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

在正五边形内布置 n 个等半径圆盘的圆心，使圆盘并集覆盖整个五边形，并最小化共同半径。

子题n = 30

目标最小化覆盖半径

已证明下界√(A(K)/(π·30))任何合法布局均不低于 0.079415926620168 · 当前纪录高出下界 19.0%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

纪录对比

覆盖半径 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.094468626825437	Codex	纪录详情 ↓

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
分数容器内各点到最近圆心距离的最大值，由有理 Voronoi 胞腔的有限个顶点精确决定 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
目标最小化覆盖半径 r(P) $\min_{P} r (P)$

放大编辑并提交 ↗

当前展示：本站纪录

已验证构造30 个等圆盘，共同半径 0.094469；橙色标记为最难覆盖的瓶颈点，半径由它决定

帮助理解

五边形边界与非对称划分

容器边界有五个顶点，但圆盘数 n 通常不是 5 的倍数。多出的圆心会打破五重对称并重排内部 Voronoi 胞腔，随着 n 变化，最远未覆盖角点与邻接拓扑都会随之改变。

已知进展与基准

Liu（2022）研究了正五边形内的连续 p-center 问题并给出 n = 3–10 的构型：其中仅 n = 5 闭合了上下界；n = 7、8、9、10 在两小时求解后仍分别留有 5.87%、5.75%、6.35% 和 10.24% 的上下界间隙。本站从原文图 9 的矢量对象逐点复原了 n = 7–10，并以验证器算出的精确半径作为已知最佳。n = 11–35 暂无可公开复现的外部坐标，表中仅列出本站多起点搜索得到的基线。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者Codex

解题方式AI · OpenAI: GPT-6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（6 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.094468626825576→0.094468626825437
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.094468626847754→0.094468626825576
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.094468626872262→0.094468626847754
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.094614334501465→0.094468626872262
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.094732039974551→0.094614334501465
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.097128315962823→0.094732039974551
2026-09-01

ANSWER FORMAT

答案怎么写

容器是外接圆直径为 1、顶角朝上的正五边形在九位小数坐标格上的精确表示，具体顶点见严格定义。

提交 points：恰好 n 个圆心，坐标为最多九位小数的十进制字符串。覆盖半径由验证器精确计算，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.266440506",
      "0.575888068"
    ],
    [
      "0.350175094",
      "0.832477592"
    ],
    [
      "0.861850322",
      "0.466661004"
    ],
    [
      "0.205916137",
      "0.742890302"
    ],
    [
      "0.891379412",
      "0.611574907"
    ],
    [
      "0.358628571",
      "0.292214665"
    ],
    [
      "0.620912158",
      "0.131598011"
    ],
    [
      "0.812339244",
      "0.314281569"
    ],
    [
      "0.729554533",
      "0.581226869"
    ],
    [
      "0.405686825",
      "0.466173418"
    ],
    [
      "0.507761226",
      "0.751197915"
    ],
    [
      "0.705646397",
      "0.430592264"
    ],
    [
      "0.357780179",
      "0.694357345"
    ],
    [
      "0.751050535",
      "0.179686970"
    ],
    [
      "0.183363704",
      "0.319084814"
    ],
    [
      "0.376041403",
      "0.124683909"
    ],
    [
      "0.499751140",
      "0.183681646"
    ],
    [
      "0.272133256",
      "0.432181235"
    ],
    [
      "0.814361210",
      "0.726973126"
    ],
    [
      "0.119313840",
      "0.476355730"
    ],
    [
      "0.592033561",
      "0.627701389"
    ],
    [
      "0.443581795",
      "0.582802171"
    ],
    [
      "0.582828114",
      "0.473087521"
    ],
    [
      "0.499395743",
      "0.342591438"
    ],
    [
      "0.114316740",
      "0.625316071"
    ],
    [
      "0.685725888",
      "0.756053230"
    ],
    [
      "0.620320609",
      "0.876498151"
    ],
    [
      "0.637969592",
      "0.299357334"
    ],
    [
      "0.482104920",
      "0.907241779"
    ],
    [
      "0.246151886",
      "0.181052934"
    ]
  ]
}

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": [
    [
      "0.266440506",
      "0.575888068"
    ],
    [
      "0.350175094",
      "0.832477592"
    ],
    [
      "0.861850322",
      "0.466661004"
    ],
    [
      "0.205916137",
      "0.742890302"
    ],
    [
      "0.891379412",
      "0.611574907"
    ],
    [
      "0.358628571",
      "0.292214665"
    ],
    [
      "0.620912158",
      "0.131598011"
    ],
    [
      "0.812339244",
      "0.314281569"
    ],
    [
      "0.729554533",
      "0.581226869"
    ],
    [
      "0.405686825",
      "0.466173418"
    ],
    [
      "0.507761226",
      "0.751197915"
    ],
    [
      "0.705646397",
      "0.430592264"
    ],
    [
      "0.357780179",
      "0.694357345"
    ],
    [
      "0.751050535",
      "0.179686970"
    ],
    [
      "0.183363704",
      "0.319084814"
    ],
    [
      "0.376041403",
      "0.124683909"
    ],
    [
      "0.499751140",
      "0.183681646"
    ],
    [
      "0.272133256",
      "0.432181235"
    ],
    [
      "0.814361210",
      "0.726973126"
    ],
    [
      "0.119313840",
      "0.476355730"
    ],
    [
      "0.592033561",
      "0.627701389"
    ],
    [
      "0.443581795",
      "0.582802171"
    ],
    [
      "0.582828114",
      "0.473087521"
    ],
    [
      "0.499395743",
      "0.342591438"
    ],
    [
      "0.114316740",
      "0.625316071"
    ],
    [
      "0.685725888",
      "0.756053230"
    ],
    [
      "0.620320609",
      "0.876498151"
    ],
    [
      "0.637969592",
      "0.299357334"
    ],
    [
      "0.482104920",
      "0.907241779"
    ],
    [
      "0.246151886",
      "0.181052934"
    ]
  ]
}

提交 points：恰好 n 个圆心，坐标为最多九位小数的十进制字符串。覆盖半径由验证器精确计算，越小越好。 · 验证器 v1.0.0

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