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 = 40

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 = 40

目标最小化覆盖半径

已证明下界√(A(K)/(π·40))任何布局都不低于 0.058700984636067 · 当前纪录高出下界 22.9%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40 个等圆盘，共同半径 0.072169；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

0.072168783986528

覆盖半径

答案来源MinMax Arena

解题方式本站参考构造

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": [
    [
      "0.937500001",
      "0.036084394"
    ],
    [
      "0.812500003",
      "0.036084392"
    ],
    [
      "0.687500003",
      "0.036084393"
    ],
    [
      "0.874999999",
      "0.14433757"
    ],
    [
      "0.562500004",
      "0.036084392"
    ],
    [
      "0.750000001",
      "0.144337568"
    ],
    [
      "0.437500004",
      "0.036084392"
    ],
    [
      "0.625000001",
      "0.144337568"
    ],
    [
      "0.812499999",
      "0.252590746"
    ],
    [
      "0.312500003",
      "0.036084391"
    ],
    [
      "0.500000001",
      "0.144337567"
    ],
    [
      "0.6875",
      "0.252590744"
    ],
    [
      "0.187500003",
      "0.036084392"
    ],
    [
      "0.375000001",
      "0.144337567"
    ],
    [
      "0.5625",
      "0.252590744"
    ],
    [
      "0.749999998",
      "0.360843921"
    ],
    [
      "0.062500001",
      "0.03608439"
    ],
    [
      "0.250000001",
      "0.144337566"
    ],
    [
      "0.4375",
      "0.252590742"
    ],
    [
      "0.624999999",
      "0.360843919"
    ],
    [
      "0.124999999",
      "0.144337565"
    ],
    [
      "0.3125",
      "0.252590742"
    ],
    [
      "0.499999999",
      "0.360843918"
    ],
    [
      "0.687499998",
      "0.469097097"
    ],
    [
      "0.187499999",
      "0.25259074"
    ],
    [
      "0.374999999",
      "0.360843917"
    ],
    [
      "0.562499999",
      "0.469097094"
    ],
    [
      "0.249999998",
      "0.360843915"
    ],
    [
      "0.437499999",
      "0.469097094"
    ],
    [
      "0.624999998",
      "0.577350272"
    ],
    [
      "0.312499998",
      "0.469097091"
    ],
    [
      "0.499999999",
      "0.577350269"
    ],
    [
      "0.374999998",
      "0.577350267"
    ],
    [
      "0.562499999",
      "0.685603447"
    ],
    [
      "0.437499999",
      "0.685603443"
    ],
    [
      "0.499999998",
      "0.79385662"
    ],
    [
      "0.125000003",
      "0"
    ],
    [
      "0.5",
      "0.866025404"
    ],
    [
      "0.562500001",
      "0.18042196"
    ],
    [
      "0.437499999",
      "0.39692831"
    ]
  ]
}

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": 40
}

当前第一名的答案

{
  "points": [
    [
      "0.937500001",
      "0.036084394"
    ],
    [
      "0.812500003",
      "0.036084392"
    ],
    [
      "0.687500003",
      "0.036084393"
    ],
    [
      "0.874999999",
      "0.14433757"
    ],
    [
      "0.562500004",
      "0.036084392"
    ],
    [
      "0.750000001",
      "0.144337568"
    ],
    [
      "0.437500004",
      "0.036084392"
    ],
    [
      "0.625000001",
      "0.144337568"
    ],
    [
      "0.812499999",
      "0.252590746"
    ],
    [
      "0.312500003",
      "0.036084391"
    ],
    [
      "0.500000001",
      "0.144337567"
    ],
    [
      "0.6875",
      "0.252590744"
    ],
    [
      "0.187500003",
      "0.036084392"
    ],
    [
      "0.375000001",
      "0.144337567"
    ],
    [
      "0.5625",
      "0.252590744"
    ],
    [
      "0.749999998",
      "0.360843921"
    ],
    [
      "0.062500001",
      "0.03608439"
    ],
    [
      "0.250000001",
      "0.144337566"
    ],
    [
      "0.4375",
      "0.252590742"
    ],
    [
      "0.624999999",
      "0.360843919"
    ],
    [
      "0.124999999",
      "0.144337565"
    ],
    [
      "0.3125",
      "0.252590742"
    ],
    [
      "0.499999999",
      "0.360843918"
    ],
    [
      "0.687499998",
      "0.469097097"
    ],
    [
      "0.187499999",
      "0.25259074"
    ],
    [
      "0.374999999",
      "0.360843917"
    ],
    [
      "0.562499999",
      "0.469097094"
    ],
    [
      "0.249999998",
      "0.360843915"
    ],
    [
      "0.437499999",
      "0.469097094"
    ],
    [
      "0.624999998",
      "0.577350272"
    ],
    [
      "0.312499998",
      "0.469097091"
    ],
    [
      "0.499999999",
      "0.577350269"
    ],
    [
      "0.374999998",
      "0.577350267"
    ],
    [
      "0.562499999",
      "0.685603447"
    ],
    [
      "0.437499999",
      "0.685603443"
    ],
    [
      "0.499999998",
      "0.79385662"
    ],
    [
      "0.125000003",
      "0"
    ],
    [
      "0.5",
      "0.866025404"
    ],
    [
      "0.562500001",
      "0.18042196"
    ],
    [
      "0.437499999",
      "0.39692831"
    ]
  ]
}

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

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