P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 96

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 = 96

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	[0.073586938569285378728736291181, 0.073586938569285378728736291183]	Eckard Specht1999–2010	Packomania ↗
本站纪录	0.073586938	condor	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596

0x2

已验证构造r = 0.073586938

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condor

解题方式AI · OpenAI: GPT-6.1 Sol

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）

condorAI · OpenAI: GPT-6.1 Sol
0.073586937→0.073586938
2026-10-27
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.071222923→0.073586937
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.057142856→0.071222923
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": [
    [
      "0.219611879",
      "0.073586939"
    ],
    [
      "0.366785756",
      "0.073586939"
    ],
    [
      "0.513959633",
      "0.073586939"
    ],
    [
      "0.661133510",
      "0.073586939"
    ],
    [
      "0.808307387",
      "0.073586939"
    ],
    [
      "0.955481264",
      "0.073586939"
    ],
    [
      "1.102655142",
      "0.073586939"
    ],
    [
      "1.337772962",
      "0.073586939"
    ],
    [
      "1.484946839",
      "0.073586939"
    ],
    [
      "1.632120716",
      "0.073586939"
    ],
    [
      "1.779294594",
      "0.073586939"
    ],
    [
      "1.926413061",
      "0.077625070"
    ],
    [
      "0.073586939",
      "0.091940875"
    ],
    [
      "1.220214052",
      "0.162131011"
    ],
    [
      "0.293198817",
      "0.201043255"
    ],
    [
      "0.440372694",
      "0.201043255"
    ],
    [
      "0.587546571",
      "0.201043255"
    ],
    [
      "0.734720449",
      "0.201043255"
    ],
    [
      "0.881894326",
      "0.201043255"
    ],
    [
      "1.029068203",
      "0.201043255"
    ],
    [
      "1.413073506",
      "0.202017364"
    ],
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      "1.560247383",
      "0.202017364"
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      "1.707421260",
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      "1.873412967",
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      "0.080352674",
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      "0.289186261"
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      "1.146627113",
      "0.289587327"
    ],
    [
      "0.204291372",
      "0.318327595"
    ],
    [
      "0.513959633",
      "0.328499571"
    ],
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      "0.661133510",
      "0.328499571"
    ],
    [
      "0.808307387",
      "0.328499571"
    ],
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      "0.955481264",
      "0.328499571"
    ],
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      "1.486660445",
      "0.329473680"
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      "0.329473680"
    ],
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      "1.781008199",
      "0.329473680"
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      "0.338537811"
    ],
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      "0.352223978"
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      "0.073586939",
      "0.385977437"
    ],
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      "1.220904387",
      "0.416642578"
    ],
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      "1.368078265",
      "0.416642578"
    ],
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      "1.073040174",
      "0.417043644"
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      "0.439682358",
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      "0.671500429"
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      "0.671555210"
    ],
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      "1.368078265",
      "0.671555210"
    ],
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      "0.925771494",
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      "0.073586939",
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      "0.779546933"
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      "0.798956745"
    ],
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      "0.321738414",
      "0.798956745"
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      "0.799357811"
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      "0.660203255",
      "0.838061987"
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      "1.794466942",
      "0.861218187"
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      "0.926413061"
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      "0.395325352",
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    ],
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      "0.926461830",
      "0.926413061"
    ],
    [
      "1.073635707",
      "0.926413061"
    ],
    [
      "1.220809585",
      "0.926413061"
    ],
    [
      "1.368173067",
      "0.926413061"
    ],
    [
      "1.515346944",
      "0.926413061"
    ],
    [
      "1.662520822",
      "0.926413061"
    ],
    [
      "1.926413061",
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    ]
  ],
  "radius": "0.073586938"
}

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": 96
}

当前第一名的答案

{
  "centers": [
    [
      "0.219611879",
      "0.073586939"
    ],
    [
      "0.366785756",
      "0.073586939"
    ],
    [
      "0.513959633",
      "0.073586939"
    ],
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      "0.661133510",
      "0.073586939"
    ],
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      "0.808307387",
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      "0.073586939"
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      "0.073586939"
    ],
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      "0.073586939"
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    ],
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      "0.542499229",
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      "0.777907281",
      "0.926413061"
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    [
      "0.926461830",
      "0.926413061"
    ],
    [
      "1.073635707",
      "0.926413061"
    ],
    [
      "1.220809585",
      "0.926413061"
    ],
    [
      "1.368173067",
      "0.926413061"
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    [
      "1.515346944",
      "0.926413061"
    ],
    [
      "1.662520822",
      "0.926413061"
    ],
    [
      "1.926413061",
      "0.926413061"
    ]
  ],
  "radius": "0.073586938"
}

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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本题n=8时用抽屉原理的证明NUE_13#313 条回复10 天前 好！！！NUE_13#310 条回复17 天前