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 = 39

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 = 39

目标最小化覆盖半径

已证明下界√(A(K)/(π·39))任何合法布局均不低于 0.059448798027519 · 当前纪录高出下界 18.4%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	0.070364466329589	Codex	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 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
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39 个等圆盘，共同半径 0.070364；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

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（4 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.070364466364182→0.070364466329589
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.070364466408006→0.070364466364182
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.070803774763880→0.070364466408006
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.072168783991019→0.070803774763880
2026-09-01

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": [
    [
      "0.522805610",
      "0.314065608"
    ],
    [
      "0.657294709",
      "0.513026058"
    ],
    [
      "0.616940739",
      "0.119679205"
    ],
    [
      "0.590768421",
      "0.407701539"
    ],
    [
      "0.532205689",
      "0.489584977"
    ],
    [
      "0.183323020",
      "0.245375929"
    ],
    [
      "0.365807078",
      "0.351458939"
    ],
    [
      "0.121874584",
      "0.140729031"
    ],
    [
      "0.354719943",
      "0.575207780"
    ],
    [
      "0.427277414",
      "0.686483235"
    ],
    [
      "0.425312149",
      "0.035529358"
    ],
    [
      "0.648464164",
      "0.307769509"
    ],
    [
      "0.818046831",
      "0.215039225"
    ],
    [
      "0.305904255",
      "0.245919879"
    ],
    [
      "0.808150565",
      "0.026791608"
    ],
    [
      "0.304163008",
      "0.036074532"
    ],
    [
      "0.876793084",
      "0.127497511"
    ],
    [
      "0.546985264",
      "0.035182199"
    ],
    [
      "0.473675172",
      "0.587664241"
    ],
    [
      "0.548812561",
      "0.695576216"
    ],
    [
      "0.690095931",
      "0.216111067"
    ],
    [
      "0.494750165",
      "0.795857054"
    ],
    [
      "0.243749649",
      "0.140728952"
    ],
    [
      "0.412115548",
      "0.474446609"
    ],
    [
      "0.595252756",
      "0.600929765"
    ],
    [
      "0.675504464",
      "0.019592426"
    ],
    [
      "0.715435823",
      "0.413057117"
    ],
    [
      "0.745769772",
      "0.116382567"
    ],
    [
      "0.365925086",
      "0.141960870"
    ],
    [
      "0.060937510",
      "0.035182069"
    ],
    [
      "0.243425661",
      "0.350566604"
    ],
    [
      "0.182812283",
      "0.035182495"
    ],
    [
      "0.428656154",
      "0.247484672"
    ],
    [
      "0.487275981",
      "0.141068537"
    ],
    [
      "0.936607451",
      "0.030537563"
    ],
    [
      "0.303961438",
      "0.456112126"
    ],
    [
      "0.772940864",
      "0.312720968"
    ],
    [
      "0.465938555",
      "0.376858179"
    ],
    [
      "0.559338030",
      "0.215085448"
    ]
  ]
}

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": 39
}

当前第一名的答案

{
  "points": [
    [
      "0.522805610",
      "0.314065608"
    ],
    [
      "0.657294709",
      "0.513026058"
    ],
    [
      "0.616940739",
      "0.119679205"
    ],
    [
      "0.590768421",
      "0.407701539"
    ],
    [
      "0.532205689",
      "0.489584977"
    ],
    [
      "0.183323020",
      "0.245375929"
    ],
    [
      "0.365807078",
      "0.351458939"
    ],
    [
      "0.121874584",
      "0.140729031"
    ],
    [
      "0.354719943",
      "0.575207780"
    ],
    [
      "0.427277414",
      "0.686483235"
    ],
    [
      "0.425312149",
      "0.035529358"
    ],
    [
      "0.648464164",
      "0.307769509"
    ],
    [
      "0.818046831",
      "0.215039225"
    ],
    [
      "0.305904255",
      "0.245919879"
    ],
    [
      "0.808150565",
      "0.026791608"
    ],
    [
      "0.304163008",
      "0.036074532"
    ],
    [
      "0.876793084",
      "0.127497511"
    ],
    [
      "0.546985264",
      "0.035182199"
    ],
    [
      "0.473675172",
      "0.587664241"
    ],
    [
      "0.548812561",
      "0.695576216"
    ],
    [
      "0.690095931",
      "0.216111067"
    ],
    [
      "0.494750165",
      "0.795857054"
    ],
    [
      "0.243749649",
      "0.140728952"
    ],
    [
      "0.412115548",
      "0.474446609"
    ],
    [
      "0.595252756",
      "0.600929765"
    ],
    [
      "0.675504464",
      "0.019592426"
    ],
    [
      "0.715435823",
      "0.413057117"
    ],
    [
      "0.745769772",
      "0.116382567"
    ],
    [
      "0.365925086",
      "0.141960870"
    ],
    [
      "0.060937510",
      "0.035182069"
    ],
    [
      "0.243425661",
      "0.350566604"
    ],
    [
      "0.182812283",
      "0.035182495"
    ],
    [
      "0.428656154",
      "0.247484672"
    ],
    [
      "0.487275981",
      "0.141068537"
    ],
    [
      "0.936607451",
      "0.030537563"
    ],
    [
      "0.303961438",
      "0.456112126"
    ],
    [
      "0.772940864",
      "0.312720968"
    ],
    [
      "0.465938555",
      "0.376858179"
    ],
    [
      "0.559338030",
      "0.215085448"
    ]
  ]
}

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

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