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 = 31

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 = 31

目标最小化覆盖半径

已证明下界√(A(K)/(π·31))任何布局都不低于 0.078124524675343 · 当前纪录高出下界 22.4%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31 个等圆盘，共同半径 0.095587；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

0.095587149715447

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": [
    [
      "0.412313787",
      "0.855036842"
    ],
    [
      "0.274402281",
      "0.528348235"
    ],
    [
      "0.296540927",
      "0.774224153"
    ],
    [
      "0.873873439",
      "0.497711041"
    ],
    [
      "0.19247563",
      "0.687938463"
    ],
    [
      "0.896066163",
      "0.643264741"
    ],
    [
      "0.371957348",
      "0.293961481"
    ],
    [
      "0.628291628",
      "0.157145031"
    ],
    [
      "0.852619598",
      "0.318274303"
    ],
    [
      "0.746134634",
      "0.428748749"
    ],
    [
      "0.425728654",
      "0.431581124"
    ],
    [
      "0.480167645",
      "0.726676485"
    ],
    [
      "0.730405788",
      "0.564025517"
    ],
    [
      "0.343926938",
      "0.667692399"
    ],
    [
      "0.753442416",
      "0.175574682"
    ],
    [
      "0.197617764",
      "0.322046432"
    ],
    [
      "0.368433321",
      "0.127544045"
    ],
    [
      "0.506792047",
      "0.177795681"
    ],
    [
      "0.296286218",
      "0.413241421"
    ],
    [
      "0.777767703",
      "0.697484159"
    ],
    [
      "0.134973987",
      "0.465704569"
    ],
    [
      "0.599528908",
      "0.635345466"
    ],
    [
      "0.450504266",
      "0.554910147"
    ],
    [
      "0.582849397",
      "0.467935231"
    ],
    [
      "0.545359331",
      "0.308958119"
    ],
    [
      "0.111001653",
      "0.614055527"
    ],
    [
      "0.691308107",
      "0.775440107"
    ],
    [
      "0.596776476",
      "0.821156745"
    ],
    [
      "0.689505551",
      "0.320611562"
    ],
    [
      "0.523811391",
      "0.931698608"
    ],
    [
      "0.242547173",
      "0.18376793"
    ]
  ]
}

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": 31
}

当前第一名的答案

{
  "points": [
    [
      "0.412313787",
      "0.855036842"
    ],
    [
      "0.274402281",
      "0.528348235"
    ],
    [
      "0.296540927",
      "0.774224153"
    ],
    [
      "0.873873439",
      "0.497711041"
    ],
    [
      "0.19247563",
      "0.687938463"
    ],
    [
      "0.896066163",
      "0.643264741"
    ],
    [
      "0.371957348",
      "0.293961481"
    ],
    [
      "0.628291628",
      "0.157145031"
    ],
    [
      "0.852619598",
      "0.318274303"
    ],
    [
      "0.746134634",
      "0.428748749"
    ],
    [
      "0.425728654",
      "0.431581124"
    ],
    [
      "0.480167645",
      "0.726676485"
    ],
    [
      "0.730405788",
      "0.564025517"
    ],
    [
      "0.343926938",
      "0.667692399"
    ],
    [
      "0.753442416",
      "0.175574682"
    ],
    [
      "0.197617764",
      "0.322046432"
    ],
    [
      "0.368433321",
      "0.127544045"
    ],
    [
      "0.506792047",
      "0.177795681"
    ],
    [
      "0.296286218",
      "0.413241421"
    ],
    [
      "0.777767703",
      "0.697484159"
    ],
    [
      "0.134973987",
      "0.465704569"
    ],
    [
      "0.599528908",
      "0.635345466"
    ],
    [
      "0.450504266",
      "0.554910147"
    ],
    [
      "0.582849397",
      "0.467935231"
    ],
    [
      "0.545359331",
      "0.308958119"
    ],
    [
      "0.111001653",
      "0.614055527"
    ],
    [
      "0.691308107",
      "0.775440107"
    ],
    [
      "0.596776476",
      "0.821156745"
    ],
    [
      "0.689505551",
      "0.320611562"
    ],
    [
      "0.523811391",
      "0.931698608"
    ],
    [
      "0.242547173",
      "0.18376793"
    ]
  ]
}

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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