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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 = 25

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 = 25
目标最大化 半径之和
已知最好（未证明）2.592537635Berthold、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 2.592537644819718。本站将同一构型保守量化到九位网格，可验证半径和为 2.592537635；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 ↗
1.03642163y0
0x0.96357837
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 = 25 的当前纪录构型，2.592537635
当前第一名

2.592537635

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": [
    [
      "0.297136912",
      "0.243977414",
      "0.103311571"
    ],
    [
      "0.854394099",
      "0.343115139",
      "0.10918427"
    ],
    [
      "0.107243083",
      "0.559533785",
      "0.107243079"
    ],
    [
      "0.103231547",
      "0.769970178",
      "0.103231545"
    ],
    [
      "0.117033543",
      "0.117033543",
      "0.117033542"
    ],
    [
      "0.666441459",
      "0.243977414",
      "0.103311571"
    ],
    [
      "0.860346824",
      "0.769970178",
      "0.103231545"
    ],
    [
      "0.481789186",
      "0.961535059",
      "0.074886569"
    ],
    [
      "0.481789186",
      "0.563489619",
      "0.108409004"
    ],
    [
      "0.846544828",
      "0.117033543",
      "0.117033542"
    ],
    [
      "0.481789186",
      "0.779273555",
      "0.10737493"
    ],
    [
      "0.298490086",
      "0.070335561",
      "0.070335554"
    ],
    [
      "0.109184272",
      "0.343115139",
      "0.10918427"
    ],
    [
      "0.665088285",
      "0.070335561",
      "0.070335554"
    ],
    [
      "0.67091125",
      "0.672425014",
      "0.109843314"
    ],
    [
      "0.286656508",
      "0.909306862",
      "0.127114766"
    ],
    [
      "0.481789186",
      "0.346962694",
      "0.108117918"
    ],
    [
      "0.881370602",
      "0.95421387",
      "0.082207758"
    ],
    [
      "0.294970973",
      "0.454935884",
      "0.107658015"
    ],
    [
      "0.082207769",
      "0.95421387",
      "0.082207758"
    ],
    [
      "0.856335288",
      "0.559533785",
      "0.107243079"
    ],
    [
      "0.668607398",
      "0.454935884",
      "0.107658015"
    ],
    [
      "0.292667121",
      "0.672425014",
      "0.109843314"
    ],
    [
      "0.676921863",
      "0.909306862",
      "0.127114766"
    ],
    [
      "0.481789186",
      "0.119422388",
      "0.119422386"
    ]
  ],
  "height": "1.03642163",
  "width": "0.96357837"
}
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": 25
}

当前第一名的答案

{
  "circles": [
    [
      "0.297136912",
      "0.243977414",
      "0.103311571"
    ],
    [
      "0.854394099",
      "0.343115139",
      "0.10918427"
    ],
    [
      "0.107243083",
      "0.559533785",
      "0.107243079"
    ],
    [
      "0.103231547",
      "0.769970178",
      "0.103231545"
    ],
    [
      "0.117033543",
      "0.117033543",
      "0.117033542"
    ],
    [
      "0.666441459",
      "0.243977414",
      "0.103311571"
    ],
    [
      "0.860346824",
      "0.769970178",
      "0.103231545"
    ],
    [
      "0.481789186",
      "0.961535059",
      "0.074886569"
    ],
    [
      "0.481789186",
      "0.563489619",
      "0.108409004"
    ],
    [
      "0.846544828",
      "0.117033543",
      "0.117033542"
    ],
    [
      "0.481789186",
      "0.779273555",
      "0.10737493"
    ],
    [
      "0.298490086",
      "0.070335561",
      "0.070335554"
    ],
    [
      "0.109184272",
      "0.343115139",
      "0.10918427"
    ],
    [
      "0.665088285",
      "0.070335561",
      "0.070335554"
    ],
    [
      "0.67091125",
      "0.672425014",
      "0.109843314"
    ],
    [
      "0.286656508",
      "0.909306862",
      "0.127114766"
    ],
    [
      "0.481789186",
      "0.346962694",
      "0.108117918"
    ],
    [
      "0.881370602",
      "0.95421387",
      "0.082207758"
    ],
    [
      "0.294970973",
      "0.454935884",
      "0.107658015"
    ],
    [
      "0.082207769",
      "0.95421387",
      "0.082207758"
    ],
    [
      "0.856335288",
      "0.559533785",
      "0.107243079"
    ],
    [
      "0.668607398",
      "0.454935884",
      "0.107658015"
    ],
    [
      "0.292667121",
      "0.672425014",
      "0.109843314"
    ],
    [
      "0.676921863",
      "0.909306862",
      "0.127114766"
    ],
    [
      "0.481789186",
      "0.119422388",
      "0.119422386"
    ]
  ],
  "height": "1.03642163",
  "width": "0.96357837"
}

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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