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 = 39

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 = 39

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	0.033489742581163	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

123456789101112131415161718192021222324252627282930313233343536373839

0x1

已验证构造虚线框占总面积的 50.5%，按比例应含 19.7 个点，实际包含 21 个，偏差 0.0335 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 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（7 次易主）

condorAI · OpenAI: GPT-6.1 Sol
0.033490320001238→0.033489742581163
2026-11-02
zzzcyAI · OpenAI: GPT-6 Astra
0.033530572345633→0.033490320001238
2026-10-05
fsbngd人工
0.033530572358975→0.033530572345633
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.045055285269432→0.033530572358975
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
0.045056111270224→0.045055285269432
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.074214248511328→0.045056111270224
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.379848783696089→0.074214248511328
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",
      "0.060573368"
    ],
    [
      "0.059130768",
      "0.508321543"
    ],
    [
      "0.084771793",
      "0.706525914"
    ],
    [
      "0.110412819",
      "0.378397009"
    ],
    [
      "0.135887140",
      "0.292459790"
    ],
    [
      "0.161694870",
      "0.930925640"
    ],
    [
      "0.173692574",
      "0.785405037"
    ],
    [
      "0.192567381",
      "0.153327937"
    ],
    [
      "0.218534966",
      "0.589997381"
    ],
    [
      "0.256323316",
      "0.232862247"
    ],
    [
      "0.283866897",
      "0.879109060"
    ],
    [
      "0.300598623",
      "0.449576760"
    ],
    [
      "0.329765682",
      "0.655183500"
    ],
    [
      "0.358932741",
      "0.097108062"
    ],
    [
      "0.388099799",
      "0.325496916"
    ],
    [
      "0.417266858",
      "0.842808466"
    ],
    [
      "0.434402676",
      "0.554185135"
    ],
    [
      "0.464752174",
      "0.185225236"
    ],
    [
      "0.495028204",
      "0.740252033"
    ],
    [
      "0.519192669",
      "0.417494785"
    ],
    [
      "0.546310255",
      "0.960298032"
    ],
    [
      "0.552878979",
      "0.038363279"
    ],
    [
      "0.586846286",
      "0.616447427"
    ],
    [
      "0.614389867",
      "0.268618518"
    ],
    [
      "0.641933448",
      "0.818396974"
    ],
    [
      "0.661914317",
      "0.478949151"
    ],
    [
      "0.694379927",
      "0.126480453"
    ],
    [
      "0.724564191",
      "0.683972156"
    ],
    [
      "0.751438460",
      "0.351946962"
    ],
    [
      "0.777079486",
      "0.905284614"
    ],
    [
      "0.792860162",
      "0.214597628"
    ],
    [
      "0.824190954",
      "0.537097288"
    ],
    [
      "0.854002563",
      "0.768391971"
    ],
    [
      "0.872963499",
      "0.000000000"
    ],
    [
      "0.905284614",
      "0.986039078"
    ],
    [
      "0.911196934",
      "0.404847055"
    ],
    [
      "0.942273916",
      "0.642897474"
    ],
    [
      "0.969413254",
      "0.185225236"
    ],
    [
      "0.996114366",
      "0.956566665"
    ]
  ]
}

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": 39
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.060573368"
    ],
    [
      "0.059130768",
      "0.508321543"
    ],
    [
      "0.084771793",
      "0.706525914"
    ],
    [
      "0.110412819",
      "0.378397009"
    ],
    [
      "0.135887140",
      "0.292459790"
    ],
    [
      "0.161694870",
      "0.930925640"
    ],
    [
      "0.173692574",
      "0.785405037"
    ],
    [
      "0.192567381",
      "0.153327937"
    ],
    [
      "0.218534966",
      "0.589997381"
    ],
    [
      "0.256323316",
      "0.232862247"
    ],
    [
      "0.283866897",
      "0.879109060"
    ],
    [
      "0.300598623",
      "0.449576760"
    ],
    [
      "0.329765682",
      "0.655183500"
    ],
    [
      "0.358932741",
      "0.097108062"
    ],
    [
      "0.388099799",
      "0.325496916"
    ],
    [
      "0.417266858",
      "0.842808466"
    ],
    [
      "0.434402676",
      "0.554185135"
    ],
    [
      "0.464752174",
      "0.185225236"
    ],
    [
      "0.495028204",
      "0.740252033"
    ],
    [
      "0.519192669",
      "0.417494785"
    ],
    [
      "0.546310255",
      "0.960298032"
    ],
    [
      "0.552878979",
      "0.038363279"
    ],
    [
      "0.586846286",
      "0.616447427"
    ],
    [
      "0.614389867",
      "0.268618518"
    ],
    [
      "0.641933448",
      "0.818396974"
    ],
    [
      "0.661914317",
      "0.478949151"
    ],
    [
      "0.694379927",
      "0.126480453"
    ],
    [
      "0.724564191",
      "0.683972156"
    ],
    [
      "0.751438460",
      "0.351946962"
    ],
    [
      "0.777079486",
      "0.905284614"
    ],
    [
      "0.792860162",
      "0.214597628"
    ],
    [
      "0.824190954",
      "0.537097288"
    ],
    [
      "0.854002563",
      "0.768391971"
    ],
    [
      "0.872963499",
      "0.000000000"
    ],
    [
      "0.905284614",
      "0.986039078"
    ],
    [
      "0.911196934",
      "0.404847055"
    ],
    [
      "0.942273916",
      "0.642897474"
    ],
    [
      "0.969413254",
      "0.185225236"
    ],
    [
      "0.996114366",
      "0.956566665"
    ]
  ]
}

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