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 = 23

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 = 23

目标最大化半径之和

已知最好（未证明）2.484239482Berthold、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 2.484239494302877。本站将同一构型保守量化到九位网格，可验证半径和为 2.484239482；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 ↗

1.007355665y0

1234567891011121314151617181920212223

0x0.992644335

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

当前第一名

2.484239482

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": [
    [
      "0.132966745",
      "0.299487604",
      "0.132966743"
    ],
    [
      "0.906814931",
      "0.921526268",
      "0.085829395"
    ],
    [
      "0.859677591",
      "0.70786807",
      "0.132966744"
    ],
    [
      "0.714626715",
      "0.107586416",
      "0.107586411"
    ],
    [
      "0.132966745",
      "0.70786807",
      "0.132966744"
    ],
    [
      "0.278017622",
      "0.899769258",
      "0.107586405"
    ],
    [
      "0.611548491",
      "0.702233449",
      "0.115226322"
    ],
    [
      "0.085829406",
      "0.921526268",
      "0.085829395"
    ],
    [
      "0.085829406",
      "0.085829406",
      "0.085829403"
    ],
    [
      "0.278017622",
      "0.107586416",
      "0.107586411"
    ],
    [
      "0.381095845",
      "0.305122225",
      "0.115226322"
    ],
    [
      "0.381095845",
      "0.702233449",
      "0.115226322"
    ],
    [
      "0.496322168",
      "0.503677837",
      "0.11434161"
    ],
    [
      "0.906814931",
      "0.085829406",
      "0.085829403"
    ],
    [
      "0.723262932",
      "0.503677837",
      "0.11259915"
    ],
    [
      "0.859677591",
      "0.299487604",
      "0.132966743"
    ],
    [
      "0.078391128",
      "0.503677837",
      "0.078391124"
    ],
    [
      "0.914253208",
      "0.503677837",
      "0.078391124"
    ],
    [
      "0.714626715",
      "0.899769258",
      "0.107586405"
    ],
    [
      "0.496322168",
      "0.110740923",
      "0.110740921"
    ],
    [
      "0.269381404",
      "0.503677837",
      "0.11259915"
    ],
    [
      "0.611548491",
      "0.305122225",
      "0.115226322"
    ],
    [
      "0.496322168",
      "0.89661475",
      "0.110740913"
    ]
  ],
  "height": "1.007355665",
  "width": "0.992644335"
}

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": 23
}

当前第一名的答案

{
  "circles": [
    [
      "0.132966745",
      "0.299487604",
      "0.132966743"
    ],
    [
      "0.906814931",
      "0.921526268",
      "0.085829395"
    ],
    [
      "0.859677591",
      "0.70786807",
      "0.132966744"
    ],
    [
      "0.714626715",
      "0.107586416",
      "0.107586411"
    ],
    [
      "0.132966745",
      "0.70786807",
      "0.132966744"
    ],
    [
      "0.278017622",
      "0.899769258",
      "0.107586405"
    ],
    [
      "0.611548491",
      "0.702233449",
      "0.115226322"
    ],
    [
      "0.085829406",
      "0.921526268",
      "0.085829395"
    ],
    [
      "0.085829406",
      "0.085829406",
      "0.085829403"
    ],
    [
      "0.278017622",
      "0.107586416",
      "0.107586411"
    ],
    [
      "0.381095845",
      "0.305122225",
      "0.115226322"
    ],
    [
      "0.381095845",
      "0.702233449",
      "0.115226322"
    ],
    [
      "0.496322168",
      "0.503677837",
      "0.11434161"
    ],
    [
      "0.906814931",
      "0.085829406",
      "0.085829403"
    ],
    [
      "0.723262932",
      "0.503677837",
      "0.11259915"
    ],
    [
      "0.859677591",
      "0.299487604",
      "0.132966743"
    ],
    [
      "0.078391128",
      "0.503677837",
      "0.078391124"
    ],
    [
      "0.914253208",
      "0.503677837",
      "0.078391124"
    ],
    [
      "0.714626715",
      "0.899769258",
      "0.107586405"
    ],
    [
      "0.496322168",
      "0.110740923",
      "0.110740921"
    ],
    [
      "0.269381404",
      "0.503677837",
      "0.11259915"
    ],
    [
      "0.611548491",
      "0.305122225",
      "0.115226322"
    ],
    [
      "0.496322168",
      "0.89661475",
      "0.110740913"
    ]
  ],
  "height": "1.007355665",
  "width": "0.992644335"
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。发帖即公开署名，与纪录同一个名字；署名后的 #编号是账号的注册序号，冒不了名。发言资格与实绩绑定：破过一次纪录，就永久拥有发言权。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？