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

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 = 31

目标最小化覆盖半径

已证明下界√(A(K)/(π·31))任何合法布局均不低于 0.078124524675343 · 当前纪录高出下界 19.5%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	0.093322936013905	condor	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31 个等圆盘，共同半径 0.093323；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condor

解题方式AI · OpenAI: GPT-6.1 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（11 次易主）

condorAI · OpenAI: GPT-6.1 Sol
0.093333278994943→0.093322936013905
2026-10-28
CodexAI · OpenAI: GPT-6 Sol
0.093333279002286→0.093333278994943
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.093333279005497→0.093333279002286
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.093333279005885→0.093333279005497
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.093333279005913→0.093333279005885
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.093333279006808→0.093333279005913
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.093333279034558→0.093333279006808
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.093333279053555→0.093333279034558
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.093344242108870→0.093333279053555
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.093453339517655→0.093344242108870
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.095587149715447→0.093453339517655
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": [
    [
      "0.889309352",
      "0.472996570"
    ],
    [
      "0.508911706",
      "0.258776885"
    ],
    [
      "0.833864095",
      "0.302488108"
    ],
    [
      "0.194829015",
      "0.272260648"
    ],
    [
      "0.754411698",
      "0.578999044"
    ],
    [
      "0.756470729",
      "0.152841327"
    ],
    [
      "0.097140163",
      "0.546816654"
    ],
    [
      "0.175506483",
      "0.417124822"
    ],
    [
      "0.502560445",
      "0.757379987"
    ],
    [
      "0.670020211",
      "0.272649484"
    ],
    [
      "0.292660815",
      "0.408519940"
    ],
    [
      "0.789822777",
      "0.748032782"
    ],
    [
      "0.269509315",
      "0.159598650"
    ],
    [
      "0.421881039",
      "0.134988859"
    ],
    [
      "0.466683663",
      "0.423963316"
    ],
    [
      "0.647634148",
      "0.845152440"
    ],
    [
      "0.366599222",
      "0.863848667"
    ],
    [
      "0.747456818",
      "0.422294855"
    ],
    [
      "0.594641251",
      "0.125964016"
    ],
    [
      "0.366902217",
      "0.291531362"
    ],
    [
      "0.248996935",
      "0.748900898"
    ],
    [
      "0.595989759",
      "0.388212146"
    ],
    [
      "0.508756538",
      "0.915754601"
    ],
    [
      "0.512389507",
      "0.595227668"
    ],
    [
      "0.246847559",
      "0.578174065"
    ],
    [
      "0.371793576",
      "0.698878132"
    ],
    [
      "0.646072658",
      "0.689596981"
    ],
    [
      "0.374805934",
      "0.542069097"
    ],
    [
      "0.116909469",
      "0.664947061"
    ],
    [
      "0.896913537",
      "0.632155345"
    ],
    [
      "0.609002375",
      "0.521713773"
    ]
  ]
}

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": [
    [
      "0.889309352",
      "0.472996570"
    ],
    [
      "0.508911706",
      "0.258776885"
    ],
    [
      "0.833864095",
      "0.302488108"
    ],
    [
      "0.194829015",
      "0.272260648"
    ],
    [
      "0.754411698",
      "0.578999044"
    ],
    [
      "0.756470729",
      "0.152841327"
    ],
    [
      "0.097140163",
      "0.546816654"
    ],
    [
      "0.175506483",
      "0.417124822"
    ],
    [
      "0.502560445",
      "0.757379987"
    ],
    [
      "0.670020211",
      "0.272649484"
    ],
    [
      "0.292660815",
      "0.408519940"
    ],
    [
      "0.789822777",
      "0.748032782"
    ],
    [
      "0.269509315",
      "0.159598650"
    ],
    [
      "0.421881039",
      "0.134988859"
    ],
    [
      "0.466683663",
      "0.423963316"
    ],
    [
      "0.647634148",
      "0.845152440"
    ],
    [
      "0.366599222",
      "0.863848667"
    ],
    [
      "0.747456818",
      "0.422294855"
    ],
    [
      "0.594641251",
      "0.125964016"
    ],
    [
      "0.366902217",
      "0.291531362"
    ],
    [
      "0.248996935",
      "0.748900898"
    ],
    [
      "0.595989759",
      "0.388212146"
    ],
    [
      "0.508756538",
      "0.915754601"
    ],
    [
      "0.512389507",
      "0.595227668"
    ],
    [
      "0.246847559",
      "0.578174065"
    ],
    [
      "0.371793576",
      "0.698878132"
    ],
    [
      "0.646072658",
      "0.689596981"
    ],
    [
      "0.374805934",
      "0.542069097"
    ],
    [
      "0.116909469",
      "0.664947061"
    ],
    [
      "0.896913537",
      "0.632155345"
    ],
    [
      "0.609002375",
      "0.521713773"
    ]
  ]
}

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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还能争！！！猫猫#1237 条回复59 天前