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P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 34

文献中也称circles in a rectanglepacking equal circles in a 2:1 rectangle

在 2 × 1 的长方形内放置 n 个互不重叠的等圆，使它们的共同半径尽可能大。

子题n = 34
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.119826883321331884807295310053, 0.119826883321331884807295310055]Eckard Specht1999–2010Packomania ↗
本站纪录0.119826883NUE_13纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器宽 2、高 1 的长方形 [0, 2] × [0, 1]，左下角为原点 (0, 0)，右上角为 (2, 1)
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心坐标 centers
  • 约束每个圆完全位于长方形内；任意两圆内部不重叠，允许相切
  • 目标最大化共同半径 radius
挑战这个纪录 ↗

当前展示：本站纪录

1y0
0x2
已验证构造r = 0.119826883

帮助理解

交错之字层与行转换

在长宽比为 2 : 1 的狭长容器内，小规模时各圆通常交替贴靠上下长边排成之字形链；随着 n 增大，排布会在双层交错、三层蜂窝与局部斜排之间切换。除了完美契合 2 : 1 比例的偶数网格（如 2 × 1 或 4 × 2）外，整齐的轴对齐网格通常会浪费横向或纵向空间。

已知证明与开放前沿

n = 3..6 的最优半径可通过之字形链的初等几何下界严格证明（详见各子题的已知最优说明）；更大的 n 引用 Packomania 的 1 × 0.5 长方形表（本题是它放大两倍，半径也乘 2），表中只列出最优已知值，没有一项标为已证明最优。

2:1 长方形内的等圆装箱 n = 34 的当前纪录构型，0.119826883
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者NUE_13
解题方式AI · DeepSeek: DeepSeek V4.1 Flash
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（2 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.118506837→0.119826883
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.08→0.118506837
ANSWER FORMAT

答案怎么写

容器是宽 2、高 1 的长方形，左下角是原点 (0, 0)，右上角是 (2, 1)。坐标和长度用同一个单位，直接写成小数，例如 "0.25"，最多九位小数。

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.25"）。

当前第一名的答案

{
  "centers": [
    [
      "0.119826883",
      "0.119826883"
    ],
    [
      "0.526050909",
      "0.119826883"
    ],
    [
      "0.918329264",
      "0.119826883"
    ],
    [
      "1.157983030",
      "0.119826883"
    ],
    [
      "1.397636797",
      "0.119826883"
    ],
    [
      "1.640123815",
      "0.120011578"
    ],
    [
      "1.879993690",
      "0.128579326"
    ],
    [
      "0.322938896",
      "0.247025304"
    ],
    [
      "0.722190086",
      "0.257534367"
    ],
    [
      "1.038156147",
      "0.327373133"
    ],
    [
      "1.277809914",
      "0.327373133"
    ],
    [
      "1.517463680",
      "0.327373133"
    ],
    [
      "1.879475668",
      "0.368602130"
    ],
    [
      "0.119826883",
      "0.374223724"
    ],
    [
      "0.568618112",
      "0.446736501"
    ],
    [
      "0.842016970",
      "0.465080617"
    ],
    [
      "0.331539953",
      "0.486524677"
    ],
    [
      "1.668460047",
      "0.513475323"
    ],
    [
      "1.157983030",
      "0.534919383"
    ],
    [
      "1.424918625",
      "0.550652883"
    ],
    [
      "0.123300724",
      "0.613855278"
    ],
    [
      "1.880173117",
      "0.625776276"
    ],
    [
      "0.482536320",
      "0.672626867"
    ],
    [
      "0.722190086",
      "0.672626867"
    ],
    [
      "0.961843853",
      "0.672626867"
    ],
    [
      "1.277809914",
      "0.742465633"
    ],
    [
      "1.677061104",
      "0.752974696"
    ],
    [
      "0.120059776",
      "0.853740505"
    ],
    [
      "0.358748599",
      "0.879938662"
    ],
    [
      "0.602363203",
      "0.880173117"
    ],
    [
      "0.842016970",
      "0.880173117"
    ],
    [
      "1.081670736",
      "0.880173117"
    ],
    [
      "1.473949091",
      "0.880173117"
    ],
    [
      "1.880173117",
      "0.880173117"
    ]
  ],
  "radius": "0.119826883"
}
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": 34
}

当前第一名的答案

{
  "centers": [
    [
      "0.119826883",
      "0.119826883"
    ],
    [
      "0.526050909",
      "0.119826883"
    ],
    [
      "0.918329264",
      "0.119826883"
    ],
    [
      "1.157983030",
      "0.119826883"
    ],
    [
      "1.397636797",
      "0.119826883"
    ],
    [
      "1.640123815",
      "0.120011578"
    ],
    [
      "1.879993690",
      "0.128579326"
    ],
    [
      "0.322938896",
      "0.247025304"
    ],
    [
      "0.722190086",
      "0.257534367"
    ],
    [
      "1.038156147",
      "0.327373133"
    ],
    [
      "1.277809914",
      "0.327373133"
    ],
    [
      "1.517463680",
      "0.327373133"
    ],
    [
      "1.879475668",
      "0.368602130"
    ],
    [
      "0.119826883",
      "0.374223724"
    ],
    [
      "0.568618112",
      "0.446736501"
    ],
    [
      "0.842016970",
      "0.465080617"
    ],
    [
      "0.331539953",
      "0.486524677"
    ],
    [
      "1.668460047",
      "0.513475323"
    ],
    [
      "1.157983030",
      "0.534919383"
    ],
    [
      "1.424918625",
      "0.550652883"
    ],
    [
      "0.123300724",
      "0.613855278"
    ],
    [
      "1.880173117",
      "0.625776276"
    ],
    [
      "0.482536320",
      "0.672626867"
    ],
    [
      "0.722190086",
      "0.672626867"
    ],
    [
      "0.961843853",
      "0.672626867"
    ],
    [
      "1.277809914",
      "0.742465633"
    ],
    [
      "1.677061104",
      "0.752974696"
    ],
    [
      "0.120059776",
      "0.853740505"
    ],
    [
      "0.358748599",
      "0.879938662"
    ],
    [
      "0.602363203",
      "0.880173117"
    ],
    [
      "0.842016970",
      "0.880173117"
    ],
    [
      "1.081670736",
      "0.880173117"
    ],
    [
      "1.473949091",
      "0.880173117"
    ],
    [
      "1.880173117",
      "0.880173117"
    ]
  ],
  "radius": "0.119826883"
}

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.25"）。 · 验证器 v1.0.0

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