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 = 35

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 = 35

目标最小化覆盖半径

已证明下界√(A(K)/(π·35))任何合法布局均不低于 0.073524861107014 · 当前纪录高出下界 18.8%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	0.087339093872681	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35 个等圆盘，共同半径 0.087339；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（9 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.087339093874447→0.087339093872681
2026-10-23
CodexAI · OpenAI: GPT-6 Sol
0.087339093874497→0.087339093874447
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.087339093879914→0.087339093874497
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.087339093889716→0.087339093879914
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.087339093905697→0.087339093889716
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.087339093930378→0.087339093905697
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.087563292066063→0.087339093930378
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.087754277588095→0.087563292066063
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.090029574630320→0.087754277588095
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": [
    [
      "0.361758204",
      "0.839058697"
    ],
    [
      "0.268073170",
      "0.749036620"
    ],
    [
      "0.915081872",
      "0.513205525"
    ],
    [
      "0.150567250",
      "0.688880468"
    ],
    [
      "0.896647273",
      "0.656046643"
    ],
    [
      "0.605169501",
      "0.170354530"
    ],
    [
      "0.474726791",
      "0.113521648"
    ],
    [
      "0.673852084",
      "0.439762633"
    ],
    [
      "0.432842376",
      "0.394699418"
    ],
    [
      "0.778176881",
      "0.540374761"
    ],
    [
      "0.364363802",
      "0.676240657"
    ],
    [
      "0.726573451",
      "0.137779228"
    ],
    [
      "0.172844433",
      "0.319183549"
    ],
    [
      "0.326092502",
      "0.155797765"
    ],
    [
      "0.821182177",
      "0.392792538"
    ],
    [
      "0.783994299",
      "0.698579488"
    ],
    [
      "0.139998650",
      "0.465621868"
    ],
    [
      "0.660731954",
      "0.595115847"
    ],
    [
      "0.391706657",
      "0.531811473"
    ],
    [
      "0.547512945",
      "0.480317244"
    ],
    [
      "0.565948886",
      "0.323016416"
    ],
    [
      "0.093301644",
      "0.600744551"
    ],
    [
      "0.716868376",
      "0.822613239"
    ],
    [
      "0.570996189",
      "0.869943750"
    ],
    [
      "0.698439082",
      "0.298340421"
    ],
    [
      "0.478805015",
      "0.923854166"
    ],
    [
      "0.227543050",
      "0.175346826"
    ],
    [
      "0.511693037",
      "0.615481685"
    ],
    [
      "0.829427394",
      "0.238693725"
    ],
    [
      "0.320199148",
      "0.293668862"
    ],
    [
      "0.626545653",
      "0.719281816"
    ],
    [
      "0.462102151",
      "0.245050274"
    ],
    [
      "0.281407560",
      "0.433898177"
    ],
    [
      "0.245459230",
      "0.581289231"
    ],
    [
      "0.481961782",
      "0.760834354"
    ]
  ]
}

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": 35
}

当前第一名的答案

{
  "points": [
    [
      "0.361758204",
      "0.839058697"
    ],
    [
      "0.268073170",
      "0.749036620"
    ],
    [
      "0.915081872",
      "0.513205525"
    ],
    [
      "0.150567250",
      "0.688880468"
    ],
    [
      "0.896647273",
      "0.656046643"
    ],
    [
      "0.605169501",
      "0.170354530"
    ],
    [
      "0.474726791",
      "0.113521648"
    ],
    [
      "0.673852084",
      "0.439762633"
    ],
    [
      "0.432842376",
      "0.394699418"
    ],
    [
      "0.778176881",
      "0.540374761"
    ],
    [
      "0.364363802",
      "0.676240657"
    ],
    [
      "0.726573451",
      "0.137779228"
    ],
    [
      "0.172844433",
      "0.319183549"
    ],
    [
      "0.326092502",
      "0.155797765"
    ],
    [
      "0.821182177",
      "0.392792538"
    ],
    [
      "0.783994299",
      "0.698579488"
    ],
    [
      "0.139998650",
      "0.465621868"
    ],
    [
      "0.660731954",
      "0.595115847"
    ],
    [
      "0.391706657",
      "0.531811473"
    ],
    [
      "0.547512945",
      "0.480317244"
    ],
    [
      "0.565948886",
      "0.323016416"
    ],
    [
      "0.093301644",
      "0.600744551"
    ],
    [
      "0.716868376",
      "0.822613239"
    ],
    [
      "0.570996189",
      "0.869943750"
    ],
    [
      "0.698439082",
      "0.298340421"
    ],
    [
      "0.478805015",
      "0.923854166"
    ],
    [
      "0.227543050",
      "0.175346826"
    ],
    [
      "0.511693037",
      "0.615481685"
    ],
    [
      "0.829427394",
      "0.238693725"
    ],
    [
      "0.320199148",
      "0.293668862"
    ],
    [
      "0.626545653",
      "0.719281816"
    ],
    [
      "0.462102151",
      "0.245050274"
    ],
    [
      "0.281407560",
      "0.433898177"
    ],
    [
      "0.245459230",
      "0.581289231"
    ],
    [
      "0.481961782",
      "0.760834354"
    ]
  ]
}

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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