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 = 38

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 = 38

目标最小化覆盖半径

已证明下界√(A(K)/(π·38))任何合法布局均不低于 0.060225939482688 · 当前纪录高出下界 18.0%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	0.071078997996647	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38 个等圆盘，共同半径 0.071079；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（7 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.071078998009464→0.071078997996647
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.071078998010254→0.071078998009464
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.071078998029110→0.071078998010254
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.071078998045961→0.071078998029110
2026-10-19
CodexAI · OpenAI: GPT-6 Sol
0.071078998080837→0.071078998045961
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.071252294789407→0.071078998080837
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.072168783991019→0.071252294789407
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": [
    [
      "0.548901060",
      "0.257014403"
    ],
    [
      "0.667930680",
      "0.491078634"
    ],
    [
      "0.611714175",
      "0.149084875"
    ],
    [
      "0.594596533",
      "0.382532733"
    ],
    [
      "0.543186877",
      "0.480565400"
    ],
    [
      "0.183538930",
      "0.246810153"
    ],
    [
      "0.365401800",
      "0.358466301"
    ],
    [
      "0.124030560",
      "0.140226043"
    ],
    [
      "0.368771089",
      "0.574799303"
    ],
    [
      "0.431459953",
      "0.685947720"
    ],
    [
      "0.432173322",
      "0.036880117"
    ],
    [
      "0.650063226",
      "0.278615828"
    ],
    [
      "0.831686854",
      "0.207988101"
    ],
    [
      "0.305988776",
      "0.248378871"
    ],
    [
      "0.800153947",
      "0.014760618"
    ],
    [
      "0.309851837",
      "0.035534699"
    ],
    [
      "0.872947283",
      "0.112044069"
    ],
    [
      "0.552354385",
      "0.039008433"
    ],
    [
      "0.487086689",
      "0.582446783"
    ],
    [
      "0.554454386",
      "0.691336717"
    ],
    [
      "0.704748574",
      "0.217927975"
    ],
    [
      "0.496888661",
      "0.795014535"
    ],
    [
      "0.247276899",
      "0.140449343"
    ],
    [
      "0.426385429",
      "0.468525700"
    ],
    [
      "0.610482574",
      "0.590037950"
    ],
    [
      "0.671198711",
      "0.038997475"
    ],
    [
      "0.712337670",
      "0.394454034"
    ],
    [
      "0.742078293",
      "0.114429922"
    ],
    [
      "0.370548362",
      "0.143493782"
    ],
    [
      "0.061637458",
      "0.035398414"
    ],
    [
      "0.244295883",
      "0.354769267"
    ],
    [
      "0.185783882",
      "0.033835678"
    ],
    [
      "0.427916310",
      "0.253551626"
    ],
    [
      "0.491525919",
      "0.146967517"
    ],
    [
      "0.934841712",
      "0.028401081"
    ],
    [
      "0.305272320",
      "0.464839623"
    ],
    [
      "0.774194765",
      "0.307643365"
    ],
    [
      "0.487336525",
      "0.363628098"
    ]
  ]
}

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": 38
}

当前第一名的答案

{
  "points": [
    [
      "0.548901060",
      "0.257014403"
    ],
    [
      "0.667930680",
      "0.491078634"
    ],
    [
      "0.611714175",
      "0.149084875"
    ],
    [
      "0.594596533",
      "0.382532733"
    ],
    [
      "0.543186877",
      "0.480565400"
    ],
    [
      "0.183538930",
      "0.246810153"
    ],
    [
      "0.365401800",
      "0.358466301"
    ],
    [
      "0.124030560",
      "0.140226043"
    ],
    [
      "0.368771089",
      "0.574799303"
    ],
    [
      "0.431459953",
      "0.685947720"
    ],
    [
      "0.432173322",
      "0.036880117"
    ],
    [
      "0.650063226",
      "0.278615828"
    ],
    [
      "0.831686854",
      "0.207988101"
    ],
    [
      "0.305988776",
      "0.248378871"
    ],
    [
      "0.800153947",
      "0.014760618"
    ],
    [
      "0.309851837",
      "0.035534699"
    ],
    [
      "0.872947283",
      "0.112044069"
    ],
    [
      "0.552354385",
      "0.039008433"
    ],
    [
      "0.487086689",
      "0.582446783"
    ],
    [
      "0.554454386",
      "0.691336717"
    ],
    [
      "0.704748574",
      "0.217927975"
    ],
    [
      "0.496888661",
      "0.795014535"
    ],
    [
      "0.247276899",
      "0.140449343"
    ],
    [
      "0.426385429",
      "0.468525700"
    ],
    [
      "0.610482574",
      "0.590037950"
    ],
    [
      "0.671198711",
      "0.038997475"
    ],
    [
      "0.712337670",
      "0.394454034"
    ],
    [
      "0.742078293",
      "0.114429922"
    ],
    [
      "0.370548362",
      "0.143493782"
    ],
    [
      "0.061637458",
      "0.035398414"
    ],
    [
      "0.244295883",
      "0.354769267"
    ],
    [
      "0.185783882",
      "0.033835678"
    ],
    [
      "0.427916310",
      "0.253551626"
    ],
    [
      "0.491525919",
      "0.146967517"
    ],
    [
      "0.934841712",
      "0.028401081"
    ],
    [
      "0.305272320",
      "0.464839623"
    ],
    [
      "0.774194765",
      "0.307643365"
    ],
    [
      "0.487336525",
      "0.363628098"
    ]
  ]
}

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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