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 = 29

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 = 29

目标最小化覆盖半径

已知最好（未证明）0.0818048133956910115Kari J. Nurmela 的构型收录于 Nurmela（2000）Figure 4；论文连续值为 0.0818048133956910115。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 0.081804813824288。

已证明下界√(A(K)/(π·29))任何布局都不低于 0.068940810167442 · 当前纪录高出下界 18.7%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29 个等圆盘，共同半径 0.081805；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

0.081804813824288

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": [
    [
      "0.930050162",
      "0.042415181"
    ],
    [
      "0.79031831",
      "0.042690726"
    ],
    [
      "0.651339765",
      "0.043633331"
    ],
    [
      "0.858510443",
      "0.166891835"
    ],
    [
      "0.514284151",
      "0.045685581"
    ],
    [
      "0.719018399",
      "0.167556918"
    ],
    [
      "0.382044278",
      "0.050469213"
    ],
    [
      "0.579839005",
      "0.169231745"
    ],
    [
      "0.786887991",
      "0.291750459"
    ],
    [
      "0.272055494",
      "0.067911253"
    ],
    [
      "0.440912072",
      "0.171836467"
    ],
    [
      "0.647082636",
      "0.291905623"
    ],
    [
      "0.154739956",
      "0.039370747"
    ],
    [
      "0.300300662",
      "0.174465459"
    ],
    [
      "0.506409815",
      "0.293531744"
    ],
    [
      "0.714494973",
      "0.416086252"
    ],
    [
      "0.041516206",
      "0.070487106"
    ],
    [
      "0.156039161",
      "0.17604797"
    ],
    [
      "0.364736127",
      "0.295150053"
    ],
    [
      "0.574309203",
      "0.415610807"
    ],
    [
      "0.221233825",
      "0.295309319"
    ],
    [
      "0.432623562",
      "0.416733974"
    ],
    [
      "0.6423198",
      "0.539807812"
    ],
    [
      "0.289477176",
      "0.416248487"
    ],
    [
      "0.501851372",
      "0.538855337"
    ],
    [
      "0.358976777",
      "0.537886545"
    ],
    [
      "0.5705983",
      "0.663066487"
    ],
    [
      "0.428940918",
      "0.660022052"
    ],
    [
      "0.498242295",
      "0.784239476"
    ]
  ]
}

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": 29
}

当前第一名的答案

{
  "points": [
    [
      "0.930050162",
      "0.042415181"
    ],
    [
      "0.79031831",
      "0.042690726"
    ],
    [
      "0.651339765",
      "0.043633331"
    ],
    [
      "0.858510443",
      "0.166891835"
    ],
    [
      "0.514284151",
      "0.045685581"
    ],
    [
      "0.719018399",
      "0.167556918"
    ],
    [
      "0.382044278",
      "0.050469213"
    ],
    [
      "0.579839005",
      "0.169231745"
    ],
    [
      "0.786887991",
      "0.291750459"
    ],
    [
      "0.272055494",
      "0.067911253"
    ],
    [
      "0.440912072",
      "0.171836467"
    ],
    [
      "0.647082636",
      "0.291905623"
    ],
    [
      "0.154739956",
      "0.039370747"
    ],
    [
      "0.300300662",
      "0.174465459"
    ],
    [
      "0.506409815",
      "0.293531744"
    ],
    [
      "0.714494973",
      "0.416086252"
    ],
    [
      "0.041516206",
      "0.070487106"
    ],
    [
      "0.156039161",
      "0.17604797"
    ],
    [
      "0.364736127",
      "0.295150053"
    ],
    [
      "0.574309203",
      "0.415610807"
    ],
    [
      "0.221233825",
      "0.295309319"
    ],
    [
      "0.432623562",
      "0.416733974"
    ],
    [
      "0.6423198",
      "0.539807812"
    ],
    [
      "0.289477176",
      "0.416248487"
    ],
    [
      "0.501851372",
      "0.538855337"
    ],
    [
      "0.358976777",
      "0.537886545"
    ],
    [
      "0.5705983",
      "0.663066487"
    ],
    [
      "0.428940918",
      "0.660022052"
    ],
    [
      "0.498242295",
      "0.784239476"
    ]
  ]
}

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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