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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 = 26

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 = 26
目标最大化 半径之和
已知最好（未证明）2.639308106Berthold、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 2.639308122181169。本站将同一构型保守量化到九位网格，可验证半径和为 2.639308106；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 ↗
1.054499721y0
0x0.945500279
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 = 26 的当前纪录构型，2.639308106
当前第一名

2.639308106

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": [
    [
      "0.278416223",
      "0.647741969",
      "0.101930058"
    ],
    [
      "0.65002608",
      "0.069404583",
      "0.069404582"
    ],
    [
      "0.284401594",
      "0.442074325",
      "0.103823255"
    ],
    [
      "0.836580074",
      "0.340827949",
      "0.10891809"
    ],
    [
      "0.26229883",
      "0.9944624",
      "0.060037107"
    ],
    [
      "0.646605868",
      "0.668718159",
      "0.111422713"
    ],
    [
      "0.103049372",
      "0.541612978",
      "0.103049371"
    ],
    [
      "0.836915902",
      "0.558330574",
      "0.108583901"
    ],
    [
      "0.288495845",
      "0.237747614",
      "0.10054447"
    ],
    [
      "0.451005938",
      "0.763895803",
      "0.10610387"
    ],
    [
      "0.650266638",
      "0.450200902",
      "0.107124173"
    ],
    [
      "0.829489564",
      "0.116010928",
      "0.116010715"
    ],
    [
      "0.85847086",
      "0.967472258",
      "0.087027249"
    ],
    [
      "0.290410783",
      "0.068606692",
      "0.068606691"
    ],
    [
      "0.838204436",
      "0.774206456",
      "0.107295807"
    ],
    [
      "0.104149746",
      "0.950349762",
      "0.104148821"
    ],
    [
      "0.469698016",
      "0.117130356",
      "0.117130355"
    ],
    [
      "0.100789078",
      "0.745439519",
      "0.100788977"
    ],
    [
      "0.113738725",
      "0.113738725",
      "0.113738724"
    ],
    [
      "0.463071951",
      "0.552118936",
      "0.106015853"
    ],
    [
      "0.105624058",
      "0.332953839",
      "0.105624057"
    ],
    [
      "0.412630967",
      "0.960393658",
      "0.094105849"
    ],
    [
      "0.26832231",
      "0.841977149",
      "0.092566721"
    ],
    [
      "0.650662011",
      "0.240941559",
      "0.102133516"
    ],
    [
      "0.63990835",
      "0.917274215",
      "0.137223559"
    ],
    [
      "0.467780697",
      "0.340203927",
      "0.105949622"
    ]
  ],
  "height": "1.054499721",
  "width": "0.945500279"
}
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": 26
}

当前第一名的答案

{
  "circles": [
    [
      "0.278416223",
      "0.647741969",
      "0.101930058"
    ],
    [
      "0.65002608",
      "0.069404583",
      "0.069404582"
    ],
    [
      "0.284401594",
      "0.442074325",
      "0.103823255"
    ],
    [
      "0.836580074",
      "0.340827949",
      "0.10891809"
    ],
    [
      "0.26229883",
      "0.9944624",
      "0.060037107"
    ],
    [
      "0.646605868",
      "0.668718159",
      "0.111422713"
    ],
    [
      "0.103049372",
      "0.541612978",
      "0.103049371"
    ],
    [
      "0.836915902",
      "0.558330574",
      "0.108583901"
    ],
    [
      "0.288495845",
      "0.237747614",
      "0.10054447"
    ],
    [
      "0.451005938",
      "0.763895803",
      "0.10610387"
    ],
    [
      "0.650266638",
      "0.450200902",
      "0.107124173"
    ],
    [
      "0.829489564",
      "0.116010928",
      "0.116010715"
    ],
    [
      "0.85847086",
      "0.967472258",
      "0.087027249"
    ],
    [
      "0.290410783",
      "0.068606692",
      "0.068606691"
    ],
    [
      "0.838204436",
      "0.774206456",
      "0.107295807"
    ],
    [
      "0.104149746",
      "0.950349762",
      "0.104148821"
    ],
    [
      "0.469698016",
      "0.117130356",
      "0.117130355"
    ],
    [
      "0.100789078",
      "0.745439519",
      "0.100788977"
    ],
    [
      "0.113738725",
      "0.113738725",
      "0.113738724"
    ],
    [
      "0.463071951",
      "0.552118936",
      "0.106015853"
    ],
    [
      "0.105624058",
      "0.332953839",
      "0.105624057"
    ],
    [
      "0.412630967",
      "0.960393658",
      "0.094105849"
    ],
    [
      "0.26832231",
      "0.841977149",
      "0.092566721"
    ],
    [
      "0.650662011",
      "0.240941559",
      "0.102133516"
    ],
    [
      "0.63990835",
      "0.917274215",
      "0.137223559"
    ],
    [
      "0.467780697",
      "0.340203927",
      "0.105949622"
    ]
  ],
  "height": "1.054499721",
  "width": "0.945500279"
}

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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