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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 = 27

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 = 27
目标最大化 半径之和
已知最好（未证明）2.690155069Berthold、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 2.690155081571631。本站将同一构型保守量化到九位网格，可验证半径和为 2.690155069；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 ↗
1.051448529y0
0x0.948551471
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 = 27 的当前纪录构型，2.690155069
当前第一名

2.690155069

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": [
    [
      "0.845094457",
      "0.544247521",
      "0.103457014"
    ],
    [
      "0.47596647",
      "0.346008008",
      "0.109457708"
    ],
    [
      "0.286440093",
      "0.450222897",
      "0.106831428"
    ],
    [
      "0.834052187",
      "0.114499284",
      "0.114499283"
    ],
    [
      "0.100821352",
      "0.54331104",
      "0.100821351"
    ],
    [
      "0.399008489",
      "0.792248447",
      "0.095572472"
    ],
    [
      "0.29383781",
      "0.069466133",
      "0.069465838"
    ],
    [
      "0.47512412",
      "0.118276585",
      "0.118275172"
    ],
    [
      "0.265555118",
      "0.653164123",
      "0.097180875"
    ],
    [
      "0.656043561",
      "0.069186155",
      "0.069186061"
    ],
    [
      "0.693855188",
      "0.954363446",
      "0.097083986"
    ],
    [
      "0.842567406",
      "0.334820922",
      "0.105984064"
    ],
    [
      "0.664430858",
      "0.444208179",
      "0.103054745"
    ],
    [
      "0.292003466",
      "0.241192131",
      "0.102269955"
    ],
    [
      "0.109872392",
      "0.941576136",
      "0.109872391"
    ],
    [
      "0.869293319",
      "0.972190376",
      "0.079257548"
    ],
    [
      "0.500454966",
      "0.955130975",
      "0.096317553"
    ],
    [
      "0.479934446",
      "0.583627995",
      "0.128193855"
    ],
    [
      "0.094145304",
      "0.738164256",
      "0.094145303"
    ],
    [
      "0.11504476",
      "0.11504476",
      "0.11504476"
    ],
    [
      "0.658177193",
      "0.239804304",
      "0.10144477"
    ],
    [
      "0.311589918",
      "0.958864831",
      "0.092583697"
    ],
    [
      "0.682192621",
      "0.624555594",
      "0.078163547"
    ],
    [
      "0.106321971",
      "0.336240125",
      "0.10632197"
    ],
    [
      "0.82271063",
      "0.772452011",
      "0.125840841"
    ],
    [
      "0.595686395",
      "0.781881989",
      "0.101377315"
    ],
    [
      "0.2367535",
      "0.81627353",
      "0.068451567"
    ]
  ],
  "height": "1.051448529",
  "width": "0.948551471"
}
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": 27
}

当前第一名的答案

{
  "circles": [
    [
      "0.845094457",
      "0.544247521",
      "0.103457014"
    ],
    [
      "0.47596647",
      "0.346008008",
      "0.109457708"
    ],
    [
      "0.286440093",
      "0.450222897",
      "0.106831428"
    ],
    [
      "0.834052187",
      "0.114499284",
      "0.114499283"
    ],
    [
      "0.100821352",
      "0.54331104",
      "0.100821351"
    ],
    [
      "0.399008489",
      "0.792248447",
      "0.095572472"
    ],
    [
      "0.29383781",
      "0.069466133",
      "0.069465838"
    ],
    [
      "0.47512412",
      "0.118276585",
      "0.118275172"
    ],
    [
      "0.265555118",
      "0.653164123",
      "0.097180875"
    ],
    [
      "0.656043561",
      "0.069186155",
      "0.069186061"
    ],
    [
      "0.693855188",
      "0.954363446",
      "0.097083986"
    ],
    [
      "0.842567406",
      "0.334820922",
      "0.105984064"
    ],
    [
      "0.664430858",
      "0.444208179",
      "0.103054745"
    ],
    [
      "0.292003466",
      "0.241192131",
      "0.102269955"
    ],
    [
      "0.109872392",
      "0.941576136",
      "0.109872391"
    ],
    [
      "0.869293319",
      "0.972190376",
      "0.079257548"
    ],
    [
      "0.500454966",
      "0.955130975",
      "0.096317553"
    ],
    [
      "0.479934446",
      "0.583627995",
      "0.128193855"
    ],
    [
      "0.094145304",
      "0.738164256",
      "0.094145303"
    ],
    [
      "0.11504476",
      "0.11504476",
      "0.11504476"
    ],
    [
      "0.658177193",
      "0.239804304",
      "0.10144477"
    ],
    [
      "0.311589918",
      "0.958864831",
      "0.092583697"
    ],
    [
      "0.682192621",
      "0.624555594",
      "0.078163547"
    ],
    [
      "0.106321971",
      "0.336240125",
      "0.10632197"
    ],
    [
      "0.82271063",
      "0.772452011",
      "0.125840841"
    ],
    [
      "0.595686395",
      "0.781881989",
      "0.101377315"
    ],
    [
      "0.2367535",
      "0.81627353",
      "0.068451567"
    ]
  ],
  "height": "1.051448529",
  "width": "0.948551471"
}

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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