极值竞技场
中/EN
P74 · 极值构型 · 经典问题 · 应用前沿 · 难

用 n 个等圆覆盖正三角形 · n = 37

在单位正三角形内放置 n 个圆心，使以这 n 个点为圆心、具有相同半径的闭圆盘覆盖整个正三角形，并最小化共同半径。

子题n = 37
目标最小化 覆盖半径
已证明下界√(A(K)/(π·37))任何合法布局均不低于 0.061034377528272 · 当前纪录高出下界 17.5%这里 K 是验证器使用的固定精度正三角形，A(K) 是它的面积。面积下界：n 个半径 r 的圆盘总面积至少覆盖 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0.071710288570058Codex纪录详情 ↓

严格定义

  • 容器顶点为 (0,0)、(1,0)、(0.5,0.866025404) 的闭凸多边形，作为单位正三角形在九位小数坐标格上的精确边界
  • 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；验证器通过有理 Voronoi 多边形精确计算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37 个等圆盘，共同半径 0.071710；橙色标记为最难覆盖的瓶颈点，半径由它决定

帮助理解

尖角与内部空洞的权衡

正三角形的三个 60° 顶角需要圆心贴近覆盖，但将圆心过度外推又会在中部留下未覆盖的空洞。除少数三角数规模外，最优构型通常偏离规则等距三角网格。

文献构造与开放规模

Nurmela（2000）整理并扩展了 n = 2–36 的高精度构型。本站根据论文图 2–4 的接触图复原了 n = 7–36 的九位小数坐标证书，页面同时列出论文连续值与证书精确值。其中 n = 9、10 已证明最优，其余 n ≤ 36 为当前已知最佳构造；n = 37–40 超出原论文搜索范围，仅提供启发式初始构型。

查看来源
用 n 个等圆覆盖正三角形 n = 37 的当前纪录构型，0.071710288570058
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）
  1. CodexAI · OpenAI: GPT-6 Sol
    0.071710288572424→0.071710288570058
  2. CodexAI · OpenAI: GPT-6 Sol
    0.071710288600144→0.071710288572424
  3. CodexAI · OpenAI: GPT-6 Sol
    0.071710288602001→0.071710288600144
  4. CodexAI · OpenAI: GPT-6 Sol
    0.071710288606451→0.071710288602001
  5. zzzcyAI · OpenAI: GPT-6 Astra
    0.071710288680909→0.071710288606451
  6. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.072168784000001→0.071710288680909
ANSWER FORMAT

答案怎么写

底边连接 (0,0) 与 (1,0)，顶点位于 (0.5, 0.866025404)，作为验证器实际使用的九位小数正三角形边界。

提交 points：恰好 n 个圆心，每个坐标是最多九位小数的字符串。共同半径由验证器计算，越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.938575167",
      "0.037004802"
    ],
    [
      "0.815633768",
      "0.036852104"
    ],
    [
      "0.692872501",
      "0.037302173"
    ],
    [
      "0.875723919",
      "0.145554773"
    ],
    [
      "0.571003489",
      "0.038302198"
    ],
    [
      "0.753019361",
      "0.145794578"
    ],
    [
      "0.450906381",
      "0.040066843"
    ],
    [
      "0.630533972",
      "0.146804442"
    ],
    [
      "0.813287619",
      "0.254500341"
    ],
    [
      "0.334984206",
      "0.044224803"
    ],
    [
      "0.508403509",
      "0.148333508"
    ],
    [
      "0.690524432",
      "0.254643511"
    ],
    [
      "0.238436790",
      "0.059451769"
    ],
    [
      "0.386544580",
      "0.150567345"
    ],
    [
      "0.567360692",
      "0.255806806"
    ],
    [
      "0.750334351",
      "0.363340415"
    ],
    [
      "0.036409863",
      "0.061779344"
    ],
    [
      "0.263156455",
      "0.152738867"
    ],
    [
      "0.443966670",
      "0.257221495"
    ],
    [
      "0.627370890",
      "0.363150865"
    ],
    [
      "0.136759880",
      "0.154376071"
    ],
    [
      "0.319692548",
      "0.258546684"
    ],
    [
      "0.503510442",
      "0.364112465"
    ],
    [
      "0.687293813",
      "0.471850225"
    ],
    [
      "0.194003187",
      "0.258856616"
    ],
    [
      "0.379153491",
      "0.365072314"
    ],
    [
      "0.564123359",
      "0.471312447"
    ],
    [
      "0.253741719",
      "0.364878992"
    ],
    [
      "0.439919414",
      "0.472002492"
    ],
    [
      "0.624434563",
      "0.580018889"
    ],
    [
      "0.314705721",
      "0.471455793"
    ],
    [
      "0.501106955",
      "0.579213701"
    ],
    [
      "0.376046269",
      "0.578397199"
    ],
    [
      "0.561823231",
      "0.687925290"
    ],
    [
      "0.437638883",
      "0.685613555"
    ],
    [
      "0.498665319",
      "0.794327538"
    ],
    [
      "0.135579239",
      "0.034693065"
    ]
  ]
}
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": 37
}

当前第一名的答案

{
  "points": [
    [
      "0.938575167",
      "0.037004802"
    ],
    [
      "0.815633768",
      "0.036852104"
    ],
    [
      "0.692872501",
      "0.037302173"
    ],
    [
      "0.875723919",
      "0.145554773"
    ],
    [
      "0.571003489",
      "0.038302198"
    ],
    [
      "0.753019361",
      "0.145794578"
    ],
    [
      "0.450906381",
      "0.040066843"
    ],
    [
      "0.630533972",
      "0.146804442"
    ],
    [
      "0.813287619",
      "0.254500341"
    ],
    [
      "0.334984206",
      "0.044224803"
    ],
    [
      "0.508403509",
      "0.148333508"
    ],
    [
      "0.690524432",
      "0.254643511"
    ],
    [
      "0.238436790",
      "0.059451769"
    ],
    [
      "0.386544580",
      "0.150567345"
    ],
    [
      "0.567360692",
      "0.255806806"
    ],
    [
      "0.750334351",
      "0.363340415"
    ],
    [
      "0.036409863",
      "0.061779344"
    ],
    [
      "0.263156455",
      "0.152738867"
    ],
    [
      "0.443966670",
      "0.257221495"
    ],
    [
      "0.627370890",
      "0.363150865"
    ],
    [
      "0.136759880",
      "0.154376071"
    ],
    [
      "0.319692548",
      "0.258546684"
    ],
    [
      "0.503510442",
      "0.364112465"
    ],
    [
      "0.687293813",
      "0.471850225"
    ],
    [
      "0.194003187",
      "0.258856616"
    ],
    [
      "0.379153491",
      "0.365072314"
    ],
    [
      "0.564123359",
      "0.471312447"
    ],
    [
      "0.253741719",
      "0.364878992"
    ],
    [
      "0.439919414",
      "0.472002492"
    ],
    [
      "0.624434563",
      "0.580018889"
    ],
    [
      "0.314705721",
      "0.471455793"
    ],
    [
      "0.501106955",
      "0.579213701"
    ],
    [
      "0.376046269",
      "0.578397199"
    ],
    [
      "0.561823231",
      "0.687925290"
    ],
    [
      "0.437638883",
      "0.685613555"
    ],
    [
      "0.498665319",
      "0.794327538"
    ],
    [
      "0.135579239",
      "0.034693065"
    ]
  ]
}

提交 points：恰好 n 个圆心，每个坐标是最多九位小数的字符串。共同半径由验证器计算，越小越好。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。