",
      "0.451244180"
    ],
    [
      "0.405180514",
      "0.193522471"
    ],
    [
      "0.429914974",
      "0.859843751"
    ],
    [
      "0.448022767",
      "0.020940658"
    ],
    [
      "0.476902238",
      "0.587802507"
    ],
    [
      "0.504431675",
      "0.366929814"
    ],
    [
      "0.529002460",
      "0.743338660"
    ],
    [
      "0.548828955",
      "0.091275485"
    ],
    [
      "0.575613360",
      "0.512839225"
    ],
    [
      "0.598220779",
      "0.884647396"
    ],
    [
      "0.612360452",
      "0.266839623"
    ],
    [
      "0.641801241",
      "0.985885017"
    ],
    [
      "0.642141008",
      "0.646927809"
    ],
    [
      "0.663585867",
      "0.215157797"
    ],
    [
      "0.694988445",
      "0.420639492"
    ],
    [
      "0.718621691",
      "0.796067476"
    ],
    [
      "0.739024217",
      "0.000000000"
    ],
    [
      "0.766025236",
      "0.562288444"
    ],
    [
      "0.789570541",
      "0.934224640"
    ],
    [
      "0.799489013",
      "0.319795312"
    ],
    [
      "0.827053492",
      "0.693709955"
    ],
    [
      "0.847212987",
      "0.165640392"
    ],
    [
      "0.876788829",
      "0.463163590"
    ],
    [
      "0.901505678",
      "0.835152885"
    ],
    [
      "0.916540981",
      "0.066585778"
    ],
    [
      "0.944995890",
      "0.600279118"
    ],
    [
      "0.972280908",
      "0.297799853"
    ],
    [
      "0.998634269",
      "0.959393167"
    ]
  ]
}

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

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

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