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 = 42

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 = 42

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	0.031408308023611	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

123456789101112131415161718192021222324252627282930313233343536373839404142

0x1

已验证构造虚线框占总面积的 49.2%，按比例应含 20.7 个点，实际包含 22 个，偏差 0.0314 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 Terra

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）

condorAI · OpenAI: GPT-6.1 Terra
0.031446541485732→0.031408308023611
2026-11-02
zzzcyAI · OpenAI: GPT-6 Astra
0.031454783748408→0.031446541485732
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.031552458756100→0.031454783748408
2026-10-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.042310211700400→0.031552458756100
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
0.045683918053955→0.042310211700400
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.161564626459184→0.045683918053955
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.379852607706917→0.161564626459184
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",
      "0.055217256"
    ],
    [
      "0.055217256",
      "0.555217632"
    ],
    [
      "0.079026779",
      "0.434061972"
    ],
    [
      "0.102836303",
      "0.719584790"
    ],
    [
      "0.121792174",
      "0.287896446"
    ],
    [
      "0.150455351",
      "0.836207058"
    ],
    [
      "0.174264875",
      "0.936431227"
    ],
    [
      "0.175998842",
      "0.147116802"
    ],
    [
      "0.208399448",
      "0.374000701"
    ],
    [
      "0.236872707",
      "0.619540190"
    ],
    [
      "0.262372320",
      "0.229026779"
    ],
    [
      "0.287798147",
      "0.785787593"
    ],
    [
      "0.308350405",
      "0.504949036"
    ],
    [
      "0.338625355",
      "0.000000000"
    ],
    [
      "0.364075629",
      "0.894517296"
    ],
    [
      "0.373272559",
      "0.330015632"
    ],
    [
      "0.403572413",
      "0.663947414"
    ],
    [
      "0.433872989",
      "0.092240115"
    ],
    [
      "0.459979160",
      "0.459979160"
    ],
    [
      "0.483788684",
      "0.758332403"
    ],
    [
      "0.506877749",
      "0.184237796"
    ],
    [
      "0.531407732",
      "0.969986653"
    ],
    [
      "0.539965640",
      "0.591460641"
    ],
    [
      "0.566034258",
      "0.262884114"
    ],
    [
      "0.592908072",
      "0.867670089"
    ],
    [
      "0.606141421",
      "0.408557533"
    ],
    [
      "0.635498333",
      "0.695966316"
    ],
    [
      "0.656934543",
      "0.125926661"
    ],
    [
      "0.692444852",
      "0.531407732"
    ],
    [
      "0.717923103",
      "0.211933267"
    ],
    [
      "0.744220046",
      "0.813847828"
    ],
    [
      "0.763553429",
      "0.357577986"
    ],
    [
      "0.793312494",
      "0.645015469"
    ],
    [
      "0.816637878",
      "0.079026779"
    ],
    [
      "0.840931541",
      "0.921363994"
    ],
    [
      "0.857745131",
      "0.483788684"
    ],
    [
      "0.886861633",
      "0.313834962"
    ],
    [
      "0.912360113",
      "0.745693446"
    ],
    [
      "0.934608069",
      "0.125926661"
    ],
    [
      "0.959979160",
      "0.996559022"
    ],
    [
      "0.963294411",
      "0.579026779"
    ],
    [
      "0.987185000",
      "0.864741065"
    ]
  ]
}

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": 42
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.055217256"
    ],
    [
      "0.055217256",
      "0.555217632"
    ],
    [
      "0.079026779",
      "0.434061972"
    ],
    [
      "0.102836303",
      "0.719584790"
    ],
    [
      "0.121792174",
      "0.287896446"
    ],
    [
      "0.150455351",
      "0.836207058"
    ],
    [
      "0.174264875",
      "0.936431227"
    ],
    [
      "0.175998842",
      "0.147116802"
    ],
    [
      "0.208399448",
      "0.374000701"
    ],
    [
      "0.236872707",
      "0.619540190"
    ],
    [
      "0.262372320",
      "0.229026779"
    ],
    [
      "0.287798147",
      "0.785787593"
    ],
    [
      "0.308350405",
      "0.504949036"
    ],
    [
      "0.338625355",
      "0.000000000"
    ],
    [
      "0.364075629",
      "0.894517296"
    ],
    [
      "0.373272559",
      "0.330015632"
    ],
    [
      "0.403572413",
      "0.663947414"
    ],
    [
      "0.433872989",
      "0.092240115"
    ],
    [
      "0.459979160",
      "0.459979160"
    ],
    [
      "0.483788684",
      "0.758332403"
    ],
    [
      "0.506877749",
      "0.184237796"
    ],
    [
      "0.531407732",
      "0.969986653"
    ],
    [
      "0.539965640",
      "0.591460641"
    ],
    [
      "0.566034258",
      "0.262884114"
    ],
    [
      "0.592908072",
      "0.867670089"
    ],
    [
      "0.606141421",
      "0.408557533"
    ],
    [
      "0.635498333",
      "0.695966316"
    ],
    [
      "0.656934543",
      "0.125926661"
    ],
    [
      "0.692444852",
      "0.531407732"
    ],
    [
      "0.717923103",
      "0.211933267"
    ],
    [
      "0.744220046",
      "0.813847828"
    ],
    [
      "0.763553429",
      "0.357577986"
    ],
    [
      "0.793312494",
      "0.645015469"
    ],
    [
      "0.816637878",
      "0.079026779"
    ],
    [
      "0.840931541",
      "0.921363994"
    ],
    [
      "0.857745131",
      "0.483788684"
    ],
    [
      "0.886861633",
      "0.313834962"
    ],
    [
      "0.912360113",
      "0.745693446"
    ],
    [
      "0.934608069",
      "0.125926661"
    ],
    [
      "0.959979160",
      "0.996559022"
    ],
    [
      "0.963294411",
      "0.579026779"
    ],
    [
      "0.987185000",
      "0.864741065"
    ]
  ]
}

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