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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 = 50

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 = 50
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 31.0 个点，实际有 50 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

0.379864000000000

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": [
    [
      "0.008",
      "0.008"
    ],
    [
      "0.024",
      "0.408"
    ],
    [
      "0.04",
      "0.208"
    ],
    [
      "0.056",
      "0.608"
    ],
    [
      "0.072",
      "0.108"
    ],
    [
      "0.088",
      "0.508"
    ],
    [
      "0.104",
      "0.308"
    ],
    [
      "0.12",
      "0.708"
    ],
    [
      "0.136",
      "0.058"
    ],
    [
      "0.152",
      "0.458"
    ],
    [
      "0.168",
      "0.258"
    ],
    [
      "0.184",
      "0.658"
    ],
    [
      "0.2",
      "0.158"
    ],
    [
      "0.216",
      "0.558"
    ],
    [
      "0.232",
      "0.358"
    ],
    [
      "0.248",
      "0.758"
    ],
    [
      "0.264",
      "0.033"
    ],
    [
      "0.28",
      "0.433"
    ],
    [
      "0.296",
      "0.233"
    ],
    [
      "0.312",
      "0.633"
    ],
    [
      "0.328",
      "0.133"
    ],
    [
      "0.344",
      "0.533"
    ],
    [
      "0.36",
      "0.333"
    ],
    [
      "0.376",
      "0.733"
    ],
    [
      "0.392",
      "0.083"
    ],
    [
      "0.408",
      "0.483"
    ],
    [
      "0.424",
      "0.283"
    ],
    [
      "0.44",
      "0.683"
    ],
    [
      "0.456",
      "0.183"
    ],
    [
      "0.472",
      "0.583"
    ],
    [
      "0.488",
      "0.383"
    ],
    [
      "0.504",
      "0.783"
    ],
    [
      "0.52",
      "0.0205"
    ],
    [
      "0.536",
      "0.4205"
    ],
    [
      "0.552",
      "0.2205"
    ],
    [
      "0.568",
      "0.6205"
    ],
    [
      "0.584",
      "0.1205"
    ],
    [
      "0.6",
      "0.5205"
    ],
    [
      "0.616",
      "0.3205"
    ],
    [
      "0.632",
      "0.7205"
    ],
    [
      "0.648",
      "0.0705"
    ],
    [
      "0.664",
      "0.4705"
    ],
    [
      "0.68",
      "0.2705"
    ],
    [
      "0.696",
      "0.6705"
    ],
    [
      "0.712",
      "0.1705"
    ],
    [
      "0.728",
      "0.5705"
    ],
    [
      "0.744",
      "0.3705"
    ],
    [
      "0.76",
      "0.7705"
    ],
    [
      "0.776",
      "0.0455"
    ],
    [
      "0.792",
      "0.4455"
    ]
  ]
}
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": [
    [
      "0.008",
      "0.008"
    ],
    [
      "0.024",
      "0.408"
    ],
    [
      "0.04",
      "0.208"
    ],
    [
      "0.056",
      "0.608"
    ],
    [
      "0.072",
      "0.108"
    ],
    [
      "0.088",
      "0.508"
    ],
    [
      "0.104",
      "0.308"
    ],
    [
      "0.12",
      "0.708"
    ],
    [
      "0.136",
      "0.058"
    ],
    [
      "0.152",
      "0.458"
    ],
    [
      "0.168",
      "0.258"
    ],
    [
      "0.184",
      "0.658"
    ],
    [
      "0.2",
      "0.158"
    ],
    [
      "0.216",
      "0.558"
    ],
    [
      "0.232",
      "0.358"
    ],
    [
      "0.248",
      "0.758"
    ],
    [
      "0.264",
      "0.033"
    ],
    [
      "0.28",
      "0.433"
    ],
    [
      "0.296",
      "0.233"
    ],
    [
      "0.312",
      "0.633"
    ],
    [
      "0.328",
      "0.133"
    ],
    [
      "0.344",
      "0.533"
    ],
    [
      "0.36",
      "0.333"
    ],
    [
      "0.376",
      "0.733"
    ],
    [
      "0.392",
      "0.083"
    ],
    [
      "0.408",
      "0.483"
    ],
    [
      "0.424",
      "0.283"
    ],
    [
      "0.44",
      "0.683"
    ],
    [
      "0.456",
      "0.183"
    ],
    [
      "0.472",
      "0.583"
    ],
    [
      "0.488",
      "0.383"
    ],
    [
      "0.504",
      "0.783"
    ],
    [
      "0.52",
      "0.0205"
    ],
    [
      "0.536",
      "0.4205"
    ],
    [
      "0.552",
      "0.2205"
    ],
    [
      "0.568",
      "0.6205"
    ],
    [
      "0.584",
      "0.1205"
    ],
    [
      "0.6",
      "0.5205"
    ],
    [
      "0.616",
      "0.3205"
    ],
    [
      "0.632",
      "0.7205"
    ],
    [
      "0.648",
      "0.0705"
    ],
    [
      "0.664",
      "0.4705"
    ],
    [
      "0.68",
      "0.2705"
    ],
    [
      "0.696",
      "0.6705"
    ],
    [
      "0.712",
      "0.1705"
    ],
    [
      "0.728",
      "0.5705"
    ],
    [
      "0.744",
      "0.3705"
    ],
    [
      "0.76",
      "0.7705"
    ],
    [
      "0.776",
      "0.0455"
    ],
    [
      "0.792",
      "0.4455"
    ]
  ]
}

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