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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 = 40

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 = 40
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 24.8 个点，实际有 40 个，差 0.3798，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

0.379850000000000

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": [
    [
      "0.01",
      "0.01"
    ],
    [
      "0.03",
      "0.41"
    ],
    [
      "0.05",
      "0.21"
    ],
    [
      "0.07",
      "0.61"
    ],
    [
      "0.09",
      "0.11"
    ],
    [
      "0.11",
      "0.51"
    ],
    [
      "0.13",
      "0.31"
    ],
    [
      "0.15",
      "0.71"
    ],
    [
      "0.17",
      "0.06"
    ],
    [
      "0.19",
      "0.46"
    ],
    [
      "0.21",
      "0.26"
    ],
    [
      "0.23",
      "0.66"
    ],
    [
      "0.25",
      "0.16"
    ],
    [
      "0.27",
      "0.56"
    ],
    [
      "0.29",
      "0.36"
    ],
    [
      "0.31",
      "0.76"
    ],
    [
      "0.33",
      "0.035"
    ],
    [
      "0.35",
      "0.435"
    ],
    [
      "0.37",
      "0.235"
    ],
    [
      "0.39",
      "0.635"
    ],
    [
      "0.41",
      "0.135"
    ],
    [
      "0.43",
      "0.535"
    ],
    [
      "0.45",
      "0.335"
    ],
    [
      "0.47",
      "0.735"
    ],
    [
      "0.49",
      "0.085"
    ],
    [
      "0.51",
      "0.485"
    ],
    [
      "0.53",
      "0.285"
    ],
    [
      "0.55",
      "0.685"
    ],
    [
      "0.57",
      "0.185"
    ],
    [
      "0.59",
      "0.585"
    ],
    [
      "0.61",
      "0.385"
    ],
    [
      "0.63",
      "0.785"
    ],
    [
      "0.65",
      "0.0225"
    ],
    [
      "0.67",
      "0.4225"
    ],
    [
      "0.69",
      "0.2225"
    ],
    [
      "0.71",
      "0.6225"
    ],
    [
      "0.73",
      "0.1225"
    ],
    [
      "0.75",
      "0.5225"
    ],
    [
      "0.77",
      "0.3225"
    ],
    [
      "0.79",
      "0.7225"
    ]
  ]
}
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": 40
}

当前第一名的答案

{
  "points": [
    [
      "0.01",
      "0.01"
    ],
    [
      "0.03",
      "0.41"
    ],
    [
      "0.05",
      "0.21"
    ],
    [
      "0.07",
      "0.61"
    ],
    [
      "0.09",
      "0.11"
    ],
    [
      "0.11",
      "0.51"
    ],
    [
      "0.13",
      "0.31"
    ],
    [
      "0.15",
      "0.71"
    ],
    [
      "0.17",
      "0.06"
    ],
    [
      "0.19",
      "0.46"
    ],
    [
      "0.21",
      "0.26"
    ],
    [
      "0.23",
      "0.66"
    ],
    [
      "0.25",
      "0.16"
    ],
    [
      "0.27",
      "0.56"
    ],
    [
      "0.29",
      "0.36"
    ],
    [
      "0.31",
      "0.76"
    ],
    [
      "0.33",
      "0.035"
    ],
    [
      "0.35",
      "0.435"
    ],
    [
      "0.37",
      "0.235"
    ],
    [
      "0.39",
      "0.635"
    ],
    [
      "0.41",
      "0.135"
    ],
    [
      "0.43",
      "0.535"
    ],
    [
      "0.45",
      "0.335"
    ],
    [
      "0.47",
      "0.735"
    ],
    [
      "0.49",
      "0.085"
    ],
    [
      "0.51",
      "0.485"
    ],
    [
      "0.53",
      "0.285"
    ],
    [
      "0.55",
      "0.685"
    ],
    [
      "0.57",
      "0.185"
    ],
    [
      "0.59",
      "0.585"
    ],
    [
      "0.61",
      "0.385"
    ],
    [
      "0.63",
      "0.785"
    ],
    [
      "0.65",
      "0.0225"
    ],
    [
      "0.67",
      "0.4225"
    ],
    [
      "0.69",
      "0.2225"
    ],
    [
      "0.71",
      "0.6225"
    ],
    [
      "0.73",
      "0.1225"
    ],
    [
      "0.75",
      "0.5225"
    ],
    [
      "0.77",
      "0.3225"
    ],
    [
      "0.79",
      "0.7225"
    ]
  ]
}

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