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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 = 39

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 = 39
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 24.2 个点，实际有 39 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

0.379848783696089

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": [
    [
      "0.01025641",
      "0.01025641"
    ],
    [
      "0.030769231",
      "0.41025641"
    ],
    [
      "0.051282051",
      "0.21025641"
    ],
    [
      "0.071794872",
      "0.61025641"
    ],
    [
      "0.092307692",
      "0.11025641"
    ],
    [
      "0.112820513",
      "0.51025641"
    ],
    [
      "0.133333333",
      "0.31025641"
    ],
    [
      "0.153846154",
      "0.71025641"
    ],
    [
      "0.174358974",
      "0.06025641"
    ],
    [
      "0.194871795",
      "0.46025641"
    ],
    [
      "0.215384615",
      "0.26025641"
    ],
    [
      "0.235897436",
      "0.66025641"
    ],
    [
      "0.256410256",
      "0.16025641"
    ],
    [
      "0.276923077",
      "0.56025641"
    ],
    [
      "0.297435897",
      "0.36025641"
    ],
    [
      "0.317948718",
      "0.76025641"
    ],
    [
      "0.338461538",
      "0.03525641"
    ],
    [
      "0.358974359",
      "0.43525641"
    ],
    [
      "0.379487179",
      "0.23525641"
    ],
    [
      "0.4",
      "0.63525641"
    ],
    [
      "0.420512821",
      "0.13525641"
    ],
    [
      "0.441025641",
      "0.53525641"
    ],
    [
      "0.461538462",
      "0.33525641"
    ],
    [
      "0.482051282",
      "0.73525641"
    ],
    [
      "0.502564103",
      "0.08525641"
    ],
    [
      "0.523076923",
      "0.48525641"
    ],
    [
      "0.543589744",
      "0.28525641"
    ],
    [
      "0.564102564",
      "0.68525641"
    ],
    [
      "0.584615385",
      "0.18525641"
    ],
    [
      "0.605128205",
      "0.58525641"
    ],
    [
      "0.625641026",
      "0.38525641"
    ],
    [
      "0.646153846",
      "0.78525641"
    ],
    [
      "0.666666667",
      "0.02275641"
    ],
    [
      "0.687179487",
      "0.42275641"
    ],
    [
      "0.707692308",
      "0.22275641"
    ],
    [
      "0.728205128",
      "0.62275641"
    ],
    [
      "0.748717949",
      "0.12275641"
    ],
    [
      "0.769230769",
      "0.52275641"
    ],
    [
      "0.78974359",
      "0.32275641"
    ]
  ]
}
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": 39
}

当前第一名的答案

{
  "points": [
    [
      "0.01025641",
      "0.01025641"
    ],
    [
      "0.030769231",
      "0.41025641"
    ],
    [
      "0.051282051",
      "0.21025641"
    ],
    [
      "0.071794872",
      "0.61025641"
    ],
    [
      "0.092307692",
      "0.11025641"
    ],
    [
      "0.112820513",
      "0.51025641"
    ],
    [
      "0.133333333",
      "0.31025641"
    ],
    [
      "0.153846154",
      "0.71025641"
    ],
    [
      "0.174358974",
      "0.06025641"
    ],
    [
      "0.194871795",
      "0.46025641"
    ],
    [
      "0.215384615",
      "0.26025641"
    ],
    [
      "0.235897436",
      "0.66025641"
    ],
    [
      "0.256410256",
      "0.16025641"
    ],
    [
      "0.276923077",
      "0.56025641"
    ],
    [
      "0.297435897",
      "0.36025641"
    ],
    [
      "0.317948718",
      "0.76025641"
    ],
    [
      "0.338461538",
      "0.03525641"
    ],
    [
      "0.358974359",
      "0.43525641"
    ],
    [
      "0.379487179",
      "0.23525641"
    ],
    [
      "0.4",
      "0.63525641"
    ],
    [
      "0.420512821",
      "0.13525641"
    ],
    [
      "0.441025641",
      "0.53525641"
    ],
    [
      "0.461538462",
      "0.33525641"
    ],
    [
      "0.482051282",
      "0.73525641"
    ],
    [
      "0.502564103",
      "0.08525641"
    ],
    [
      "0.523076923",
      "0.48525641"
    ],
    [
      "0.543589744",
      "0.28525641"
    ],
    [
      "0.564102564",
      "0.68525641"
    ],
    [
      "0.584615385",
      "0.18525641"
    ],
    [
      "0.605128205",
      "0.58525641"
    ],
    [
      "0.625641026",
      "0.38525641"
    ],
    [
      "0.646153846",
      "0.78525641"
    ],
    [
      "0.666666667",
      "0.02275641"
    ],
    [
      "0.687179487",
      "0.42275641"
    ],
    [
      "0.707692308",
      "0.22275641"
    ],
    [
      "0.728205128",
      "0.62275641"
    ],
    [
      "0.748717949",
      "0.12275641"
    ],
    [
      "0.769230769",
      "0.52275641"
    ],
    [
      "0.78974359",
      "0.32275641"
    ]
  ]
}

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