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 = 22

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 = 22

目标最大化半径之和

已知最好（未证明）2.425137213Berthold、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 2.425137224684899。本站将同一构型保守量化到九位网格，可验证半径和为 2.425137213；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 ↗

1.030056012y0

12345678910111213141516171819202122

0x0.969943988

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

2.425137213

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": [
    [
      "0.125261331",
      "0.125261331",
      "0.125261329"
    ],
    [
      "0.709388272",
      "0.711997073",
      "0.110271025"
    ],
    [
      "0.894799567",
      "0.319299367",
      "0.075144417"
    ],
    [
      "0.842490407",
      "0.51502801",
      "0.127453581"
    ],
    [
      "0.484971995",
      "0.316807059",
      "0.114148739"
    ],
    [
      "0.260555718",
      "0.711997073",
      "0.110271025"
    ],
    [
      "0.484971995",
      "0.713248961",
      "0.11414874"
    ],
    [
      "0.260555718",
      "0.318058946",
      "0.110271025"
    ],
    [
      "0.60226217",
      "0.117290176",
      "0.117290174"
    ],
    [
      "0.36993958",
      "0.51502801",
      "0.115032413"
    ],
    [
      "0.709388272",
      "0.318058946",
      "0.110271025"
    ],
    [
      "0.844682659",
      "0.904794689",
      "0.12526132"
    ],
    [
      "0.36768182",
      "0.117290176",
      "0.117290174"
    ],
    [
      "0.36768182",
      "0.912765844",
      "0.117290165"
    ],
    [
      "0.127453583",
      "0.51502801",
      "0.127453581"
    ],
    [
      "0.075144423",
      "0.710756653",
      "0.075144418"
    ],
    [
      "0.075144423",
      "0.319299367",
      "0.075144417"
    ],
    [
      "0.125261331",
      "0.904794689",
      "0.12526132"
    ],
    [
      "0.60000441",
      "0.51502801",
      "0.115032413"
    ],
    [
      "0.894799567",
      "0.710756653",
      "0.075144418"
    ],
    [
      "0.60226217",
      "0.912765844",
      "0.117290165"
    ],
    [
      "0.844682659",
      "0.125261331",
      "0.125261329"
    ]
  ],
  "height": "1.030056012",
  "width": "0.969943988"
}

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": 22
}

当前第一名的答案

{
  "circles": [
    [
      "0.125261331",
      "0.125261331",
      "0.125261329"
    ],
    [
      "0.709388272",
      "0.711997073",
      "0.110271025"
    ],
    [
      "0.894799567",
      "0.319299367",
      "0.075144417"
    ],
    [
      "0.842490407",
      "0.51502801",
      "0.127453581"
    ],
    [
      "0.484971995",
      "0.316807059",
      "0.114148739"
    ],
    [
      "0.260555718",
      "0.711997073",
      "0.110271025"
    ],
    [
      "0.484971995",
      "0.713248961",
      "0.11414874"
    ],
    [
      "0.260555718",
      "0.318058946",
      "0.110271025"
    ],
    [
      "0.60226217",
      "0.117290176",
      "0.117290174"
    ],
    [
      "0.36993958",
      "0.51502801",
      "0.115032413"
    ],
    [
      "0.709388272",
      "0.318058946",
      "0.110271025"
    ],
    [
      "0.844682659",
      "0.904794689",
      "0.12526132"
    ],
    [
      "0.36768182",
      "0.117290176",
      "0.117290174"
    ],
    [
      "0.36768182",
      "0.912765844",
      "0.117290165"
    ],
    [
      "0.127453583",
      "0.51502801",
      "0.127453581"
    ],
    [
      "0.075144423",
      "0.710756653",
      "0.075144418"
    ],
    [
      "0.075144423",
      "0.319299367",
      "0.075144417"
    ],
    [
      "0.125261331",
      "0.904794689",
      "0.12526132"
    ],
    [
      "0.60000441",
      "0.51502801",
      "0.115032413"
    ],
    [
      "0.894799567",
      "0.710756653",
      "0.075144418"
    ],
    [
      "0.60226217",
      "0.912765844",
      "0.117290165"
    ],
    [
      "0.844682659",
      "0.125261331",
      "0.125261329"
    ]
  ],
  "height": "1.030056012",
  "width": "0.969943988"
}

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？