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 = 32

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 = 32

目标最小化三角形面积

已知最好（未证明）127.07490417923377167Erich 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32 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

127.07490417923377167

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": [
    [
      "6.815936727",
      "12.743782415"
    ],
    [
      "7.986641417",
      "11.122224357"
    ],
    [
      "5.996978600",
      "10.919143384"
    ],
    [
      "9.157346107",
      "9.500666298"
    ],
    [
      "7.167683290",
      "9.297585326"
    ],
    [
      "5.178020473",
      "9.094504353"
    ],
    [
      "10.328050797",
      "7.879108240"
    ],
    [
      "8.338387980",
      "7.676027267"
    ],
    [
      "6.348725163",
      "7.472946295"
    ],
    [
      "4.359062345",
      "7.269865322"
    ],
    [
      "11.498755487",
      "6.257550182"
    ],
    [
      "9.509092670",
      "6.054469209"
    ],
    [
      "7.519429853",
      "5.851388236"
    ],
    [
      "5.529767035",
      "5.648307264"
    ],
    [
      "3.540104218",
      "5.445226291"
    ],
    [
      "12.669460177",
      "4.635992123"
    ],
    [
      "10.679797360",
      "4.432911150"
    ],
    [
      "8.690134543",
      "4.229830178"
    ],
    [
      "6.700471725",
      "4.026749205"
    ],
    [
      "2.721146091",
      "3.620587260"
    ],
    [
      "4.710329755",
      "3.412521954"
    ],
    [
      "12.040009645",
      "2.737626501"
    ],
    [
      "14.040009647",
      "2.737626501"
    ],
    [
      "9.860839233",
      "2.608272119"
    ],
    [
      "6.966365549",
      "2.044502883"
    ],
    [
      "3.402861305",
      "1.740357387"
    ],
    [
      "15.104494893",
      "1.044404221"
    ],
    [
      "1.544940169",
      "1.000000001"
    ],
    [
      "5.260782441",
      "1.000000001"
    ],
    [
      "8.671948657",
      "1.000000001"
    ],
    [
      "11.049729808",
      "1.000000001"
    ],
    [
      "13.049729810",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.249855980",
      "0.000000000"
    ],
    [
      "6.612855755",
      "14.733445233"
    ]
  ]
}

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": 32
}

当前第一名的答案

{
  "centers": [
    [
      "6.815936727",
      "12.743782415"
    ],
    [
      "7.986641417",
      "11.122224357"
    ],
    [
      "5.996978600",
      "10.919143384"
    ],
    [
      "9.157346107",
      "9.500666298"
    ],
    [
      "7.167683290",
      "9.297585326"
    ],
    [
      "5.178020473",
      "9.094504353"
    ],
    [
      "10.328050797",
      "7.879108240"
    ],
    [
      "8.338387980",
      "7.676027267"
    ],
    [
      "6.348725163",
      "7.472946295"
    ],
    [
      "4.359062345",
      "7.269865322"
    ],
    [
      "11.498755487",
      "6.257550182"
    ],
    [
      "9.509092670",
      "6.054469209"
    ],
    [
      "7.519429853",
      "5.851388236"
    ],
    [
      "5.529767035",
      "5.648307264"
    ],
    [
      "3.540104218",
      "5.445226291"
    ],
    [
      "12.669460177",
      "4.635992123"
    ],
    [
      "10.679797360",
      "4.432911150"
    ],
    [
      "8.690134543",
      "4.229830178"
    ],
    [
      "6.700471725",
      "4.026749205"
    ],
    [
      "2.721146091",
      "3.620587260"
    ],
    [
      "4.710329755",
      "3.412521954"
    ],
    [
      "12.040009645",
      "2.737626501"
    ],
    [
      "14.040009647",
      "2.737626501"
    ],
    [
      "9.860839233",
      "2.608272119"
    ],
    [
      "6.966365549",
      "2.044502883"
    ],
    [
      "3.402861305",
      "1.740357387"
    ],
    [
      "15.104494893",
      "1.044404221"
    ],
    [
      "1.544940169",
      "1.000000001"
    ],
    [
      "5.260782441",
      "1.000000001"
    ],
    [
      "8.671948657",
      "1.000000001"
    ],
    [
      "11.049729808",
      "1.000000001"
    ],
    [
      "13.049729810",
      "1.000000001"
    ]
  ],
  "triangle": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "17.249855980",
      "0.000000000"
    ],
    [
      "6.612855755",
      "14.733445233"
    ]
  ]
}

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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