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 = 31

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 = 31

目标最小化覆盖半径

已知最好（未证明）0.0798972448089536737Kari J. Nurmela 的构型收录于 Nurmela（2000）Figure 4；论文连续值为 0.0798972448089536737。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 0.079897245252974。

已证明下界√(A(K)/(π·31))任何布局都不低于 0.066679837812019 · 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31 个等圆盘，共同半径 0.079897；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

0.079897245252974

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": [
    [
      "0.066536758",
      "0.044231545"
    ],
    [
      "0.135125444",
      "0.172392833"
    ],
    [
      "0.19662494",
      "0.048422993"
    ],
    [
      "0.202904252",
      "0.306991484"
    ],
    [
      "0.258857525",
      "0.183948964"
    ],
    [
      "0.289260482",
      "0.42773466"
    ],
    [
      "0.315837995",
      "0.057317994"
    ],
    [
      "0.333858458",
      "0.310690074"
    ],
    [
      "0.360963237",
      "0.54959827"
    ],
    [
      "0.387586589",
      "0.184435744"
    ],
    [
      "0.425854293",
      "0.428933068"
    ],
    [
      "0.435685161",
      "0.04757927"
    ],
    [
      "0.433815504",
      "0.667242853"
    ],
    [
      "0.460320367",
      "0.31409198"
    ],
    [
      "0.499919954",
      "0.172947087"
    ],
    [
      "0.500021728",
      "0.546526385"
    ],
    [
      "0.501052829",
      "0.786135096"
    ],
    [
      "0.564071573",
      "0.424479538"
    ],
    [
      "0.565557804",
      "0.045483772"
    ],
    [
      "0.572189576",
      "0.665419299"
    ],
    [
      "0.625092192",
      "0.176983626"
    ],
    [
      "0.637458511",
      "0.543492286"
    ],
    [
      "0.663782084",
      "0.312491598"
    ],
    [
      "0.692231615",
      "0.051615809"
    ],
    [
      "0.704170244",
      "0.426054819"
    ],
    [
      "0.743260434",
      "0.184755325"
    ],
    [
      "0.790189063",
      "0.310942119"
    ],
    [
      "0.812413718",
      "0.053660734"
    ],
    [
      "0.8675136",
      "0.18060447"
    ],
    [
      "0.935804559",
      "0.047565902"
    ],
    [
      "0.571245582",
      "0.298654683"
    ]
  ]
}

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": 31
}

当前第一名的答案

{
  "points": [
    [
      "0.066536758",
      "0.044231545"
    ],
    [
      "0.135125444",
      "0.172392833"
    ],
    [
      "0.19662494",
      "0.048422993"
    ],
    [
      "0.202904252",
      "0.306991484"
    ],
    [
      "0.258857525",
      "0.183948964"
    ],
    [
      "0.289260482",
      "0.42773466"
    ],
    [
      "0.315837995",
      "0.057317994"
    ],
    [
      "0.333858458",
      "0.310690074"
    ],
    [
      "0.360963237",
      "0.54959827"
    ],
    [
      "0.387586589",
      "0.184435744"
    ],
    [
      "0.425854293",
      "0.428933068"
    ],
    [
      "0.435685161",
      "0.04757927"
    ],
    [
      "0.433815504",
      "0.667242853"
    ],
    [
      "0.460320367",
      "0.31409198"
    ],
    [
      "0.499919954",
      "0.172947087"
    ],
    [
      "0.500021728",
      "0.546526385"
    ],
    [
      "0.501052829",
      "0.786135096"
    ],
    [
      "0.564071573",
      "0.424479538"
    ],
    [
      "0.565557804",
      "0.045483772"
    ],
    [
      "0.572189576",
      "0.665419299"
    ],
    [
      "0.625092192",
      "0.176983626"
    ],
    [
      "0.637458511",
      "0.543492286"
    ],
    [
      "0.663782084",
      "0.312491598"
    ],
    [
      "0.692231615",
      "0.051615809"
    ],
    [
      "0.704170244",
      "0.426054819"
    ],
    [
      "0.743260434",
      "0.184755325"
    ],
    [
      "0.790189063",
      "0.310942119"
    ],
    [
      "0.812413718",
      "0.053660734"
    ],
    [
      "0.8675136",
      "0.18060447"
    ],
    [
      "0.935804559",
      "0.047565902"
    ],
    [
      "0.571245582",
      "0.298654683"
    ]
  ]
}

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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