P66 · 装箱与覆盖 · 经典问题 · 难

最小面积三角形内的单位圆装箱 · n = 31

文献中也称circles in arbitrary trianglesunit circles in a minimum-area trianglecircle packing in a triangle of variable shape

在任意三角形中放置 n 个半径为 1、内部互不重叠的圆；三角形的三个顶点也是答案的一部分。让这个三角形的面积尽可能小。

子题n = 31

目标最小化三角形面积

已知最好（未证明）122.58804487656187252Erich Friedman 汇总表所载 Tej Stead 构形；本站从公布图形重建坐标并在九位网格上精确验证

严格定义

容器容器是形状不预先固定的非退化三角形；三个顶点和所有圆心都位于 0 到 200 的坐标框内
提交三个按逆时针顺序排列的三角形顶点，以及恰好 n 个圆心
圆每个圆的半径固定为 1；圆的内部两两不相交，相切允许
容纳每个圆都完整落在三角形内，可以与边相切
目标让三角形面积尽可能小

挑战这个纪录 ↗

已验证构造31 个半径为 1 的圆

帮助理解

容器也是变量

普通装箱只移动圆；这里连三条边的方向都能改变。一条边转动一点，可能同时释放多个接触，也可能让整套构形失稳。

难点是接触结构

候选最优解通常由圆—圆、圆—边相切关系卡死。真正的搜索不是把圆平均铺开，而是在大量可能的接触图之间找到对的那一张。

从非平凡实例开始

Friedman 的公开表给出了 n≤50 的当前最好已知构形。n=1、2 已证明，n=3 仍是简单的等边三角形排布，因此本站从 n=4 开放；匹配公开构形只是到达前沿，打破它们才是新纪录。

查看来源

当前第一名

122.58804487656187252

三角形面积

已追平已知最好

答案来源Tej Stead

解题方式公开参考构造

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

坐标单位就是圆的半径：每个圆半径固定为 1。三个顶点和所有圆心都写在 0≤x,y≤200 的坐标框内，顶点必须逆时针排列。

提交按逆时针排列的三个顶点 triangle，以及恰好 n 个圆心 centers。圆的半径固定为 1；所有数写成最多九位小数的字符串。

当前第一名的答案

{
  "centers": [
    [
      "11.712230595",
      "12.293207294"
    ],
    [
      "10.420323202",
      "10.766454107"
    ],
    [
      "12.388483944",
      "10.411006079"
    ],
    [
      "9.128415809",
      "9.239700919"
    ],
    [
      "11.096576551",
      "8.884252891"
    ],
    [
      "13.064737293",
      "8.528804863"
    ],
    [
      "7.836508416",
      "7.712947732"
    ],
    [
      "9.804669158",
      "7.357499704"
    ],
    [
      "11.772829900",
      "7.002051676"
    ],
    [
      "13.740990642",
      "6.646603648"
    ],
    [
      "6.544601023",
      "6.186194545"
    ],
    [
      "8.512761765",
      "5.830746516"
    ],
    [
      "10.480922507",
      "5.475298488"
    ],
    [
      "12.449083249",
      "5.119850460"
    ],
    [
      "14.417243991",
      "4.764402432"
    ],
    [
      "5.252693629",
      "4.659441357"
    ],
    [
      "7.177938668",
      "4.033795876"
    ],
    [
      "9.189015114",
      "3.948545301"
    ],
    [
      "11.152080137",
      "3.565950833"
    ],
    [
      "13.125336598",
      "3.237649245"
    ],
    [
      "15.093497341",
      "2.882201217"
    ],
    [
      "3.648402369",
      "2.763518469"
    ],
    [
      "5.647353913",
      "2.698767170"
    ],
    [
      "8.351465855",
      "2.132364900"
    ],
    [
      "11.884888221",
      "1.668800039"
    ],
    [
      "2.579117888",
      "1.055341269"
    ],
    [
      "4.591801871",
      "1.000000001"
    ],
    [
      "6.702905955",
      "1.000000001"
    ],
    [
      "10.000025755",
      "1.000000001"
    ],
    [
      "13.769750688",
      "1.000000001"
    ],
    [
      "15.769750690",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.191624893",
      "0.000000000"
    ],
    [
      "12.067678623",
      "14.261368037"
    ]
  ]
}

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

{
  "centers": [
    [
      "11.712230595",
      "12.293207294"
    ],
    [
      "10.420323202",
      "10.766454107"
    ],
    [
      "12.388483944",
      "10.411006079"
    ],
    [
      "9.128415809",
      "9.239700919"
    ],
    [
      "11.096576551",
      "8.884252891"
    ],
    [
      "13.064737293",
      "8.528804863"
    ],
    [
      "7.836508416",
      "7.712947732"
    ],
    [
      "9.804669158",
      "7.357499704"
    ],
    [
      "11.772829900",
      "7.002051676"
    ],
    [
      "13.740990642",
      "6.646603648"
    ],
    [
      "6.544601023",
      "6.186194545"
    ],
    [
      "8.512761765",
      "5.830746516"
    ],
    [
      "10.480922507",
      "5.475298488"
    ],
    [
      "12.449083249",
      "5.119850460"
    ],
    [
      "14.417243991",
      "4.764402432"
    ],
    [
      "5.252693629",
      "4.659441357"
    ],
    [
      "7.177938668",
      "4.033795876"
    ],
    [
      "9.189015114",
      "3.948545301"
    ],
    [
      "11.152080137",
      "3.565950833"
    ],
    [
      "13.125336598",
      "3.237649245"
    ],
    [
      "15.093497341",
      "2.882201217"
    ],
    [
      "3.648402369",
      "2.763518469"
    ],
    [
      "5.647353913",
      "2.698767170"
    ],
    [
      "8.351465855",
      "2.132364900"
    ],
    [
      "11.884888221",
      "1.668800039"
    ],
    [
      "2.579117888",
      "1.055341269"
    ],
    [
      "4.591801871",
      "1.000000001"
    ],
    [
      "6.702905955",
      "1.000000001"
    ],
    [
      "10.000025755",
      "1.000000001"
    ],
    [
      "13.769750688",
      "1.000000001"
    ],
    [
      "15.769750690",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.191624893",
      "0.000000000"
    ],
    [
      "12.067678623",
      "14.261368037"
    ]
  ]
}

提交按逆时针排列的三个顶点 triangle，以及恰好 n 个圆心 centers。圆的半径固定为 1；所有数写成最多九位小数的字符串。 · 验证器 v1.0.0

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