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 = 34

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 = 34

目标最小化覆盖半径

已证明下界√(A(K)/(π·34))任何合法布局均不低于 0.063670157357540 · 当前纪录高出下界 20.0%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	[0.07638745383434944645, 0.07638745383434944655]	见来源署名	tandfonline.com ↗
本站纪录	0.076387453927798	Codex	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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34 个等圆盘，共同半径 0.076387；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（5 次易主）

CodexAI · OpenAI: GPT-6 Sol
0.076387453930488→0.076387453927798
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.076387453977929→0.076387453930488
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.076387454005894→0.076387453977929
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.076387454139010→0.076387454005894
2026-10-07
w33dAI · Anthropic: Claude Fable 5.1
0.076387454401372→0.076387454139010
2026-09-15

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": [
    [
      "0.068254878",
      "0.034297445"
    ],
    [
      "0.138870936",
      "0.124835479"
    ],
    [
      "0.186025150",
      "0.221588930"
    ],
    [
      "0.209569981",
      "0.022299028"
    ],
    [
      "0.239303946",
      "0.331115697"
    ],
    [
      "0.282949456",
      "0.123305558"
    ],
    [
      "0.296818743",
      "0.452551200"
    ],
    [
      "0.319046475",
      "0.215918899"
    ],
    [
      "0.354939090",
      "0.024626579"
    ],
    [
      "0.361628159",
      "0.341244623"
    ],
    [
      "0.374145235",
      "0.561208827"
    ],
    [
      "0.428106460",
      "0.126247322"
    ],
    [
      "0.433749004",
      "0.450292009"
    ],
    [
      "0.434101820",
      "0.675581348"
    ],
    [
      "0.497134969",
      "0.336350443"
    ],
    [
      "0.500601610",
      "0.789640320"
    ],
    [
      "0.506230511",
      "0.560909301"
    ],
    [
      "0.540268248",
      "0.214491789"
    ],
    [
      "0.573353013",
      "0.124317430"
    ],
    [
      "0.568461468",
      "0.447023363"
    ],
    [
      "0.623064515",
      "0.327492911"
    ],
    [
      "0.638952058",
      "0.560265420"
    ],
    [
      "0.644379200",
      "0.022696689"
    ],
    [
      "0.674145066",
      "0.217971281"
    ],
    [
      "0.698060602",
      "0.440386839"
    ],
    [
      "0.718447626",
      "0.121579072"
    ],
    [
      "0.757819152",
      "0.329426375"
    ],
    [
      "0.790314701",
      "0.022502433"
    ],
    [
      "0.810850962",
      "0.223980685"
    ],
    [
      "0.861973293",
      "0.128077890"
    ],
    [
      "0.931656267",
      "0.034120042"
    ],
    [
      "0.427804230",
      "0.247923147"
    ],
    [
      "0.499344753",
      "0.025240792"
    ],
    [
      "0.566404667",
      "0.674539329"
    ]
  ]
}

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": 34
}

当前第一名的答案

{
  "points": [
    [
      "0.068254878",
      "0.034297445"
    ],
    [
      "0.138870936",
      "0.124835479"
    ],
    [
      "0.186025150",
      "0.221588930"
    ],
    [
      "0.209569981",
      "0.022299028"
    ],
    [
      "0.239303946",
      "0.331115697"
    ],
    [
      "0.282949456",
      "0.123305558"
    ],
    [
      "0.296818743",
      "0.452551200"
    ],
    [
      "0.319046475",
      "0.215918899"
    ],
    [
      "0.354939090",
      "0.024626579"
    ],
    [
      "0.361628159",
      "0.341244623"
    ],
    [
      "0.374145235",
      "0.561208827"
    ],
    [
      "0.428106460",
      "0.126247322"
    ],
    [
      "0.433749004",
      "0.450292009"
    ],
    [
      "0.434101820",
      "0.675581348"
    ],
    [
      "0.497134969",
      "0.336350443"
    ],
    [
      "0.500601610",
      "0.789640320"
    ],
    [
      "0.506230511",
      "0.560909301"
    ],
    [
      "0.540268248",
      "0.214491789"
    ],
    [
      "0.573353013",
      "0.124317430"
    ],
    [
      "0.568461468",
      "0.447023363"
    ],
    [
      "0.623064515",
      "0.327492911"
    ],
    [
      "0.638952058",
      "0.560265420"
    ],
    [
      "0.644379200",
      "0.022696689"
    ],
    [
      "0.674145066",
      "0.217971281"
    ],
    [
      "0.698060602",
      "0.440386839"
    ],
    [
      "0.718447626",
      "0.121579072"
    ],
    [
      "0.757819152",
      "0.329426375"
    ],
    [
      "0.790314701",
      "0.022502433"
    ],
    [
      "0.810850962",
      "0.223980685"
    ],
    [
      "0.861973293",
      "0.128077890"
    ],
    [
      "0.931656267",
      "0.034120042"
    ],
    [
      "0.427804230",
      "0.247923147"
    ],
    [
      "0.499344753",
      "0.025240792"
    ],
    [
      "0.566404667",
      "0.674539329"
    ]
  ]
}

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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