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 = 27

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 = 27

目标最小化覆盖半径

已证明下界√(A(K)/(π·27))任何布局都不低于 0.083711736870843 · 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27 个等圆盘，共同半径 0.102950；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

0.102949780104476

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": [
    [
      "0.498030525",
      "0.924364963"
    ],
    [
      "0.276379031",
      "0.516551874"
    ],
    [
      "0.313141627",
      "0.861517462"
    ],
    [
      "0.850828771",
      "0.506196285"
    ],
    [
      "0.21567276",
      "0.700995647"
    ],
    [
      "0.900102943",
      "0.646348929"
    ],
    [
      "0.39256985",
      "0.291019995"
    ],
    [
      "0.603078859",
      "0.138172281"
    ],
    [
      "0.858619034",
      "0.319954038"
    ],
    [
      "0.720292569",
      "0.365306265"
    ],
    [
      "0.432617911",
      "0.41506431"
    ],
    [
      "0.4638713",
      "0.77104076"
    ],
    [
      "0.733927551",
      "0.559906133"
    ],
    [
      "0.340108574",
      "0.681061508"
    ],
    [
      "0.739113016",
      "0.182332382"
    ],
    [
      "0.161402753",
      "0.246196498"
    ],
    [
      "0.291077734",
      "0.151644873"
    ],
    [
      "0.444800873",
      "0.170950797"
    ],
    [
      "0.265443814",
      "0.35086146"
    ],
    [
      "0.755110548",
      "0.733691826"
    ],
    [
      "0.127604912",
      "0.444595841"
    ],
    [
      "0.589046548",
      "0.646539545"
    ],
    [
      "0.447685607",
      "0.577526922"
    ],
    [
      "0.593526707",
      "0.461598053"
    ],
    [
      "0.586098205",
      "0.298857265"
    ],
    [
      "0.11382366",
      "0.617463509"
    ],
    [
      "0.626714502",
      "0.826977"
    ]
  ]
}

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": 27
}

当前第一名的答案

{
  "points": [
    [
      "0.498030525",
      "0.924364963"
    ],
    [
      "0.276379031",
      "0.516551874"
    ],
    [
      "0.313141627",
      "0.861517462"
    ],
    [
      "0.850828771",
      "0.506196285"
    ],
    [
      "0.21567276",
      "0.700995647"
    ],
    [
      "0.900102943",
      "0.646348929"
    ],
    [
      "0.39256985",
      "0.291019995"
    ],
    [
      "0.603078859",
      "0.138172281"
    ],
    [
      "0.858619034",
      "0.319954038"
    ],
    [
      "0.720292569",
      "0.365306265"
    ],
    [
      "0.432617911",
      "0.41506431"
    ],
    [
      "0.4638713",
      "0.77104076"
    ],
    [
      "0.733927551",
      "0.559906133"
    ],
    [
      "0.340108574",
      "0.681061508"
    ],
    [
      "0.739113016",
      "0.182332382"
    ],
    [
      "0.161402753",
      "0.246196498"
    ],
    [
      "0.291077734",
      "0.151644873"
    ],
    [
      "0.444800873",
      "0.170950797"
    ],
    [
      "0.265443814",
      "0.35086146"
    ],
    [
      "0.755110548",
      "0.733691826"
    ],
    [
      "0.127604912",
      "0.444595841"
    ],
    [
      "0.589046548",
      "0.646539545"
    ],
    [
      "0.447685607",
      "0.577526922"
    ],
    [
      "0.593526707",
      "0.461598053"
    ],
    [
      "0.586098205",
      "0.298857265"
    ],
    [
      "0.11382366",
      "0.617463509"
    ],
    [
      "0.626714502",
      "0.826977"
    ]
  ]
}

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

讨论区

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。

还没有帖子。第一个聊聊这道题的思路？