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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 = 31

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 = 31
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 60.0% 的面积，照这个比例应该有 18.6 个点，实际有 31 个，差 0.3995，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

0.399521331941208

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": [
    [
      "0.012903226",
      "0.012903226"
    ],
    [
      "0.038709677",
      "0.412903226"
    ],
    [
      "0.064516129",
      "0.212903226"
    ],
    [
      "0.090322581",
      "0.612903226"
    ],
    [
      "0.116129032",
      "0.112903226"
    ],
    [
      "0.141935484",
      "0.512903226"
    ],
    [
      "0.167741935",
      "0.312903226"
    ],
    [
      "0.193548387",
      "0.712903226"
    ],
    [
      "0.219354839",
      "0.062903226"
    ],
    [
      "0.24516129",
      "0.462903226"
    ],
    [
      "0.270967742",
      "0.262903226"
    ],
    [
      "0.296774194",
      "0.662903226"
    ],
    [
      "0.322580645",
      "0.162903226"
    ],
    [
      "0.348387097",
      "0.562903226"
    ],
    [
      "0.374193548",
      "0.362903226"
    ],
    [
      "0.4",
      "0.762903226"
    ],
    [
      "0.425806452",
      "0.037903226"
    ],
    [
      "0.451612903",
      "0.437903226"
    ],
    [
      "0.477419355",
      "0.237903226"
    ],
    [
      "0.503225806",
      "0.637903226"
    ],
    [
      "0.529032258",
      "0.137903226"
    ],
    [
      "0.55483871",
      "0.537903226"
    ],
    [
      "0.580645161",
      "0.337903226"
    ],
    [
      "0.606451613",
      "0.737903226"
    ],
    [
      "0.632258065",
      "0.087903226"
    ],
    [
      "0.658064516",
      "0.487903226"
    ],
    [
      "0.683870968",
      "0.287903226"
    ],
    [
      "0.709677419",
      "0.687903226"
    ],
    [
      "0.735483871",
      "0.187903226"
    ],
    [
      "0.761290323",
      "0.587903226"
    ],
    [
      "0.787096774",
      "0.387903226"
    ]
  ]
}
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": 31
}

当前第一名的答案

{
  "points": [
    [
      "0.012903226",
      "0.012903226"
    ],
    [
      "0.038709677",
      "0.412903226"
    ],
    [
      "0.064516129",
      "0.212903226"
    ],
    [
      "0.090322581",
      "0.612903226"
    ],
    [
      "0.116129032",
      "0.112903226"
    ],
    [
      "0.141935484",
      "0.512903226"
    ],
    [
      "0.167741935",
      "0.312903226"
    ],
    [
      "0.193548387",
      "0.712903226"
    ],
    [
      "0.219354839",
      "0.062903226"
    ],
    [
      "0.24516129",
      "0.462903226"
    ],
    [
      "0.270967742",
      "0.262903226"
    ],
    [
      "0.296774194",
      "0.662903226"
    ],
    [
      "0.322580645",
      "0.162903226"
    ],
    [
      "0.348387097",
      "0.562903226"
    ],
    [
      "0.374193548",
      "0.362903226"
    ],
    [
      "0.4",
      "0.762903226"
    ],
    [
      "0.425806452",
      "0.037903226"
    ],
    [
      "0.451612903",
      "0.437903226"
    ],
    [
      "0.477419355",
      "0.237903226"
    ],
    [
      "0.503225806",
      "0.637903226"
    ],
    [
      "0.529032258",
      "0.137903226"
    ],
    [
      "0.55483871",
      "0.537903226"
    ],
    [
      "0.580645161",
      "0.337903226"
    ],
    [
      "0.606451613",
      "0.737903226"
    ],
    [
      "0.632258065",
      "0.087903226"
    ],
    [
      "0.658064516",
      "0.487903226"
    ],
    [
      "0.683870968",
      "0.287903226"
    ],
    [
      "0.709677419",
      "0.687903226"
    ],
    [
      "0.735483871",
      "0.187903226"
    ],
    [
      "0.761290323",
      "0.587903226"
    ],
    [
      "0.787096774",
      "0.387903226"
    ]
  ]
}

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