极值竞技场
中/EN
P54 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破

单位正方形内的最低星偏差 · n = 43

在单位正方形里放 n 个采样点。从原点量起、边平行于坐标轴的每一块矩形，占了多少面积，就该分到多少比例的点；你的分数，是所有这类矩形里最大的那个偏差。把它压到最低。

子题n = 43
目标最小化 最大误差 D*

严格定义

  • 容器单位正方形，左下角是原点 (0, 0)，右上角是 (1, 1)
  • 提交恰好 n 个点的坐标，十进制小数，最多九位；两点不得重合
  • 矩形被比较的矩形是半开的 [0, u) × [0, v)，右上角可落在正方形内任何位置；恰好压在上边或右边上的点算在外面
  • 目标让所有矩形中最大的偏差 D* 尽可能小。上确界在提交坐标的网格上取得，以整数精确计分
放大来摆，然后提交 ↗
1y0
0x1
已验证构造虚线框占 62.0% 的面积，照这个比例应该有 26.7 个点，实际有 43 个，差 0.3799，这就是分数

帮助理解

一个比喻：采样预算

把正方形当成一帧要渲染的画面，这 n 个点就是你全部的采样预算。哪块矩形分到的点比面积应得的多，是预算浪费在同一处；少了，是那块的细节被漏掉。

哪里有优化空间

格点和随机撒点都会在某些矩形上系统性偏置；低偏差构造（Hammersley、van der Corput）压得低得多。但对每个具体的 n，没人知道还能压到哪里。

前沿在哪里

n ≤ 21 的最优解已由 Clément、Doerr、Klamroth 与 Paquete 在 2025 年证明（Proc. Amer. Math. Soc. Ser. B 12: 78–90），所以本站从 n = 22 起；再往上，每一个 n 都是开放的。

当前第一名

0.379853975119011

最大误差 D*

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形：左下角是原点 (0, 0)，右上角是 (1, 1)。被比较的矩形永远从原点量起，右上角可以落在正方形里的任何位置，所以一共有无穷多块矩形要同时满足，而不是某几块。坐标写成小数，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。榜上的数字是 D*，也就是所有矩形里最大的那个误差；分数越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.009302326",
      "0.009302326"
    ],
    [
      "0.027906977",
      "0.409302326"
    ],
    [
      "0.046511628",
      "0.209302326"
    ],
    [
      "0.065116279",
      "0.609302326"
    ],
    [
      "0.08372093",
      "0.109302326"
    ],
    [
      "0.102325581",
      "0.509302326"
    ],
    [
      "0.120930233",
      "0.309302326"
    ],
    [
      "0.139534884",
      "0.709302326"
    ],
    [
      "0.158139535",
      "0.059302326"
    ],
    [
      "0.176744186",
      "0.459302326"
    ],
    [
      "0.195348837",
      "0.259302326"
    ],
    [
      "0.213953488",
      "0.659302326"
    ],
    [
      "0.23255814",
      "0.159302326"
    ],
    [
      "0.251162791",
      "0.559302326"
    ],
    [
      "0.269767442",
      "0.359302326"
    ],
    [
      "0.288372093",
      "0.759302326"
    ],
    [
      "0.306976744",
      "0.034302326"
    ],
    [
      "0.325581395",
      "0.434302326"
    ],
    [
      "0.344186047",
      "0.234302326"
    ],
    [
      "0.362790698",
      "0.634302326"
    ],
    [
      "0.381395349",
      "0.134302326"
    ],
    [
      "0.4",
      "0.534302326"
    ],
    [
      "0.418604651",
      "0.334302326"
    ],
    [
      "0.437209302",
      "0.734302326"
    ],
    [
      "0.455813953",
      "0.084302326"
    ],
    [
      "0.474418605",
      "0.484302326"
    ],
    [
      "0.493023256",
      "0.284302326"
    ],
    [
      "0.511627907",
      "0.684302326"
    ],
    [
      "0.530232558",
      "0.184302326"
    ],
    [
      "0.548837209",
      "0.584302326"
    ],
    [
      "0.56744186",
      "0.384302326"
    ],
    [
      "0.586046512",
      "0.784302326"
    ],
    [
      "0.604651163",
      "0.021802326"
    ],
    [
      "0.623255814",
      "0.421802326"
    ],
    [
      "0.641860465",
      "0.221802326"
    ],
    [
      "0.660465116",
      "0.621802326"
    ],
    [
      "0.679069767",
      "0.121802326"
    ],
    [
      "0.697674419",
      "0.521802326"
    ],
    [
      "0.71627907",
      "0.321802326"
    ],
    [
      "0.734883721",
      "0.721802326"
    ],
    [
      "0.753488372",
      "0.071802326"
    ],
    [
      "0.772093023",
      "0.471802326"
    ],
    [
      "0.790697674",
      "0.271802326"
    ]
  ]
}
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": 43
}

当前第一名的答案

{
  "points": [
    [
      "0.009302326",
      "0.009302326"
    ],
    [
      "0.027906977",
      "0.409302326"
    ],
    [
      "0.046511628",
      "0.209302326"
    ],
    [
      "0.065116279",
      "0.609302326"
    ],
    [
      "0.08372093",
      "0.109302326"
    ],
    [
      "0.102325581",
      "0.509302326"
    ],
    [
      "0.120930233",
      "0.309302326"
    ],
    [
      "0.139534884",
      "0.709302326"
    ],
    [
      "0.158139535",
      "0.059302326"
    ],
    [
      "0.176744186",
      "0.459302326"
    ],
    [
      "0.195348837",
      "0.259302326"
    ],
    [
      "0.213953488",
      "0.659302326"
    ],
    [
      "0.23255814",
      "0.159302326"
    ],
    [
      "0.251162791",
      "0.559302326"
    ],
    [
      "0.269767442",
      "0.359302326"
    ],
    [
      "0.288372093",
      "0.759302326"
    ],
    [
      "0.306976744",
      "0.034302326"
    ],
    [
      "0.325581395",
      "0.434302326"
    ],
    [
      "0.344186047",
      "0.234302326"
    ],
    [
      "0.362790698",
      "0.634302326"
    ],
    [
      "0.381395349",
      "0.134302326"
    ],
    [
      "0.4",
      "0.534302326"
    ],
    [
      "0.418604651",
      "0.334302326"
    ],
    [
      "0.437209302",
      "0.734302326"
    ],
    [
      "0.455813953",
      "0.084302326"
    ],
    [
      "0.474418605",
      "0.484302326"
    ],
    [
      "0.493023256",
      "0.284302326"
    ],
    [
      "0.511627907",
      "0.684302326"
    ],
    [
      "0.530232558",
      "0.184302326"
    ],
    [
      "0.548837209",
      "0.584302326"
    ],
    [
      "0.56744186",
      "0.384302326"
    ],
    [
      "0.586046512",
      "0.784302326"
    ],
    [
      "0.604651163",
      "0.021802326"
    ],
    [
      "0.623255814",
      "0.421802326"
    ],
    [
      "0.641860465",
      "0.221802326"
    ],
    [
      "0.660465116",
      "0.621802326"
    ],
    [
      "0.679069767",
      "0.121802326"
    ],
    [
      "0.697674419",
      "0.521802326"
    ],
    [
      "0.71627907",
      "0.321802326"
    ],
    [
      "0.734883721",
      "0.721802326"
    ],
    [
      "0.753488372",
      "0.071802326"
    ],
    [
      "0.772093023",
      "0.471802326"
    ],
    [
      "0.790697674",
      "0.271802326"
    ]
  ]
}

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。榜上的数字是 D*，也就是所有矩形里最大的那个误差；分数越小越好。 · 验证器 v1.0.0