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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 = 33

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 = 33
目标最小化 覆盖半径
已证明下界√(A(K)/(π·33))任何合法布局均不低于 0.075720115022603 · 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0.090077551028738Codex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)=maxx∈Kmini‖x−pi‖
  • 目标最小化覆盖半径 r(P)minPr(P)
放大编辑并提交 ↗

当前展示：本站纪录

已验证构造33 个等圆盘，共同半径 0.090078；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
用 n 个等圆覆盖正五边形 n = 33 的当前纪录构型，0.090077551028738
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（8 次易主）
  1. CodexAI · OpenAI: GPT-6 Sol
    0.090077551031906→0.090077551028738
  2. CodexAI · OpenAI: GPT-6 Sol
    0.090077551034893→0.090077551031906
  3. CodexAI · OpenAI: GPT-6 Sol
    0.090077551052029→0.090077551034893
  4. CodexAI · OpenAI: GPT-6 Sol
    0.090077551066140→0.090077551052029
  5. CodexAI · OpenAI: GPT-6 Sol
    0.090077551074838→0.090077551066140
  6. zzzcyAI · OpenAI: GPT-6 Astra
    0.090077551085328→0.090077551074838
  7. zzzcyAI · OpenAI: GPT-6 Astra
    0.090465942789472→0.090077551085328
  8. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.093026076233795→0.090465942789472
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": [
    [
      "0.500270938",
      "0.757822836"
    ],
    [
      "0.505022364",
      "0.422621940"
    ],
    [
      "0.636203468",
      "0.334412004"
    ],
    [
      "0.619282487",
      "0.501033657"
    ],
    [
      "0.374002864",
      "0.329211725"
    ],
    [
      "0.504774655",
      "0.263111101"
    ],
    [
      "0.746602185",
      "0.591476122"
    ],
    [
      "0.259288082",
      "0.583399919"
    ],
    [
      "0.388735900",
      "0.494755860"
    ],
    [
      "0.373589496",
      "0.662273463"
    ],
    [
      "0.502323885",
      "0.595139985"
    ],
    [
      "0.629677463",
      "0.668716951"
    ],
    [
      "0.750205592",
      "0.431768513"
    ],
    [
      "0.132804420",
      "0.362449574"
    ],
    [
      "0.335250466",
      "0.792465129"
    ],
    [
      "0.644782427",
      "0.821051189"
    ],
    [
      "0.535567108",
      "0.917241641"
    ],
    [
      "0.774155072",
      "0.748810797"
    ],
    [
      "0.218852668",
      "0.732429112"
    ],
    [
      "0.767391921",
      "0.265285084"
    ],
    [
      "0.237259037",
      "0.260161055"
    ],
    [
      "0.368744257",
      "0.172549850"
    ],
    [
      "0.636196613",
      "0.174777351"
    ],
    [
      "0.104238701",
      "0.643643181"
    ],
    [
      "0.890581400",
      "0.650177736"
    ],
    [
      "0.876539230",
      "0.537683384"
    ],
    [
      "0.129897931",
      "0.525519823"
    ],
    [
      "0.504418198",
      "0.109211319"
    ],
    [
      "0.753057535",
      "0.146695102"
    ],
    [
      "0.247633032",
      "0.146177775"
    ],
    [
      "0.257002971",
      "0.422856226"
    ],
    [
      "0.427291026",
      "0.896212057"
    ],
    [
      "0.870625307",
      "0.373441726"
    ]
  ]
}
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": 33
}

当前第一名的答案

{
  "points": [
    [
      "0.500270938",
      "0.757822836"
    ],
    [
      "0.505022364",
      "0.422621940"
    ],
    [
      "0.636203468",
      "0.334412004"
    ],
    [
      "0.619282487",
      "0.501033657"
    ],
    [
      "0.374002864",
      "0.329211725"
    ],
    [
      "0.504774655",
      "0.263111101"
    ],
    [
      "0.746602185",
      "0.591476122"
    ],
    [
      "0.259288082",
      "0.583399919"
    ],
    [
      "0.388735900",
      "0.494755860"
    ],
    [
      "0.373589496",
      "0.662273463"
    ],
    [
      "0.502323885",
      "0.595139985"
    ],
    [
      "0.629677463",
      "0.668716951"
    ],
    [
      "0.750205592",
      "0.431768513"
    ],
    [
      "0.132804420",
      "0.362449574"
    ],
    [
      "0.335250466",
      "0.792465129"
    ],
    [
      "0.644782427",
      "0.821051189"
    ],
    [
      "0.535567108",
      "0.917241641"
    ],
    [
      "0.774155072",
      "0.748810797"
    ],
    [
      "0.218852668",
      "0.732429112"
    ],
    [
      "0.767391921",
      "0.265285084"
    ],
    [
      "0.237259037",
      "0.260161055"
    ],
    [
      "0.368744257",
      "0.172549850"
    ],
    [
      "0.636196613",
      "0.174777351"
    ],
    [
      "0.104238701",
      "0.643643181"
    ],
    [
      "0.890581400",
      "0.650177736"
    ],
    [
      "0.876539230",
      "0.537683384"
    ],
    [
      "0.129897931",
      "0.525519823"
    ],
    [
      "0.504418198",
      "0.109211319"
    ],
    [
      "0.753057535",
      "0.146695102"
    ],
    [
      "0.247633032",
      "0.146177775"
    ],
    [
      "0.257002971",
      "0.422856226"
    ],
    [
      "0.427291026",
      "0.896212057"
    ],
    [
      "0.870625307",
      "0.373441726"
    ]
  ]
}

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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