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P67 · 装箱与覆盖 · 经典问题 · 应用前沿 · 难

周长固定矩形内的可变半径圆 · n = 19

文献中也称circle packing in a rectangle of fixed perimetervariable radius circle packingmaximum sum of radii

在一个周长为 4、长宽均可改变的矩形内放入 n 个内部互不重叠的圆。每个圆的半径也可独立改变，使半径之和最大。

子题n = 19
目标最大化 半径之和
已知最好（未证明）2.24156519Berthold、Kamp、Mexi、Pokutta 与 Polik 公开仓库中的周长 4 矩形构型；连续构型的半径和为 2.241565197559671。本站将同一构型保守量化到九位网格，可验证半径和为 2.24156519；最优性未知

严格定义

  • 容器width>0、height>0，且 width+height=2；左下角固定为 (0,0)
  • 提交恰好 n 个圆，每个写成 [x,y,r]，半径严格为正
  • 约束每个圆完整位于矩形内；圆的内部两两不相交，相切允许
  • 目标最大化所有半径的精确整数和
挑战这个纪录 ↗
1.033357309y0
0x0.966642691
已验证构造计分是全部半径之和

帮助理解

容器和货物一起优化

固定正方形时只能调整圆；这里把一点周长从高度移给宽度，会改变所有贴边圆和空隙。它是容器选型与装箱设计耦合的最小模型。

为什么不平均

相同半径通常浪费曲边之间的空隙。大圆决定矩形比例，小圆填补接触图中的孔洞；增加一个圆甚至可能令整个长宽比跳到另一种结构。

公开前沿

2026 年的公开计算项目给出了 n≤60 的完整坐标。本站从第一个非平凡实例 n=3 开始展示其可验证九位网格版本；这些是最好已知构型，不是最优性证明。

查看来源
周长固定矩形内的可变半径圆 n = 19 的当前纪录构型，2.24156519
当前第一名

2.24156519

半径之和

已追平已知最好
答案来源Timo Berthold, Dominik Kamp, Gonzalo Mexi, Sebastian Pokutta and Ilja Polik
解题方式公开参考构造
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

矩形左下角为 (0,0)，宽为 width、高为 height；周长 2(width+height)=4。

提交 width、height，以及恰好 n 个 [x, y, r]。width + height 必须精确等于 2；所有数是最多九位小数的字符串。

当前第一名的答案

{
  "circles": [
    [
      "0.805270654",
      "0.678545523",
      "0.161372035"
    ],
    [
      "0.121957356",
      "0.626230747",
      "0.121957346"
    ],
    [
      "0.309356263",
      "0.726427496",
      "0.090546156"
    ],
    [
      "0.404552258",
      "0.913765707",
      "0.119591601"
    ],
    [
      "0.37393459",
      "0.107908095",
      "0.107908094"
    ],
    [
      "0.133703735",
      "0.133703735",
      "0.133703726"
    ],
    [
      "0.645563445",
      "0.911930741",
      "0.121426565"
    ],
    [
      "0.669107127",
      "0.280972093",
      "0.082641384"
    ],
    [
      "0.521589779",
      "0.701045569",
      "0.123199732"
    ],
    [
      "0.831302479",
      "0.135340212",
      "0.135340212"
    ],
    [
      "0.842093559",
      "0.395005425",
      "0.124549122"
    ],
    [
      "0.866313424",
      "0.933028041",
      "0.100329267"
    ],
    [
      "0.143003181",
      "0.890354127",
      "0.14300317"
    ],
    [
      "0.351739913",
      "0.49919889",
      "0.140601418"
    ],
    [
      "0.611116651",
      "0.475032822",
      "0.11989864"
    ],
    [
      "0.589682456",
      "0.10783978",
      "0.10783978"
    ],
    [
      "0.481933891",
      "0.291169965",
      "0.104809454"
    ],
    [
      "0.293158918",
      "0.282186045",
      "0.084179159"
    ],
    [
      "0.118668339",
      "0.385627532",
      "0.118668329"
    ]
  ],
  "height": "1.033357309",
  "width": "0.966642691"
}
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": 19
}

当前第一名的答案

{
  "circles": [
    [
      "0.805270654",
      "0.678545523",
      "0.161372035"
    ],
    [
      "0.121957356",
      "0.626230747",
      "0.121957346"
    ],
    [
      "0.309356263",
      "0.726427496",
      "0.090546156"
    ],
    [
      "0.404552258",
      "0.913765707",
      "0.119591601"
    ],
    [
      "0.37393459",
      "0.107908095",
      "0.107908094"
    ],
    [
      "0.133703735",
      "0.133703735",
      "0.133703726"
    ],
    [
      "0.645563445",
      "0.911930741",
      "0.121426565"
    ],
    [
      "0.669107127",
      "0.280972093",
      "0.082641384"
    ],
    [
      "0.521589779",
      "0.701045569",
      "0.123199732"
    ],
    [
      "0.831302479",
      "0.135340212",
      "0.135340212"
    ],
    [
      "0.842093559",
      "0.395005425",
      "0.124549122"
    ],
    [
      "0.866313424",
      "0.933028041",
      "0.100329267"
    ],
    [
      "0.143003181",
      "0.890354127",
      "0.14300317"
    ],
    [
      "0.351739913",
      "0.49919889",
      "0.140601418"
    ],
    [
      "0.611116651",
      "0.475032822",
      "0.11989864"
    ],
    [
      "0.589682456",
      "0.10783978",
      "0.10783978"
    ],
    [
      "0.481933891",
      "0.291169965",
      "0.104809454"
    ],
    [
      "0.293158918",
      "0.282186045",
      "0.084179159"
    ],
    [
      "0.118668339",
      "0.385627532",
      "0.118668329"
    ]
  ],
  "height": "1.033357309",
  "width": "0.966642691"
}

提交 width、height，以及恰好 n 个 [x, y, r]。width + height 必须精确等于 2；所有数是最多九位小数的字符串。 · 验证器 v1.0.0

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