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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 = 45

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 = 45
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 27.9 个点，实际有 45 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

0.379856790122655

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": [
    [
      "0.008888889",
      "0.008888889"
    ],
    [
      "0.026666667",
      "0.408888889"
    ],
    [
      "0.044444444",
      "0.208888889"
    ],
    [
      "0.062222222",
      "0.608888889"
    ],
    [
      "0.08",
      "0.108888889"
    ],
    [
      "0.097777778",
      "0.508888889"
    ],
    [
      "0.115555556",
      "0.308888889"
    ],
    [
      "0.133333333",
      "0.708888889"
    ],
    [
      "0.151111111",
      "0.058888889"
    ],
    [
      "0.168888889",
      "0.458888889"
    ],
    [
      "0.186666667",
      "0.258888889"
    ],
    [
      "0.204444444",
      "0.658888889"
    ],
    [
      "0.222222222",
      "0.158888889"
    ],
    [
      "0.24",
      "0.558888889"
    ],
    [
      "0.257777778",
      "0.358888889"
    ],
    [
      "0.275555556",
      "0.758888889"
    ],
    [
      "0.293333333",
      "0.033888889"
    ],
    [
      "0.311111111",
      "0.433888889"
    ],
    [
      "0.328888889",
      "0.233888889"
    ],
    [
      "0.346666667",
      "0.633888889"
    ],
    [
      "0.364444444",
      "0.133888889"
    ],
    [
      "0.382222222",
      "0.533888889"
    ],
    [
      "0.4",
      "0.333888889"
    ],
    [
      "0.417777778",
      "0.733888889"
    ],
    [
      "0.435555556",
      "0.083888889"
    ],
    [
      "0.453333333",
      "0.483888889"
    ],
    [
      "0.471111111",
      "0.283888889"
    ],
    [
      "0.488888889",
      "0.683888889"
    ],
    [
      "0.506666667",
      "0.183888889"
    ],
    [
      "0.524444444",
      "0.583888889"
    ],
    [
      "0.542222222",
      "0.383888889"
    ],
    [
      "0.56",
      "0.783888889"
    ],
    [
      "0.577777778",
      "0.021388889"
    ],
    [
      "0.595555556",
      "0.421388889"
    ],
    [
      "0.613333333",
      "0.221388889"
    ],
    [
      "0.631111111",
      "0.621388889"
    ],
    [
      "0.648888889",
      "0.121388889"
    ],
    [
      "0.666666667",
      "0.521388889"
    ],
    [
      "0.684444444",
      "0.321388889"
    ],
    [
      "0.702222222",
      "0.721388889"
    ],
    [
      "0.72",
      "0.071388889"
    ],
    [
      "0.737777778",
      "0.471388889"
    ],
    [
      "0.755555556",
      "0.271388889"
    ],
    [
      "0.773333333",
      "0.671388889"
    ],
    [
      "0.791111111",
      "0.171388889"
    ]
  ]
}
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": 45
}

当前第一名的答案

{
  "points": [
    [
      "0.008888889",
      "0.008888889"
    ],
    [
      "0.026666667",
      "0.408888889"
    ],
    [
      "0.044444444",
      "0.208888889"
    ],
    [
      "0.062222222",
      "0.608888889"
    ],
    [
      "0.08",
      "0.108888889"
    ],
    [
      "0.097777778",
      "0.508888889"
    ],
    [
      "0.115555556",
      "0.308888889"
    ],
    [
      "0.133333333",
      "0.708888889"
    ],
    [
      "0.151111111",
      "0.058888889"
    ],
    [
      "0.168888889",
      "0.458888889"
    ],
    [
      "0.186666667",
      "0.258888889"
    ],
    [
      "0.204444444",
      "0.658888889"
    ],
    [
      "0.222222222",
      "0.158888889"
    ],
    [
      "0.24",
      "0.558888889"
    ],
    [
      "0.257777778",
      "0.358888889"
    ],
    [
      "0.275555556",
      "0.758888889"
    ],
    [
      "0.293333333",
      "0.033888889"
    ],
    [
      "0.311111111",
      "0.433888889"
    ],
    [
      "0.328888889",
      "0.233888889"
    ],
    [
      "0.346666667",
      "0.633888889"
    ],
    [
      "0.364444444",
      "0.133888889"
    ],
    [
      "0.382222222",
      "0.533888889"
    ],
    [
      "0.4",
      "0.333888889"
    ],
    [
      "0.417777778",
      "0.733888889"
    ],
    [
      "0.435555556",
      "0.083888889"
    ],
    [
      "0.453333333",
      "0.483888889"
    ],
    [
      "0.471111111",
      "0.283888889"
    ],
    [
      "0.488888889",
      "0.683888889"
    ],
    [
      "0.506666667",
      "0.183888889"
    ],
    [
      "0.524444444",
      "0.583888889"
    ],
    [
      "0.542222222",
      "0.383888889"
    ],
    [
      "0.56",
      "0.783888889"
    ],
    [
      "0.577777778",
      "0.021388889"
    ],
    [
      "0.595555556",
      "0.421388889"
    ],
    [
      "0.613333333",
      "0.221388889"
    ],
    [
      "0.631111111",
      "0.621388889"
    ],
    [
      "0.648888889",
      "0.121388889"
    ],
    [
      "0.666666667",
      "0.521388889"
    ],
    [
      "0.684444444",
      "0.321388889"
    ],
    [
      "0.702222222",
      "0.721388889"
    ],
    [
      "0.72",
      "0.071388889"
    ],
    [
      "0.737777778",
      "0.471388889"
    ],
    [
      "0.755555556",
      "0.271388889"
    ],
    [
      "0.773333333",
      "0.671388889"
    ],
    [
      "0.791111111",
      "0.171388889"
    ]
  ]
}

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