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 = 30

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 = 30

目标最小化覆盖半径

已证明下界√(A(K)/(π·30))任何布局都不低于 0.079415926620168 · 当前纪录高出下界 22.3%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30 个等圆盘，共同半径 0.097128；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

0.097128315962823

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": [
    [
      "0.409183522",
      "0.852083729"
    ],
    [
      "0.266881662",
      "0.513442501"
    ],
    [
      "0.290515053",
      "0.769633171"
    ],
    [
      "0.872498141",
      "0.495263328"
    ],
    [
      "0.181186381",
      "0.681253949"
    ],
    [
      "0.895018631",
      "0.642966727"
    ],
    [
      "0.353414578",
      "0.267006088"
    ],
    [
      "0.605971395",
      "0.147936545"
    ],
    [
      "0.84925991",
      "0.313981785"
    ],
    [
      "0.744973596",
      "0.431813668"
    ],
    [
      "0.414843248",
      "0.413993182"
    ],
    [
      "0.476969828",
      "0.72179375"
    ],
    [
      "0.727244895",
      "0.561804762"
    ],
    [
      "0.327284914",
      "0.656233986"
    ],
    [
      "0.749413441",
      "0.170319171"
    ],
    [
      "0.161941034",
      "0.255132511"
    ],
    [
      "0.281587653",
      "0.151501545"
    ],
    [
      "0.442616805",
      "0.148001247"
    ],
    [
      "0.260360156",
      "0.360152534"
    ],
    [
      "0.773927658",
      "0.701371698"
    ],
    [
      "0.133552122",
      "0.441731545"
    ],
    [
      "0.59620595",
      "0.630612203"
    ],
    [
      "0.444522793",
      "0.547241897"
    ],
    [
      "0.57792581",
      "0.46055474"
    ],
    [
      "0.52989208",
      "0.29705009"
    ],
    [
      "0.107666917",
      "0.604419548"
    ],
    [
      "0.684309033",
      "0.774489765"
    ],
    [
      "0.596276299",
      "0.819418281"
    ],
    [
      "0.680265029",
      "0.313501979"
    ],
    [
      "0.523084879",
      "0.930581969"
    ]
  ]
}

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": 30
}

当前第一名的答案

{
  "points": [
    [
      "0.409183522",
      "0.852083729"
    ],
    [
      "0.266881662",
      "0.513442501"
    ],
    [
      "0.290515053",
      "0.769633171"
    ],
    [
      "0.872498141",
      "0.495263328"
    ],
    [
      "0.181186381",
      "0.681253949"
    ],
    [
      "0.895018631",
      "0.642966727"
    ],
    [
      "0.353414578",
      "0.267006088"
    ],
    [
      "0.605971395",
      "0.147936545"
    ],
    [
      "0.84925991",
      "0.313981785"
    ],
    [
      "0.744973596",
      "0.431813668"
    ],
    [
      "0.414843248",
      "0.413993182"
    ],
    [
      "0.476969828",
      "0.72179375"
    ],
    [
      "0.727244895",
      "0.561804762"
    ],
    [
      "0.327284914",
      "0.656233986"
    ],
    [
      "0.749413441",
      "0.170319171"
    ],
    [
      "0.161941034",
      "0.255132511"
    ],
    [
      "0.281587653",
      "0.151501545"
    ],
    [
      "0.442616805",
      "0.148001247"
    ],
    [
      "0.260360156",
      "0.360152534"
    ],
    [
      "0.773927658",
      "0.701371698"
    ],
    [
      "0.133552122",
      "0.441731545"
    ],
    [
      "0.59620595",
      "0.630612203"
    ],
    [
      "0.444522793",
      "0.547241897"
    ],
    [
      "0.57792581",
      "0.46055474"
    ],
    [
      "0.52989208",
      "0.29705009"
    ],
    [
      "0.107666917",
      "0.604419548"
    ],
    [
      "0.684309033",
      "0.774489765"
    ],
    [
      "0.596276299",
      "0.819418281"
    ],
    [
      "0.680265029",
      "0.313501979"
    ],
    [
      "0.523084879",
      "0.930581969"
    ]
  ]
}

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

DISCUSSION

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聊思路、贴方法、问为什么卡住。发帖即公开署名，与纪录同一个名字；署名后的 #编号是账号的注册序号，冒不了名。发言资格与实绩绑定：破过一次纪录，就永久拥有发言权。新发言经自动审核后公开。

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