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 = 50

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 = 50

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	0.026776859504586	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

0x1

已验证构造虚线框占总面积的 61.3%，按比例应含 30.7 个点，实际包含 32 个，偏差 0.0268 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（17 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.027361111546058→0.026776859504586
2026-10-03
邱仲普AI · OpenAI: GPT-5.6 Sol
0.027361111712255→0.027361111546058
2026-09-08
游侠AI · OpenAI: GPT-5.6 Sol
0.028917160240331→0.027361111712255
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.030397729049012→0.028917160240331
2026-09-04
?人工
0.030397784569043→0.030397729049012
2026-09-02
215590053人工
0.030459522958837→0.030397784569043
2026-09-02
215590053人工
0.030460397096053→0.030459522958837
2026-09-02
215590053人工
0.030462507414057→0.030460397096053
2026-09-02
215590053人工
0.030463381551273→0.030462507414057
2026-09-02
215590053人工
0.030470276223310→0.030463381551273
2026-09-02
215590053人工
0.030512674859877→0.030470276223310
2026-09-02
deeppig人工
0.034723891723803→0.030512674859877
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.034724599305834→0.034723891723803
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.043984280308044→0.034724599305834
2026-09-01
邱仲普人工
0.046031472264744→0.043984280308044
2026-09-01
邱仲普人工
0.046437222412476→0.046031472264744
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.379864000000000→0.046437222412476
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": [
    [
      "0.026776859",
      "0.562146681"
    ],
    [
      "0.046776859",
      "0.815394652"
    ],
    [
      "0.047633225",
      "0.135491765"
    ],
    [
      "0.081895140",
      "0.345656452"
    ],
    [
      "0.106423140",
      "0.698915611"
    ],
    [
      "0.118789030",
      "0.106423496"
    ],
    [
      "0.146776859",
      "0.473892369"
    ],
    [
      "0.166776859",
      "0.263586229"
    ],
    [
      "0.186776859",
      "0.896721916"
    ],
    [
      "0.204535139",
      "0.959295937"
    ],
    [
      "0.204535140",
      "0.619425787"
    ],
    [
      "0.225318799",
      "0.401208381"
    ],
    [
      "0.251049442",
      "0.036341994"
    ],
    [
      "0.275198871",
      "0.746935441"
    ],
    [
      "0.295853828",
      "0.203331341"
    ],
    [
      "0.319793765",
      "0.522418730"
    ],
    [
      "0.340642388",
      "0.867687707"
    ],
    [
      "0.351703140",
      "0.309901376"
    ],
    [
      "0.376231140",
      "0.662393633"
    ],
    [
      "0.394133503",
      "0.058219513"
    ],
    [
      "0.422706068",
      "0.442196313"
    ],
    [
      "0.444885507",
      "0.791628878"
    ],
    [
      "0.462972103",
      "0.241925225"
    ],
    [
      "0.486582754",
      "0.584941502"
    ],
    [
      "0.506776859",
      "0.179686489"
    ],
    [
      "0.526776859",
      "0.933320263"
    ],
    [
      "0.537954906",
      "0.368312478"
    ],
    [
      "0.561004674",
      "0.721710433"
    ],
    [
      "0.581301767",
      "0.014464476"
    ],
    [
      "0.606776859",
      "0.495769888"
    ],
    [
      "0.626776859",
      "0.845810188"
    ],
    [
      "0.639242925",
      "0.283417716"
    ],
    [
      "0.665434112",
      "0.636103887"
    ],
    [
      "0.686776859",
      "0.979769500"
    ],
    [
      "0.688035476",
      "0.080097032"
    ],
    [
      "0.714413782",
      "0.415655761"
    ],
    [
      "0.735842654",
      "0.768111300"
    ],
    [
      "0.753563815",
      "0.158940244"
    ],
    [
      "0.778463486",
      "0.539524926"
    ],
    [
      "0.799312109",
      "0.911442744"
    ],
    [
      "0.810481874",
      "0.324910207"
    ],
    [
      "0.832785333",
      "0.677596378"
    ],
    [
      "0.850427719",
      "0.221178980"
    ],
    [
      "0.877392250",
      "0.452014851"
    ],
    [
      "0.899695708",
      "0.824152153"
    ],
    [
      "0.914180453",
      "0.000000000"
    ],
    [
      "0.940375139",
      "0.594611396"
    ],
    [
      "0.964029860",
      "0.138193998"
    ],
    [
      "0.986739084",
      "0.722808240"
    ],
    [
      "0.999999999",
      "0.933320263"
    ]
  ]
}

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": 50
}

当前第一名的答案

{
  "points": [
    [
      "0.026776859",
      "0.562146681"
    ],
    [
      "0.046776859",
      "0.815394652"
    ],
    [
      "0.047633225",
      "0.135491765"
    ],
    [
      "0.081895140",
      "0.345656452"
    ],
    [
      "0.106423140",
      "0.698915611"
    ],
    [
      "0.118789030",
      "0.106423496"
    ],
    [
      "0.146776859",
      "0.473892369"
    ],
    [
      "0.166776859",
      "0.263586229"
    ],
    [
      "0.186776859",
      "0.896721916"
    ],
    [
      "0.204535139",
      "0.959295937"
    ],
    [
      "0.204535140",
      "0.619425787"
    ],
    [
      "0.225318799",
      "0.401208381"
    ],
    [
      "0.251049442",
      "0.036341994"
    ],
    [
      "0.275198871",
      "0.746935441"
    ],
    [
      "0.295853828",
      "0.203331341"
    ],
    [
      "0.319793765",
      "0.522418730"
    ],
    [
      "0.340642388",
      "0.867687707"
    ],
    [
      "0.351703140",
      "0.309901376"
    ],
    [
      "0.376231140",
      "0.662393633"
    ],
    [
      "0.394133503",
      "0.058219513"
    ],
    [
      "0.422706068",
      "0.442196313"
    ],
    [
      "0.444885507",
      "0.791628878"
    ],
    [
      "0.462972103",
      "0.241925225"
    ],
    [
      "0.486582754",
      "0.584941502"
    ],
    [
      "0.506776859",
      "0.179686489"
    ],
    [
      "0.526776859",
      "0.933320263"
    ],
    [
      "0.537954906",
      "0.368312478"
    ],
    [
      "0.561004674",
      "0.721710433"
    ],
    [
      "0.581301767",
      "0.014464476"
    ],
    [
      "0.606776859",
      "0.495769888"
    ],
    [
      "0.626776859",
      "0.845810188"
    ],
    [
      "0.639242925",
      "0.283417716"
    ],
    [
      "0.665434112",
      "0.636103887"
    ],
    [
      "0.686776859",
      "0.979769500"
    ],
    [
      "0.688035476",
      "0.080097032"
    ],
    [
      "0.714413782",
      "0.415655761"
    ],
    [
      "0.735842654",
      "0.768111300"
    ],
    [
      "0.753563815",
      "0.158940244"
    ],
    [
      "0.778463486",
      "0.539524926"
    ],
    [
      "0.799312109",
      "0.911442744"
    ],
    [
      "0.810481874",
      "0.324910207"
    ],
    [
      "0.832785333",
      "0.677596378"
    ],
    [
      "0.850427719",
      "0.221178980"
    ],
    [
      "0.877392250",
      "0.452014851"
    ],
    [
      "0.899695708",
      "0.824152153"
    ],
    [
      "0.914180453",
      "0.000000000"
    ],
    [
      "0.940375139",
      "0.594611396"
    ],
    [
      "0.964029860",
      "0.138193998"
    ],
    [
      "0.986739084",
      "0.722808240"
    ],
    [
      "0.999999999",
      "0.933320263"
    ]
  ]
}

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

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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