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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 = 21

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 = 21
目标最大化 半径之和
已知最好（未证明）2.365832315Berthold、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 2.365832326862653。本站将同一构型保守量化到九位网格，可验证半径和为 2.365832315；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 ↗
1.023268893y0
0x0.976731107
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 = 21 的当前纪录构型，2.365832315
当前第一名

2.365832315

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": [
    [
      "0.8527695",
      "0.123961609",
      "0.123961607"
    ],
    [
      "0.298203233",
      "0.731338595",
      "0.086412063"
    ],
    [
      "0.71756247",
      "0.314526273",
      "0.109695898"
    ],
    [
      "0.385895121",
      "0.112921699",
      "0.112921697"
    ],
    [
      "0.901989604",
      "0.316472144",
      "0.074741499"
    ],
    [
      "0.385895121",
      "0.910347202",
      "0.112921689"
    ],
    [
      "0.71756247",
      "0.708742629",
      "0.109695898"
    ],
    [
      "0.347821024",
      "0.511634451",
      "0.138825223"
    ],
    [
      "0.901989604",
      "0.706796757",
      "0.074741499"
    ],
    [
      "0.613889701",
      "0.115083127",
      "0.115083125"
    ],
    [
      "0.298203233",
      "0.291930306",
      "0.086412062"
    ],
    [
      "0.8527695",
      "0.899307293",
      "0.123961598"
    ],
    [
      "0.495899584",
      "0.309144341",
      "0.11203231"
    ],
    [
      "0.849330917",
      "0.511634451",
      "0.12740019"
    ],
    [
      "0.604288487",
      "0.511634451",
      "0.117642236"
    ],
    [
      "0.137071297",
      "0.886197604",
      "0.137071286"
    ],
    [
      "0.119062857",
      "0.392571594",
      "0.119062856"
    ],
    [
      "0.613889701",
      "0.908185774",
      "0.115083117"
    ],
    [
      "0.495899584",
      "0.71412456",
      "0.112032311"
    ],
    [
      "0.119062857",
      "0.630697308",
      "0.119062855"
    ],
    [
      "0.137071297",
      "0.137071297",
      "0.137071296"
    ]
  ],
  "height": "1.023268893",
  "width": "0.976731107"
}
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": 21
}

当前第一名的答案

{
  "circles": [
    [
      "0.8527695",
      "0.123961609",
      "0.123961607"
    ],
    [
      "0.298203233",
      "0.731338595",
      "0.086412063"
    ],
    [
      "0.71756247",
      "0.314526273",
      "0.109695898"
    ],
    [
      "0.385895121",
      "0.112921699",
      "0.112921697"
    ],
    [
      "0.901989604",
      "0.316472144",
      "0.074741499"
    ],
    [
      "0.385895121",
      "0.910347202",
      "0.112921689"
    ],
    [
      "0.71756247",
      "0.708742629",
      "0.109695898"
    ],
    [
      "0.347821024",
      "0.511634451",
      "0.138825223"
    ],
    [
      "0.901989604",
      "0.706796757",
      "0.074741499"
    ],
    [
      "0.613889701",
      "0.115083127",
      "0.115083125"
    ],
    [
      "0.298203233",
      "0.291930306",
      "0.086412062"
    ],
    [
      "0.8527695",
      "0.899307293",
      "0.123961598"
    ],
    [
      "0.495899584",
      "0.309144341",
      "0.11203231"
    ],
    [
      "0.849330917",
      "0.511634451",
      "0.12740019"
    ],
    [
      "0.604288487",
      "0.511634451",
      "0.117642236"
    ],
    [
      "0.137071297",
      "0.886197604",
      "0.137071286"
    ],
    [
      "0.119062857",
      "0.392571594",
      "0.119062856"
    ],
    [
      "0.613889701",
      "0.908185774",
      "0.115083117"
    ],
    [
      "0.495899584",
      "0.71412456",
      "0.112032311"
    ],
    [
      "0.119062857",
      "0.630697308",
      "0.119062855"
    ],
    [
      "0.137071297",
      "0.137071297",
      "0.137071296"
    ]
  ],
  "height": "1.023268893",
  "width": "0.976731107"
}

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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