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 = 33

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 = 33

目标最小化覆盖半径

已证明下界√(A(K)/(π·33))任何布局都不低于 0.075720115022603 · 当前纪录高出下界 22.9%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33 个等圆盘，共同半径 0.093026；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

0.093026076233795

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": [
    [
      "0.362120193",
      "0.885556097"
    ],
    [
      "0.266855609",
      "0.532871891"
    ],
    [
      "0.267790701",
      "0.7486512"
    ],
    [
      "0.877445486",
      "0.515189668"
    ],
    [
      "0.178258085",
      "0.671787522"
    ],
    [
      "0.895582445",
      "0.649924286"
    ],
    [
      "0.357040475",
      "0.292192498"
    ],
    [
      "0.597853094",
      "0.161669742"
    ],
    [
      "0.817894753",
      "0.383589735"
    ],
    [
      "0.710147993",
      "0.441183339"
    ],
    [
      "0.434229342",
      "0.430328198"
    ],
    [
      "0.43853556",
      "0.76858291"
    ],
    [
      "0.734359617",
      "0.564972599"
    ],
    [
      "0.351822134",
      "0.630716613"
    ],
    [
      "0.728557633",
      "0.161642503"
    ],
    [
      "0.191952648",
      "0.324230362"
    ],
    [
      "0.357015917",
      "0.128588195"
    ],
    [
      "0.488229148",
      "0.177282908"
    ],
    [
      "0.305785652",
      "0.43721636"
    ],
    [
      "0.774469242",
      "0.706269687"
    ],
    [
      "0.138575583",
      "0.466252211"
    ],
    [
      "0.612244346",
      "0.619133733"
    ],
    [
      "0.446458512",
      "0.542892983"
    ],
    [
      "0.571719184",
      "0.478398251"
    ],
    [
      "0.514579135",
      "0.319617111"
    ],
    [
      "0.103677777",
      "0.610366545"
    ],
    [
      "0.689657642",
      "0.765529035"
    ],
    [
      "0.607174004",
      "0.827708962"
    ],
    [
      "0.644643293",
      "0.318941836"
    ],
    [
      "0.516442112",
      "0.915162499"
    ],
    [
      "0.235164442",
      "0.181714851"
    ],
    [
      "0.525731817",
      "0.681485186"
    ],
    [
      "0.777532067",
      "0.259366158"
    ]
  ]
}

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": 33
}

当前第一名的答案

{
  "points": [
    [
      "0.362120193",
      "0.885556097"
    ],
    [
      "0.266855609",
      "0.532871891"
    ],
    [
      "0.267790701",
      "0.7486512"
    ],
    [
      "0.877445486",
      "0.515189668"
    ],
    [
      "0.178258085",
      "0.671787522"
    ],
    [
      "0.895582445",
      "0.649924286"
    ],
    [
      "0.357040475",
      "0.292192498"
    ],
    [
      "0.597853094",
      "0.161669742"
    ],
    [
      "0.817894753",
      "0.383589735"
    ],
    [
      "0.710147993",
      "0.441183339"
    ],
    [
      "0.434229342",
      "0.430328198"
    ],
    [
      "0.43853556",
      "0.76858291"
    ],
    [
      "0.734359617",
      "0.564972599"
    ],
    [
      "0.351822134",
      "0.630716613"
    ],
    [
      "0.728557633",
      "0.161642503"
    ],
    [
      "0.191952648",
      "0.324230362"
    ],
    [
      "0.357015917",
      "0.128588195"
    ],
    [
      "0.488229148",
      "0.177282908"
    ],
    [
      "0.305785652",
      "0.43721636"
    ],
    [
      "0.774469242",
      "0.706269687"
    ],
    [
      "0.138575583",
      "0.466252211"
    ],
    [
      "0.612244346",
      "0.619133733"
    ],
    [
      "0.446458512",
      "0.542892983"
    ],
    [
      "0.571719184",
      "0.478398251"
    ],
    [
      "0.514579135",
      "0.319617111"
    ],
    [
      "0.103677777",
      "0.610366545"
    ],
    [
      "0.689657642",
      "0.765529035"
    ],
    [
      "0.607174004",
      "0.827708962"
    ],
    [
      "0.644643293",
      "0.318941836"
    ],
    [
      "0.516442112",
      "0.915162499"
    ],
    [
      "0.235164442",
      "0.181714851"
    ],
    [
      "0.525731817",
      "0.681485186"
    ],
    [
      "0.777532067",
      "0.259366158"
    ]
  ]
}

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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