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 = 47

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 = 47

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	0.028368794743443	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647

0x1

已验证构造虚线框占总面积的 39.7%，按比例应含 18.7 个点，实际包含 20 个，偏差 0.0284 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（8 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.028368794797276→0.028368794743443
2026-10-08
zzzcyAI · OpenAI: GPT-6 Astra
0.028486020750653→0.028368794797276
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.028994217164475→0.028486020750653
2026-10-03
游侠AI · OpenAI: GPT-5.6 Sol
0.030266831187877→0.028994217164475
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.035925903240671→0.030266831187877
2026-09-04
邱仲普AI · Z.ai: GLM 5.3 Flash
0.035926336001207→0.035925903240671
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.053859635613503→0.035926336001207
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.379859665009168→0.053859635613503
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": [
    [
      "0.028368794",
      "0.440008774"
    ],
    [
      "0.049645390",
      "0.762886598"
    ],
    [
      "0.064473247",
      "0.194349537"
    ],
    [
      "0.092198581",
      "0.066096366"
    ],
    [
      "0.113475177",
      "0.960049718"
    ],
    [
      "0.118197188",
      "0.651070949"
    ],
    [
      "0.140359161",
      "0.515463918"
    ],
    [
      "0.161183117",
      "0.319393296"
    ],
    [
      "0.184683107",
      "0.843971631"
    ],
    [
      "0.204523673",
      "0.252654398"
    ],
    [
      "0.229007052",
      "0.578343204"
    ],
    [
      "0.251169025",
      "0.041310229"
    ],
    [
      "0.273330998",
      "0.894007870"
    ],
    [
      "0.285714286",
      "0.375295658"
    ],
    [
      "0.310924370",
      "0.701030928"
    ],
    [
      "0.336134454",
      "0.115668641"
    ],
    [
      "0.361344538",
      "0.487460700"
    ],
    [
      "0.384140862",
      "0.809680485"
    ],
    [
      "0.399750815",
      "0.295411296"
    ],
    [
      "0.427640406",
      "0.621918518"
    ],
    [
      "0.449132009",
      "0.140454778"
    ],
    [
      "0.472788753",
      "0.933611172"
    ],
    [
      "0.481759075",
      "0.417635446"
    ],
    [
      "0.507714445",
      "0.738528240"
    ],
    [
      "0.524572151",
      "0.225445463"
    ],
    [
      "0.554546166",
      "0.551952685"
    ],
    [
      "0.577335735",
      "0.028368794"
    ],
    [
      "0.600125303",
      "0.865248227"
    ],
    [
      "0.613147914",
      "0.350515464"
    ],
    [
      "0.638655462",
      "0.671848327"
    ],
    [
      "0.661417060",
      "0.165240915"
    ],
    [
      "0.687943262",
      "0.462674563"
    ],
    [
      "0.709219858",
      "0.785172129"
    ],
    [
      "0.729839514",
      "0.983183447"
    ],
    [
      "0.729839514",
      "0.090882503"
    ],
    [
      "0.758648969",
      "0.600031074"
    ],
    [
      "0.781162465",
      "0.272089352"
    ],
    [
      "0.805218345",
      "0.909848651"
    ],
    [
      "0.815126050",
      "0.395262119"
    ],
    [
      "0.840336134",
      "0.716312057"
    ],
    [
      "0.858407080",
      "0.028368794"
    ],
    [
      "0.888500278",
      "0.529502084"
    ],
    [
      "0.912299393",
      "0.202123518"
    ],
    [
      "0.936098506",
      "0.822695036"
    ],
    [
      "0.950980392",
      "0.328142137"
    ],
    [
      "0.981036662",
      "0.556655334"
    ],
    [
      "1.000000000",
      "0.701030928"
    ]
  ]
}

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": 47
}

当前第一名的答案

{
  "points": [
    [
      "0.028368794",
      "0.440008774"
    ],
    [
      "0.049645390",
      "0.762886598"
    ],
    [
      "0.064473247",
      "0.194349537"
    ],
    [
      "0.092198581",
      "0.066096366"
    ],
    [
      "0.113475177",
      "0.960049718"
    ],
    [
      "0.118197188",
      "0.651070949"
    ],
    [
      "0.140359161",
      "0.515463918"
    ],
    [
      "0.161183117",
      "0.319393296"
    ],
    [
      "0.184683107",
      "0.843971631"
    ],
    [
      "0.204523673",
      "0.252654398"
    ],
    [
      "0.229007052",
      "0.578343204"
    ],
    [
      "0.251169025",
      "0.041310229"
    ],
    [
      "0.273330998",
      "0.894007870"
    ],
    [
      "0.285714286",
      "0.375295658"
    ],
    [
      "0.310924370",
      "0.701030928"
    ],
    [
      "0.336134454",
      "0.115668641"
    ],
    [
      "0.361344538",
      "0.487460700"
    ],
    [
      "0.384140862",
      "0.809680485"
    ],
    [
      "0.399750815",
      "0.295411296"
    ],
    [
      "0.427640406",
      "0.621918518"
    ],
    [
      "0.449132009",
      "0.140454778"
    ],
    [
      "0.472788753",
      "0.933611172"
    ],
    [
      "0.481759075",
      "0.417635446"
    ],
    [
      "0.507714445",
      "0.738528240"
    ],
    [
      "0.524572151",
      "0.225445463"
    ],
    [
      "0.554546166",
      "0.551952685"
    ],
    [
      "0.577335735",
      "0.028368794"
    ],
    [
      "0.600125303",
      "0.865248227"
    ],
    [
      "0.613147914",
      "0.350515464"
    ],
    [
      "0.638655462",
      "0.671848327"
    ],
    [
      "0.661417060",
      "0.165240915"
    ],
    [
      "0.687943262",
      "0.462674563"
    ],
    [
      "0.709219858",
      "0.785172129"
    ],
    [
      "0.729839514",
      "0.983183447"
    ],
    [
      "0.729839514",
      "0.090882503"
    ],
    [
      "0.758648969",
      "0.600031074"
    ],
    [
      "0.781162465",
      "0.272089352"
    ],
    [
      "0.805218345",
      "0.909848651"
    ],
    [
      "0.815126050",
      "0.395262119"
    ],
    [
      "0.840336134",
      "0.716312057"
    ],
    [
      "0.858407080",
      "0.028368794"
    ],
    [
      "0.888500278",
      "0.529502084"
    ],
    [
      "0.912299393",
      "0.202123518"
    ],
    [
      "0.936098506",
      "0.822695036"
    ],
    [
      "0.950980392",
      "0.328142137"
    ],
    [
      "0.981036662",
      "0.556655334"
    ],
    [
      "1.000000000",
      "0.701030928"
    ]
  ]
}

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