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 = 29

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 = 29

目标最小化三角形面积

已知最好（未证明）115.292243152047020714Erich 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29 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

115.292243152047020714

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": [
    [
      "10.189470088",
      "12.017621136"
    ],
    [
      "8.995480979",
      "10.413130558"
    ],
    [
      "10.982005134",
      "10.181350947"
    ],
    [
      "7.801491870",
      "8.808639980"
    ],
    [
      "9.788016026",
      "8.576860369"
    ],
    [
      "11.774540181",
      "8.345080758"
    ],
    [
      "6.607502761",
      "7.204149401"
    ],
    [
      "8.594026917",
      "6.972369790"
    ],
    [
      "10.580551072",
      "6.740590180"
    ],
    [
      "12.567075228",
      "6.508810569"
    ],
    [
      "5.413513653",
      "5.599658823"
    ],
    [
      "7.400037808",
      "5.367879212"
    ],
    [
      "9.386561963",
      "5.136099601"
    ],
    [
      "11.373086119",
      "4.904319990"
    ],
    [
      "13.359610274",
      "4.672540379"
    ],
    [
      "4.219524544",
      "3.995168245"
    ],
    [
      "6.206048699",
      "3.763388634"
    ],
    [
      "8.192572855",
      "3.531609023"
    ],
    [
      "10.179022178",
      "3.299187845"
    ],
    [
      "12.165621165",
      "3.068049801"
    ],
    [
      "14.152145321",
      "2.836270190"
    ],
    [
      "3.680965196",
      "2.069044200"
    ],
    [
      "7.208418964",
      "1.790505600"
    ],
    [
      "10.995122050",
      "1.446343346"
    ],
    [
      "1.990656198",
      "1.000000001"
    ],
    [
      "5.371274194",
      "1.000000001"
    ],
    [
      "9.045563734",
      "1.000000001"
    ],
    [
      "12.944680365",
      "1.000000001"
    ],
    [
      "14.944680367",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "16.465445159",
      "0.000000000"
    ],
    [
      "10.421249699",
      "14.004145292"
    ]
  ]
}

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

{
  "centers": [
    [
      "10.189470088",
      "12.017621136"
    ],
    [
      "8.995480979",
      "10.413130558"
    ],
    [
      "10.982005134",
      "10.181350947"
    ],
    [
      "7.801491870",
      "8.808639980"
    ],
    [
      "9.788016026",
      "8.576860369"
    ],
    [
      "11.774540181",
      "8.345080758"
    ],
    [
      "6.607502761",
      "7.204149401"
    ],
    [
      "8.594026917",
      "6.972369790"
    ],
    [
      "10.580551072",
      "6.740590180"
    ],
    [
      "12.567075228",
      "6.508810569"
    ],
    [
      "5.413513653",
      "5.599658823"
    ],
    [
      "7.400037808",
      "5.367879212"
    ],
    [
      "9.386561963",
      "5.136099601"
    ],
    [
      "11.373086119",
      "4.904319990"
    ],
    [
      "13.359610274",
      "4.672540379"
    ],
    [
      "4.219524544",
      "3.995168245"
    ],
    [
      "6.206048699",
      "3.763388634"
    ],
    [
      "8.192572855",
      "3.531609023"
    ],
    [
      "10.179022178",
      "3.299187845"
    ],
    [
      "12.165621165",
      "3.068049801"
    ],
    [
      "14.152145321",
      "2.836270190"
    ],
    [
      "3.680965196",
      "2.069044200"
    ],
    [
      "7.208418964",
      "1.790505600"
    ],
    [
      "10.995122050",
      "1.446343346"
    ],
    [
      "1.990656198",
      "1.000000001"
    ],
    [
      "5.371274194",
      "1.000000001"
    ],
    [
      "9.045563734",
      "1.000000001"
    ],
    [
      "12.944680365",
      "1.000000001"
    ],
    [
      "14.944680367",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "16.465445159",
      "0.000000000"
    ],
    [
      "10.421249699",
      "14.004145292"
    ]
  ]
}

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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