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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 = 24

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 = 24
目标最大化 半径之和
已知最好（未证明）2.53534404Berthold、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 2.535344050819014。本站将同一构型保守量化到九位网格，可验证半径和为 2.53534404；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 ↗
1.012408783y0
0x0.987591217
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 = 24 的当前纪录构型，2.53534404
当前第一名

2.53534404

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": [
    [
      "0.499356224",
      "0.263855601",
      "0.110245354"
    ],
    [
      "0.90449121",
      "0.08310001",
      "0.083100006"
    ],
    [
      "0.090809377",
      "0.921599413",
      "0.090809366"
    ],
    [
      "0.498078326",
      "0.076807307",
      "0.076807303"
    ],
    [
      "0.713792959",
      "0.949687978",
      "0.062720801"
    ],
    [
      "0.311451265",
      "0.608825191",
      "0.116819776"
    ],
    [
      "0.88265459",
      "0.690151612",
      "0.104936627"
    ],
    [
      "0.878914572",
      "0.903732142",
      "0.108676637"
    ],
    [
      "0.516584665",
      "0.708314633",
      "0.111166758"
    ],
    [
      "0.693839282",
      "0.369278465",
      "0.110973175"
    ],
    [
      "0.319067855",
      "0.868971142",
      "0.143437637"
    ],
    [
      "0.697709403",
      "0.587031573",
      "0.106814319"
    ],
    [
      "0.30528949",
      "0.377459885",
      "0.114627563"
    ],
    [
      "0.107413718",
      "0.276725041",
      "0.107413712"
    ],
    [
      "0.297091097",
      "0.131484455",
      "0.131484452"
    ],
    [
      "0.112121945",
      "0.496210208",
      "0.112121942"
    ],
    [
      "0.55667328",
      "0.914010044",
      "0.098398734"
    ],
    [
      "0.697284909",
      "0.129165004",
      "0.129165001"
    ],
    [
      "0.085292601",
      "0.085292601",
      "0.085292598"
    ],
    [
      "0.707625637",
      "0.790345349",
      "0.096741132"
    ],
    [
      "0.504042966",
      "0.485776239",
      "0.111724762"
    ],
    [
      "0.881338205",
      "0.478966073",
      "0.106253012"
    ],
    [
      "0.883750714",
      "0.268886408",
      "0.1038405"
    ],
    [
      "0.111772875",
      "0.720104753",
      "0.111772873"
    ]
  ],
  "height": "1.012408783",
  "width": "0.987591217"
}
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": 24
}

当前第一名的答案

{
  "circles": [
    [
      "0.499356224",
      "0.263855601",
      "0.110245354"
    ],
    [
      "0.90449121",
      "0.08310001",
      "0.083100006"
    ],
    [
      "0.090809377",
      "0.921599413",
      "0.090809366"
    ],
    [
      "0.498078326",
      "0.076807307",
      "0.076807303"
    ],
    [
      "0.713792959",
      "0.949687978",
      "0.062720801"
    ],
    [
      "0.311451265",
      "0.608825191",
      "0.116819776"
    ],
    [
      "0.88265459",
      "0.690151612",
      "0.104936627"
    ],
    [
      "0.878914572",
      "0.903732142",
      "0.108676637"
    ],
    [
      "0.516584665",
      "0.708314633",
      "0.111166758"
    ],
    [
      "0.693839282",
      "0.369278465",
      "0.110973175"
    ],
    [
      "0.319067855",
      "0.868971142",
      "0.143437637"
    ],
    [
      "0.697709403",
      "0.587031573",
      "0.106814319"
    ],
    [
      "0.30528949",
      "0.377459885",
      "0.114627563"
    ],
    [
      "0.107413718",
      "0.276725041",
      "0.107413712"
    ],
    [
      "0.297091097",
      "0.131484455",
      "0.131484452"
    ],
    [
      "0.112121945",
      "0.496210208",
      "0.112121942"
    ],
    [
      "0.55667328",
      "0.914010044",
      "0.098398734"
    ],
    [
      "0.697284909",
      "0.129165004",
      "0.129165001"
    ],
    [
      "0.085292601",
      "0.085292601",
      "0.085292598"
    ],
    [
      "0.707625637",
      "0.790345349",
      "0.096741132"
    ],
    [
      "0.504042966",
      "0.485776239",
      "0.111724762"
    ],
    [
      "0.881338205",
      "0.478966073",
      "0.106253012"
    ],
    [
      "0.883750714",
      "0.268886408",
      "0.1038405"
    ],
    [
      "0.111772875",
      "0.720104753",
      "0.111772873"
    ]
  ],
  "height": "1.012408783",
  "width": "0.987591217"
}

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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