P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 55

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 = 55

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	[0.096359755403975087422419813613, 0.096359755403975087422419813615]	Eckard Specht2010-08-10	Packomania ↗
本站纪录	0.096359755	NUE_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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455

0x2

已验证构造r = 0.096359755

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。

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（3 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.096359754→0.096359755
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.0940334→0.096359754
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.066666666→0.0940334
2026-10-17

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": [
    [
      "0.229046001",
      "0.096359755"
    ],
    [
      "0.421917423",
      "0.096359755"
    ],
    [
      "0.614636934",
      "0.096359755"
    ],
    [
      "0.807356445",
      "0.096359755"
    ],
    [
      "1.000227867",
      "0.096359755"
    ],
    [
      "1.266531687",
      "0.096359755"
    ],
    [
      "1.459251198",
      "0.096359755"
    ],
    [
      "1.651970708",
      "0.096359755"
    ],
    [
      "1.902054169",
      "0.097945831"
    ],
    [
      "1.133379777",
      "0.235684479"
    ],
    [
      "0.096359755",
      "0.236128028"
    ],
    [
      "0.325481712",
      "0.263215871"
    ],
    [
      "0.903792156",
      "0.263215871"
    ],
    [
      "0.518277179",
      "0.263259748"
    ],
    [
      "0.710996689",
      "0.263259748"
    ],
    [
      "1.362583635",
      "0.263437082"
    ],
    [
      "1.555303146",
      "0.263437082"
    ],
    [
      "1.748022657",
      "0.263437082"
    ],
    [
      "1.903640245",
      "0.377120749"
    ],
    [
      "1.036712215",
      "0.402761806"
    ],
    [
      "1.229431726",
      "0.402761806"
    ],
    [
      "0.192795466",
      "0.402984144"
    ],
    [
      "0.421841468",
      "0.430115864"
    ],
    [
      "0.807432401",
      "0.430115864"
    ],
    [
      "0.614636934",
      "0.430159740"
    ],
    [
      "1.458943391",
      "0.430337074"
    ],
    [
      "1.651662902",
      "0.430337074"
    ],
    [
      "1.807280489",
      "0.544020742"
    ],
    [
      "0.940352460",
      "0.569661798"
    ],
    [
      "1.133071970",
      "0.569661798"
    ],
    [
      "1.325791481",
      "0.569661798"
    ],
    [
      "0.096359755",
      "0.569840260"
    ],
    [
      "0.289155222",
      "0.569884136"
    ],
    [
      "0.518201223",
      "0.597015856"
    ],
    [
      "0.711072645",
      "0.597015856"
    ],
    [
      "1.555303146",
      "0.597237066"
    ],
    [
      "1.710920734",
      "0.710920734"
    ],
    [
      "1.903640245",
      "0.710920734"
    ],
    [
      "0.843992704",
      "0.736561790"
    ],
    [
      "1.036712215",
      "0.736561790"
    ],
    [
      "1.229431726",
      "0.736561790"
    ],
    [
      "1.422151236",
      "0.736561790"
    ],
    [
      "0.192719511",
      "0.736740252"
    ],
    [
      "0.385514977",
      "0.736784129"
    ],
    [
      "0.614636934",
      "0.763871972"
    ],
    [
      "0.940042691",
      "0.903640245"
    ],
    [
      "0.096359755",
      "0.903640245"
    ],
    [
      "0.289079266",
      "0.903640245"
    ],
    [
      "0.481950688",
      "0.903640245"
    ],
    [
      "0.747323180",
      "0.903640245"
    ],
    [
      "1.132762201",
      "0.903640245"
    ],
    [
      "1.325481712",
      "0.903640245"
    ],
    [
      "1.518201223",
      "0.903640245"
    ],
    [
      "1.710920734",
      "0.903640245"
    ],
    [
      "1.903640245",
      "0.903640245"
    ]
  ],
  "radius": "0.096359755"
}

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": 55
}

当前第一名的答案

{
  "centers": [
    [
      "0.229046001",
      "0.096359755"
    ],
    [
      "0.421917423",
      "0.096359755"
    ],
    [
      "0.614636934",
      "0.096359755"
    ],
    [
      "0.807356445",
      "0.096359755"
    ],
    [
      "1.000227867",
      "0.096359755"
    ],
    [
      "1.266531687",
      "0.096359755"
    ],
    [
      "1.459251198",
      "0.096359755"
    ],
    [
      "1.651970708",
      "0.096359755"
    ],
    [
      "1.902054169",
      "0.097945831"
    ],
    [
      "1.133379777",
      "0.235684479"
    ],
    [
      "0.096359755",
      "0.236128028"
    ],
    [
      "0.325481712",
      "0.263215871"
    ],
    [
      "0.903792156",
      "0.263215871"
    ],
    [
      "0.518277179",
      "0.263259748"
    ],
    [
      "0.710996689",
      "0.263259748"
    ],
    [
      "1.362583635",
      "0.263437082"
    ],
    [
      "1.555303146",
      "0.263437082"
    ],
    [
      "1.748022657",
      "0.263437082"
    ],
    [
      "1.903640245",
      "0.377120749"
    ],
    [
      "1.036712215",
      "0.402761806"
    ],
    [
      "1.229431726",
      "0.402761806"
    ],
    [
      "0.192795466",
      "0.402984144"
    ],
    [
      "0.421841468",
      "0.430115864"
    ],
    [
      "0.807432401",
      "0.430115864"
    ],
    [
      "0.614636934",
      "0.430159740"
    ],
    [
      "1.458943391",
      "0.430337074"
    ],
    [
      "1.651662902",
      "0.430337074"
    ],
    [
      "1.807280489",
      "0.544020742"
    ],
    [
      "0.940352460",
      "0.569661798"
    ],
    [
      "1.133071970",
      "0.569661798"
    ],
    [
      "1.325791481",
      "0.569661798"
    ],
    [
      "0.096359755",
      "0.569840260"
    ],
    [
      "0.289155222",
      "0.569884136"
    ],
    [
      "0.518201223",
      "0.597015856"
    ],
    [
      "0.711072645",
      "0.597015856"
    ],
    [
      "1.555303146",
      "0.597237066"
    ],
    [
      "1.710920734",
      "0.710920734"
    ],
    [
      "1.903640245",
      "0.710920734"
    ],
    [
      "0.843992704",
      "0.736561790"
    ],
    [
      "1.036712215",
      "0.736561790"
    ],
    [
      "1.229431726",
      "0.736561790"
    ],
    [
      "1.422151236",
      "0.736561790"
    ],
    [
      "0.192719511",
      "0.736740252"
    ],
    [
      "0.385514977",
      "0.736784129"
    ],
    [
      "0.614636934",
      "0.763871972"
    ],
    [
      "0.940042691",
      "0.903640245"
    ],
    [
      "0.096359755",
      "0.903640245"
    ],
    [
      "0.289079266",
      "0.903640245"
    ],
    [
      "0.481950688",
      "0.903640245"
    ],
    [
      "0.747323180",
      "0.903640245"
    ],
    [
      "1.132762201",
      "0.903640245"
    ],
    [
      "1.325481712",
      "0.903640245"
    ],
    [
      "1.518201223",
      "0.903640245"
    ],
    [
      "1.710920734",
      "0.903640245"
    ],
    [
      "1.903640245",
      "0.903640245"
    ]
  ],
  "radius": "0.096359755"
}

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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