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 = 32

把 n 个圆心放在单位正三角形中。每个圆心拥有相同覆盖半径；要求三角形内任何位置都至少落入一个圆盘，并让所需半径尽可能小。

子题n = 32

目标最小化覆盖半径

已知最好（未证明）0.0788506226168764215Kari J. Nurmela 的构型收录于 Nurmela（2000）Figure 4；论文连续值为 0.0788506226168764215。其最优性尚未证明；本站从论文接触图复原的九位坐标证书可验证到 0.078850623071391。

已证明下界√(A(K)/(π·32))任何布局都不低于 0.065629695996209 · 当前纪录高出下界 20.1%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

严格定义

容器顶点为 (0,0)、(1,0)、(0.5,0.866025404) 的闭凸多边形；这是本站九位坐标格式下对正三角形的唯一明确定义
提交恰好 n 个互不重合、且位于容器内的点
分数对容器中每个位置取最近圆心距离，再取其中最大值；验证器用有理 Voronoi 多边形精确计算 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
目标让覆盖半径尽可能小 $\min_{P} r (P)$

放大来摆，然后提交 ↗

已验证构造32 个等圆盘，共同半径 0.078851；标出的那一点是最难够到的位置，半径由它决定

帮助理解

角落和内部在争夺圆心

三个尖角必须被照顾，但把圆心都推向边界又会在中央留下空洞。最优构型通常不是简单的等距三角网格。

论文纪录与可验证构型已经接入

Nurmela（2000）汇总并扩展了 n=2–36 的高精度构型。本站已从论文 Figures 2–4 的接触图复原 n=7–36：页面同时显示论文连续值与九位坐标证书实际值。n=9、10 已证明最优，其余 n≤36 是可被挑战的 known best。论文明确没有搜索 n>36，因此 n=37–40 只提供从 n=36 逐个填补最远空洞的本站起点，不虚构外部纪录。

查看来源

当前第一名

0.078850623071391

覆盖半径

答案来源Kari J. Nurmela

解题方式公开参考构造

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

底边从 (0,0) 到 (1,0)，顶点在 (0.5, 0.866025404)。这是正三角形在九位坐标格上的明确代表，也是验证器实际使用的边界。

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": [
    [
      "0.931730113",
      "0.039454319"
    ],
    [
      "0.790422794",
      "0.029714543"
    ],
    [
      "0.858659181",
      "0.148019477"
    ],
    [
      "0.654002172",
      "0.046904073"
    ],
    [
      "0.730139943",
      "0.155408263"
    ],
    [
      "0.534869057",
      "0.055761693"
    ],
    [
      "0.807798951",
      "0.275294717"
    ],
    [
      "0.433844213",
      "0.064556971"
    ],
    [
      "0.601352584",
      "0.181455414"
    ],
    [
      "0.326205499",
      "0.048251192"
    ],
    [
      "0.503969768",
      "0.197847769"
    ],
    [
      "0.199059194",
      "0.044951611"
    ],
    [
      "0.720241",
      "0.374977897"
    ],
    [
      "0.382021293",
      "0.184924993"
    ],
    [
      "0.582788163",
      "0.326567825"
    ],
    [
      "0.067138343",
      "0.041350497"
    ],
    [
      "0.2636644",
      "0.172021562"
    ],
    [
      "0.455930034",
      "0.314122534"
    ],
    [
      "0.660599599",
      "0.455701372"
    ],
    [
      "0.13416226",
      "0.165120867"
    ],
    [
      "0.328888301",
      "0.303263591"
    ],
    [
      "0.532910531",
      "0.441281472"
    ],
    [
      "0.203064041",
      "0.291951162"
    ],
    [
      "0.40338132",
      "0.430376754"
    ],
    [
      "0.623877739",
      "0.557152476"
    ],
    [
      "0.273891621",
      "0.420983766"
    ],
    [
      "0.485484735",
      "0.551364616"
    ],
    [
      "0.351056131",
      "0.545742555"
    ],
    [
      "0.562345455",
      "0.670227763"
    ],
    [
      "0.425791064",
      "0.667956072"
    ],
    [
      "0.498688439",
      "0.78718569"
    ],
    [
      "0.675471262",
      "0.264090563"
    ]
  ]
}

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": 32
}

当前第一名的答案

{
  "points": [
    [
      "0.931730113",
      "0.039454319"
    ],
    [
      "0.790422794",
      "0.029714543"
    ],
    [
      "0.858659181",
      "0.148019477"
    ],
    [
      "0.654002172",
      "0.046904073"
    ],
    [
      "0.730139943",
      "0.155408263"
    ],
    [
      "0.534869057",
      "0.055761693"
    ],
    [
      "0.807798951",
      "0.275294717"
    ],
    [
      "0.433844213",
      "0.064556971"
    ],
    [
      "0.601352584",
      "0.181455414"
    ],
    [
      "0.326205499",
      "0.048251192"
    ],
    [
      "0.503969768",
      "0.197847769"
    ],
    [
      "0.199059194",
      "0.044951611"
    ],
    [
      "0.720241",
      "0.374977897"
    ],
    [
      "0.382021293",
      "0.184924993"
    ],
    [
      "0.582788163",
      "0.326567825"
    ],
    [
      "0.067138343",
      "0.041350497"
    ],
    [
      "0.2636644",
      "0.172021562"
    ],
    [
      "0.455930034",
      "0.314122534"
    ],
    [
      "0.660599599",
      "0.455701372"
    ],
    [
      "0.13416226",
      "0.165120867"
    ],
    [
      "0.328888301",
      "0.303263591"
    ],
    [
      "0.532910531",
      "0.441281472"
    ],
    [
      "0.203064041",
      "0.291951162"
    ],
    [
      "0.40338132",
      "0.430376754"
    ],
    [
      "0.623877739",
      "0.557152476"
    ],
    [
      "0.273891621",
      "0.420983766"
    ],
    [
      "0.485484735",
      "0.551364616"
    ],
    [
      "0.351056131",
      "0.545742555"
    ],
    [
      "0.562345455",
      "0.670227763"
    ],
    [
      "0.425791064",
      "0.667956072"
    ],
    [
      "0.498688439",
      "0.78718569"
    ],
    [
      "0.675471262",
      "0.264090563"
    ]
  ]
}

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

聊思路、贴方法、问为什么卡住。发帖即公开署名，与纪录同一个名字；署名后的 #编号是账号的注册序号，冒不了名。发言资格与实绩绑定：破过一次纪录，就永久拥有发言权。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？