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 = 35

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 = 35

目标最小化覆盖半径

已知最好（未证明）0.0751604548962267707Kari J. Nurmela 的构型收录于 Nurmela（2000）Figure 4；论文连续值为 0.0751604548962267707。其最优性尚未证明；论文没有发布可直接导入的坐标；本站从接触图复原的九位坐标证书可验证到 0.075163717460255，与论文值的差异被明确保留而未冒充原始数据。

已证明下界√(A(K)/(π·35))任何布局都不低于 0.062753992221270 · 当前纪录高出下界 19.8%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35 个等圆盘，共同半径 0.075164；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

0.075163717460255

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": [
    [
      "0.072858584",
      "0.018471899"
    ],
    [
      "0.123935797",
      "0.11395678"
    ],
    [
      "0.172538116",
      "0.22637384"
    ],
    [
      "0.21722654",
      "0.023151547"
    ],
    [
      "0.234283567",
      "0.346527778"
    ],
    [
      "0.253734882",
      "0.115985611"
    ],
    [
      "0.307855118",
      "0.459429469"
    ],
    [
      "0.298229976",
      "0.234496429"
    ],
    [
      "0.359319112",
      "0.025837882"
    ],
    [
      "0.364470566",
      "0.347908201"
    ],
    [
      "0.376135086",
      "0.567675122"
    ],
    [
      "0.382117906",
      "0.129439793"
    ],
    [
      "0.427869709",
      "0.234921536"
    ],
    [
      "0.440739447",
      "0.455459367"
    ],
    [
      "0.436740124",
      "0.679183823"
    ],
    [
      "0.501541976",
      "0.022745176"
    ],
    [
      "0.497000595",
      "0.342960088"
    ],
    [
      "0.507226079",
      "0.567136566"
    ],
    [
      "0.501204673",
      "0.790871341"
    ],
    [
      "0.549211668",
      "0.224875603"
    ],
    [
      "0.566910779",
      "0.677097268"
    ],
    [
      "0.570909586",
      "0.45334087"
    ],
    [
      "0.601113731",
      "0.114263604"
    ],
    [
      "0.638162707",
      "0.56360369"
    ],
    [
      "0.624506622",
      "0.335100299"
    ],
    [
      "0.645175891",
      "0.021596574"
    ],
    [
      "0.680994151",
      "0.223784529"
    ],
    [
      "0.698593896",
      "0.445147729"
    ],
    [
      "0.73591312",
      "0.119179123"
    ],
    [
      "0.756467389",
      "0.333675873"
    ],
    [
      "0.788019967",
      "0.025097548"
    ],
    [
      "0.815971823",
      "0.228366697"
    ],
    [
      "0.871448598",
      "0.125039613"
    ],
    [
      "0.929434895",
      "0.025887261"
    ],
    [
      "0.471213621",
      "0.1120585"
    ]
  ]
}

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": 35
}

当前第一名的答案

{
  "points": [
    [
      "0.072858584",
      "0.018471899"
    ],
    [
      "0.123935797",
      "0.11395678"
    ],
    [
      "0.172538116",
      "0.22637384"
    ],
    [
      "0.21722654",
      "0.023151547"
    ],
    [
      "0.234283567",
      "0.346527778"
    ],
    [
      "0.253734882",
      "0.115985611"
    ],
    [
      "0.307855118",
      "0.459429469"
    ],
    [
      "0.298229976",
      "0.234496429"
    ],
    [
      "0.359319112",
      "0.025837882"
    ],
    [
      "0.364470566",
      "0.347908201"
    ],
    [
      "0.376135086",
      "0.567675122"
    ],
    [
      "0.382117906",
      "0.129439793"
    ],
    [
      "0.427869709",
      "0.234921536"
    ],
    [
      "0.440739447",
      "0.455459367"
    ],
    [
      "0.436740124",
      "0.679183823"
    ],
    [
      "0.501541976",
      "0.022745176"
    ],
    [
      "0.497000595",
      "0.342960088"
    ],
    [
      "0.507226079",
      "0.567136566"
    ],
    [
      "0.501204673",
      "0.790871341"
    ],
    [
      "0.549211668",
      "0.224875603"
    ],
    [
      "0.566910779",
      "0.677097268"
    ],
    [
      "0.570909586",
      "0.45334087"
    ],
    [
      "0.601113731",
      "0.114263604"
    ],
    [
      "0.638162707",
      "0.56360369"
    ],
    [
      "0.624506622",
      "0.335100299"
    ],
    [
      "0.645175891",
      "0.021596574"
    ],
    [
      "0.680994151",
      "0.223784529"
    ],
    [
      "0.698593896",
      "0.445147729"
    ],
    [
      "0.73591312",
      "0.119179123"
    ],
    [
      "0.756467389",
      "0.333675873"
    ],
    [
      "0.788019967",
      "0.025097548"
    ],
    [
      "0.815971823",
      "0.228366697"
    ],
    [
      "0.871448598",
      "0.125039613"
    ],
    [
      "0.929434895",
      "0.025887261"
    ],
    [
      "0.471213621",
      "0.1120585"
    ]
  ]
}

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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