P75 · 极值构型 · 本站原创 · 应用前沿 · 难

用 n 个等圆覆盖正五边形 · n = 32

把 n 个等半径圆盘的圆心放进正五边形，使圆盘并集覆盖整个五边形，并把共同半径压到最小。

子题n = 32

目标最小化覆盖半径

已证明下界√(A(K)/(π·32))任何布局都不低于 0.076894140305885 · 当前纪录高出下界 23.0%这里 K 是验证器使用的固定精度正五边形，A(K) 是它的面积。面积下界：nπr² 至少等于 A(K)，所以 r ≥ √(A(K)/(πn))；页面数值按 A(K) 的精确有理值向下取整

严格定义

容器顶点依次为 (0.5,1)、(0.024471742,0.654508497)、(0.206107374,0.095491503)、(0.793892626,0.095491503)、(0.975528258,0.654508497) 的闭凸多边形
提交恰好 n 个互不重合、且位于容器内的点
分数容器中最难覆盖的位置到最近圆心的距离；由有理 Voronoi 多边形的有限顶点精确决定 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
目标让覆盖半径尽可能小 $\min_{P} r (P)$

放大来摆，然后提交 ↗

已验证构造32 个等圆盘，共同半径 0.094614；标出的那一点是最难够到的位置，半径由它决定

帮助理解

五重对称不等于答案也五重对称

边界有五个角，但 n 往往不是 5 的倍数。额外圆心放在哪一侧，会迫使内部 Voronoi 结构重新分配；随着 n 改变，最远空洞的位置和邻接拓扑也会改变，并不存在一套显然可以反复复制的周期图案。

公开研究只解决了一个小规模

Liu（2022）研究了正五边形中的连续 p-center，并公开展示了 n=3–10 的构型。只有 n=5 的上下界闭合；n=7、8、9、10 在两小时计算后仍分别留下 5.87%、5.75%、6.35%、10.24% 的差距。本站已从 Figure 9 的矢量对象逐点复原 n=7–10，并以验证器实际算出的半径作为 known best。n=11–35 暂未找到可公开复现的外部构型，因此只提供本站多起点搜索基准，不冒充 SOTA。

查看来源

当前第一名

0.094614485368856

覆盖半径

答案来源MinMax Arena

解题方式本站离线搜索

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

容器是外接圆直径为 1、一个顶点朝上的正五边形在九位坐标格上的明确代表。页面严格定义列出验证器实际使用的五个顶点。

提交 points：恰好 n 个圆心，坐标是最多九位小数的字符串。半径由验证器精确算出，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.394490728",
      "0.845022438"
    ],
    [
      "0.269486728",
      "0.527660347"
    ],
    [
      "0.2850262",
      "0.761805656"
    ],
    [
      "0.876387436",
      "0.492493735"
    ],
    [
      "0.184561501",
      "0.681622949"
    ],
    [
      "0.899665809",
      "0.641764223"
    ],
    [
      "0.371619271",
      "0.290741294"
    ],
    [
      "0.626650927",
      "0.154598487"
    ],
    [
      "0.851314615",
      "0.314166949"
    ],
    [
      "0.745342063",
      "0.436640848"
    ],
    [
      "0.432731227",
      "0.407537997"
    ],
    [
      "0.469992123",
      "0.726173941"
    ],
    [
      "0.74028538",
      "0.551886882"
    ],
    [
      "0.348911323",
      "0.64539321"
    ],
    [
      "0.752132893",
      "0.174791106"
    ],
    [
      "0.198494346",
      "0.318856193"
    ],
    [
      "0.369053475",
      "0.125190364"
    ],
    [
      "0.505801706",
      "0.177595927"
    ],
    [
      "0.307509752",
      "0.432788341"
    ],
    [
      "0.779856562",
      "0.70166486"
    ],
    [
      "0.134435959",
      "0.460887609"
    ],
    [
      "0.617876208",
      "0.610814112"
    ],
    [
      "0.451467809",
      "0.541410982"
    ],
    [
      "0.578318173",
      "0.460988957"
    ],
    [
      "0.546073473",
      "0.303409749"
    ],
    [
      "0.108334306",
      "0.610863029"
    ],
    [
      "0.690708808",
      "0.766434494"
    ],
    [
      "0.599945421",
      "0.836740691"
    ],
    [
      "0.688479337",
      "0.317971261"
    ],
    [
      "0.50341375",
      "0.906875046"
    ],
    [
      "0.242741436",
      "0.182677653"
    ],
    [
      "0.52958655",
      "0.681111193"
    ]
  ]
}

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": 32
}

当前第一名的答案

{
  "points": [
    [
      "0.394490728",
      "0.845022438"
    ],
    [
      "0.269486728",
      "0.527660347"
    ],
    [
      "0.2850262",
      "0.761805656"
    ],
    [
      "0.876387436",
      "0.492493735"
    ],
    [
      "0.184561501",
      "0.681622949"
    ],
    [
      "0.899665809",
      "0.641764223"
    ],
    [
      "0.371619271",
      "0.290741294"
    ],
    [
      "0.626650927",
      "0.154598487"
    ],
    [
      "0.851314615",
      "0.314166949"
    ],
    [
      "0.745342063",
      "0.436640848"
    ],
    [
      "0.432731227",
      "0.407537997"
    ],
    [
      "0.469992123",
      "0.726173941"
    ],
    [
      "0.74028538",
      "0.551886882"
    ],
    [
      "0.348911323",
      "0.64539321"
    ],
    [
      "0.752132893",
      "0.174791106"
    ],
    [
      "0.198494346",
      "0.318856193"
    ],
    [
      "0.369053475",
      "0.125190364"
    ],
    [
      "0.505801706",
      "0.177595927"
    ],
    [
      "0.307509752",
      "0.432788341"
    ],
    [
      "0.779856562",
      "0.70166486"
    ],
    [
      "0.134435959",
      "0.460887609"
    ],
    [
      "0.617876208",
      "0.610814112"
    ],
    [
      "0.451467809",
      "0.541410982"
    ],
    [
      "0.578318173",
      "0.460988957"
    ],
    [
      "0.546073473",
      "0.303409749"
    ],
    [
      "0.108334306",
      "0.610863029"
    ],
    [
      "0.690708808",
      "0.766434494"
    ],
    [
      "0.599945421",
      "0.836740691"
    ],
    [
      "0.688479337",
      "0.317971261"
    ],
    [
      "0.50341375",
      "0.906875046"
    ],
    [
      "0.242741436",
      "0.182677653"
    ],
    [
      "0.52958655",
      "0.681111193"
    ]
  ]
}

提交 points：恰好 n 个圆心，坐标是最多九位小数的字符串。半径由验证器精确算出，越小越好。 · 验证器 v1.0.0

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