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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 = 41

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 = 41
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 25.4 个点，实际有 41 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

0.379851278998186

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": [
    [
      "0.009756098",
      "0.009756098"
    ],
    [
      "0.029268293",
      "0.409756098"
    ],
    [
      "0.048780488",
      "0.209756098"
    ],
    [
      "0.068292683",
      "0.609756098"
    ],
    [
      "0.087804878",
      "0.109756098"
    ],
    [
      "0.107317073",
      "0.509756098"
    ],
    [
      "0.126829268",
      "0.309756098"
    ],
    [
      "0.146341463",
      "0.709756098"
    ],
    [
      "0.165853659",
      "0.059756098"
    ],
    [
      "0.185365854",
      "0.459756098"
    ],
    [
      "0.204878049",
      "0.259756098"
    ],
    [
      "0.224390244",
      "0.659756098"
    ],
    [
      "0.243902439",
      "0.159756098"
    ],
    [
      "0.263414634",
      "0.559756098"
    ],
    [
      "0.282926829",
      "0.359756098"
    ],
    [
      "0.302439024",
      "0.759756098"
    ],
    [
      "0.32195122",
      "0.034756098"
    ],
    [
      "0.341463415",
      "0.434756098"
    ],
    [
      "0.36097561",
      "0.234756098"
    ],
    [
      "0.380487805",
      "0.634756098"
    ],
    [
      "0.4",
      "0.134756098"
    ],
    [
      "0.419512195",
      "0.534756098"
    ],
    [
      "0.43902439",
      "0.334756098"
    ],
    [
      "0.458536585",
      "0.734756098"
    ],
    [
      "0.47804878",
      "0.084756098"
    ],
    [
      "0.497560976",
      "0.484756098"
    ],
    [
      "0.517073171",
      "0.284756098"
    ],
    [
      "0.536585366",
      "0.684756098"
    ],
    [
      "0.556097561",
      "0.184756098"
    ],
    [
      "0.575609756",
      "0.584756098"
    ],
    [
      "0.595121951",
      "0.384756098"
    ],
    [
      "0.614634146",
      "0.784756098"
    ],
    [
      "0.634146341",
      "0.022256098"
    ],
    [
      "0.653658537",
      "0.422256098"
    ],
    [
      "0.673170732",
      "0.222256098"
    ],
    [
      "0.692682927",
      "0.622256098"
    ],
    [
      "0.712195122",
      "0.122256098"
    ],
    [
      "0.731707317",
      "0.522256098"
    ],
    [
      "0.751219512",
      "0.322256098"
    ],
    [
      "0.770731707",
      "0.722256098"
    ],
    [
      "0.790243902",
      "0.072256098"
    ]
  ]
}
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": 41
}

当前第一名的答案

{
  "points": [
    [
      "0.009756098",
      "0.009756098"
    ],
    [
      "0.029268293",
      "0.409756098"
    ],
    [
      "0.048780488",
      "0.209756098"
    ],
    [
      "0.068292683",
      "0.609756098"
    ],
    [
      "0.087804878",
      "0.109756098"
    ],
    [
      "0.107317073",
      "0.509756098"
    ],
    [
      "0.126829268",
      "0.309756098"
    ],
    [
      "0.146341463",
      "0.709756098"
    ],
    [
      "0.165853659",
      "0.059756098"
    ],
    [
      "0.185365854",
      "0.459756098"
    ],
    [
      "0.204878049",
      "0.259756098"
    ],
    [
      "0.224390244",
      "0.659756098"
    ],
    [
      "0.243902439",
      "0.159756098"
    ],
    [
      "0.263414634",
      "0.559756098"
    ],
    [
      "0.282926829",
      "0.359756098"
    ],
    [
      "0.302439024",
      "0.759756098"
    ],
    [
      "0.32195122",
      "0.034756098"
    ],
    [
      "0.341463415",
      "0.434756098"
    ],
    [
      "0.36097561",
      "0.234756098"
    ],
    [
      "0.380487805",
      "0.634756098"
    ],
    [
      "0.4",
      "0.134756098"
    ],
    [
      "0.419512195",
      "0.534756098"
    ],
    [
      "0.43902439",
      "0.334756098"
    ],
    [
      "0.458536585",
      "0.734756098"
    ],
    [
      "0.47804878",
      "0.084756098"
    ],
    [
      "0.497560976",
      "0.484756098"
    ],
    [
      "0.517073171",
      "0.284756098"
    ],
    [
      "0.536585366",
      "0.684756098"
    ],
    [
      "0.556097561",
      "0.184756098"
    ],
    [
      "0.575609756",
      "0.584756098"
    ],
    [
      "0.595121951",
      "0.384756098"
    ],
    [
      "0.614634146",
      "0.784756098"
    ],
    [
      "0.634146341",
      "0.022256098"
    ],
    [
      "0.653658537",
      "0.422256098"
    ],
    [
      "0.673170732",
      "0.222256098"
    ],
    [
      "0.692682927",
      "0.622256098"
    ],
    [
      "0.712195122",
      "0.122256098"
    ],
    [
      "0.731707317",
      "0.522256098"
    ],
    [
      "0.751219512",
      "0.322256098"
    ],
    [
      "0.770731707",
      "0.722256098"
    ],
    [
      "0.790243902",
      "0.072256098"
    ]
  ]
}

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