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 = 29

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 = 29

目标最小化覆盖半径

已证明下界√(A(K)/(π·29))任何布局都不低于 0.080773562171047 · 当前纪录高出下界 21.6%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29 个等圆盘，共同半径 0.098259；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

0.098258830191613

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": [
    [
      "0.454523417",
      "0.914214892"
    ],
    [
      "0.269633578",
      "0.520832749"
    ],
    [
      "0.311314534",
      "0.810167551"
    ],
    [
      "0.873353305",
      "0.501591276"
    ],
    [
      "0.197730012",
      "0.695896754"
    ],
    [
      "0.896077719",
      "0.650505822"
    ],
    [
      "0.353500309",
      "0.271294699"
    ],
    [
      "0.60670096",
      "0.149538524"
    ],
    [
      "0.850902373",
      "0.31798419"
    ],
    [
      "0.745898361",
      "0.438811806"
    ],
    [
      "0.415224676",
      "0.421411817"
    ],
    [
      "0.466374548",
      "0.748655571"
    ],
    [
      "0.724837356",
      "0.575403804"
    ],
    [
      "0.329691542",
      "0.651492052"
    ],
    [
      "0.749657354",
      "0.172309294"
    ],
    [
      "0.160331151",
      "0.258492675"
    ],
    [
      "0.281752369",
      "0.153324658"
    ],
    [
      "0.442929043",
      "0.149625214"
    ],
    [
      "0.253117414",
      "0.367132002"
    ],
    [
      "0.759377868",
      "0.730484672"
    ],
    [
      "0.126819578",
      "0.447680705"
    ],
    [
      "0.590318732",
      "0.642444294"
    ],
    [
      "0.442110625",
      "0.569924187"
    ],
    [
      "0.578720441",
      "0.468276143"
    ],
    [
      "0.530692621",
      "0.301637268"
    ],
    [
      "0.112081298",
      "0.612185738"
    ],
    [
      "0.651167278",
      "0.78214754"
    ],
    [
      "0.568727545",
      "0.890854946"
    ],
    [
      "0.680766862",
      "0.318057715"
    ]
  ]
}

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": 29
}

当前第一名的答案

{
  "points": [
    [
      "0.454523417",
      "0.914214892"
    ],
    [
      "0.269633578",
      "0.520832749"
    ],
    [
      "0.311314534",
      "0.810167551"
    ],
    [
      "0.873353305",
      "0.501591276"
    ],
    [
      "0.197730012",
      "0.695896754"
    ],
    [
      "0.896077719",
      "0.650505822"
    ],
    [
      "0.353500309",
      "0.271294699"
    ],
    [
      "0.60670096",
      "0.149538524"
    ],
    [
      "0.850902373",
      "0.31798419"
    ],
    [
      "0.745898361",
      "0.438811806"
    ],
    [
      "0.415224676",
      "0.421411817"
    ],
    [
      "0.466374548",
      "0.748655571"
    ],
    [
      "0.724837356",
      "0.575403804"
    ],
    [
      "0.329691542",
      "0.651492052"
    ],
    [
      "0.749657354",
      "0.172309294"
    ],
    [
      "0.160331151",
      "0.258492675"
    ],
    [
      "0.281752369",
      "0.153324658"
    ],
    [
      "0.442929043",
      "0.149625214"
    ],
    [
      "0.253117414",
      "0.367132002"
    ],
    [
      "0.759377868",
      "0.730484672"
    ],
    [
      "0.126819578",
      "0.447680705"
    ],
    [
      "0.590318732",
      "0.642444294"
    ],
    [
      "0.442110625",
      "0.569924187"
    ],
    [
      "0.578720441",
      "0.468276143"
    ],
    [
      "0.530692621",
      "0.301637268"
    ],
    [
      "0.112081298",
      "0.612185738"
    ],
    [
      "0.651167278",
      "0.78214754"
    ],
    [
      "0.568727545",
      "0.890854946"
    ],
    [
      "0.680766862",
      "0.318057715"
    ]
  ]
}

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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