P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 81

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 = 81

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	[0.081320272797708039669543105639, 0.081320272797708039669543105641]	Eckard Specht1999–2010	Packomania ↗
本站纪录	0.081320272	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081

0x2

已验证构造r = 0.081320272

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（2 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.07717803→0.081320272
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.057142856→0.07717803
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": [
    [
      "0.081320273",
      "0.081320273"
    ],
    [
      "0.248352950",
      "0.081320273"
    ],
    [
      "0.415385628",
      "0.081320273"
    ],
    [
      "0.582418306",
      "0.081320273"
    ],
    [
      "0.749450983",
      "0.081320273"
    ],
    [
      "0.916483661",
      "0.081320273"
    ],
    [
      "1.083516339",
      "0.081320273"
    ],
    [
      "1.250549017",
      "0.081320273"
    ],
    [
      "1.417581694",
      "0.081320273"
    ],
    [
      "1.584614372",
      "0.081320273"
    ],
    [
      "1.751647050",
      "0.081320273"
    ],
    [
      "1.918679727",
      "0.081320273"
    ],
    [
      "0.164836612",
      "0.220880182"
    ],
    [
      "0.331869289",
      "0.220880182"
    ],
    [
      "0.498901967",
      "0.220880182"
    ],
    [
      "0.665934645",
      "0.220880182"
    ],
    [
      "0.832967322",
      "0.220880182"
    ],
    [
      "1.000000000",
      "0.220880182"
    ],
    [
      "1.167032678",
      "0.220880182"
    ],
    [
      "1.334065355",
      "0.220880182"
    ],
    [
      "1.501098033",
      "0.220880182"
    ],
    [
      "1.668130711",
      "0.220880182"
    ],
    [
      "1.835163388",
      "0.220880182"
    ],
    [
      "0.081320273",
      "0.360440091"
    ],
    [
      "0.248352950",
      "0.360440091"
    ],
    [
      "0.415385628",
      "0.360440091"
    ],
    [
      "0.582418306",
      "0.360440091"
    ],
    [
      "0.749450983",
      "0.360440091"
    ],
    [
      "0.916483661",
      "0.360440091"
    ],
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      "0.360440091"
    ],
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      "1.250549017",
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    ],
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    ],
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    ],
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      "0.360440091"
    ],
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      "0.360440091"
    ],
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      "0.164836612",
      "0.500000000"
    ],
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      "0.500000000"
    ],
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      "0.498901967",
      "0.500000000"
    ],
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      "0.500000000"
    ],
    [
      "0.832967322",
      "0.500000000"
    ],
    [
      "1.000000000",
      "0.500000000"
    ],
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      "1.167032678",
      "0.500000000"
    ],
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      "1.334065355",
      "0.500000000"
    ],
    [
      "1.501098033",
      "0.500000000"
    ],
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      "1.668130711",
      "0.500000000"
    ],
    [
      "1.835163388",
      "0.500000000"
    ],
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      "0.639559909"
    ],
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      "0.639559909"
    ],
    [
      "0.415385628",
      "0.639559909"
    ],
    [
      "0.582418306",
      "0.639559909"
    ],
    [
      "0.749450983",
      "0.639559909"
    ],
    [
      "0.916483661",
      "0.639559909"
    ],
    [
      "1.083516339",
      "0.639559909"
    ],
    [
      "1.250549017",
      "0.639559909"
    ],
    [
      "1.417581694",
      "0.639559909"
    ],
    [
      "1.584614372",
      "0.639559909"
    ],
    [
      "1.751647050",
      "0.639559909"
    ],
    [
      "1.918679727",
      "0.639559909"
    ],
    [
      "0.164836612",
      "0.779119818"
    ],
    [
      "0.331869289",
      "0.779119818"
    ],
    [
      "0.498901967",
      "0.779119818"
    ],
    [
      "0.665934645",
      "0.779119818"
    ],
    [
      "0.832967322",
      "0.779119818"
    ],
    [
      "1.000000000",
      "0.779119818"
    ],
    [
      "1.167032678",
      "0.779119818"
    ],
    [
      "1.334065355",
      "0.779119818"
    ],
    [
      "1.501098033",
      "0.779119818"
    ],
    [
      "1.668130711",
      "0.779119818"
    ],
    [
      "1.835163388",
      "0.779119818"
    ],
    [
      "0.081320273",
      "0.918679727"
    ],
    [
      "0.248352950",
      "0.918679727"
    ],
    [
      "0.415385628",
      "0.918679727"
    ],
    [
      "0.582418306",
      "0.918679727"
    ],
    [
      "0.749450983",
      "0.918679727"
    ],
    [
      "0.916483661",
      "0.918679727"
    ],
    [
      "1.083516339",
      "0.918679727"
    ],
    [
      "1.250549017",
      "0.918679727"
    ],
    [
      "1.417581694",
      "0.918679727"
    ],
    [
      "1.584614372",
      "0.918679727"
    ],
    [
      "1.751647050",
      "0.918679727"
    ],
    [
      "1.918679727",
      "0.918679727"
    ]
  ],
  "radius": "0.081320272"
}

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": 81
}

当前第一名的答案

{
  "centers": [
    [
      "0.081320273",
      "0.081320273"
    ],
    [
      "0.248352950",
      "0.081320273"
    ],
    [
      "0.415385628",
      "0.081320273"
    ],
    [
      "0.582418306",
      "0.081320273"
    ],
    [
      "0.749450983",
      "0.081320273"
    ],
    [
      "0.916483661",
      "0.081320273"
    ],
    [
      "1.083516339",
      "0.081320273"
    ],
    [
      "1.250549017",
      "0.081320273"
    ],
    [
      "1.417581694",
      "0.081320273"
    ],
    [
      "1.584614372",
      "0.081320273"
    ],
    [
      "1.751647050",
      "0.081320273"
    ],
    [
      "1.918679727",
      "0.081320273"
    ],
    [
      "0.164836612",
      "0.220880182"
    ],
    [
      "0.331869289",
      "0.220880182"
    ],
    [
      "0.498901967",
      "0.220880182"
    ],
    [
      "0.665934645",
      "0.220880182"
    ],
    [
      "0.832967322",
      "0.220880182"
    ],
    [
      "1.000000000",
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    ],
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      "0.220880182"
    ],
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      "0.220880182"
    ],
    [
      "1.501098033",
      "0.220880182"
    ],
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      "1.668130711",
      "0.220880182"
    ],
    [
      "1.835163388",
      "0.220880182"
    ],
    [
      "0.081320273",
      "0.360440091"
    ],
    [
      "0.248352950",
      "0.360440091"
    ],
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      "0.415385628",
      "0.360440091"
    ],
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      "0.582418306",
      "0.360440091"
    ],
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      "0.749450983",
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    ],
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      "0.916483661",
      "0.360440091"
    ],
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    ],
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      "0.360440091"
    ],
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    ],
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      "1.584614372",
      "0.360440091"
    ],
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      "0.360440091"
    ],
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      "0.360440091"
    ],
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      "0.500000000"
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      "0.500000000"
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      "0.500000000"
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      "0.639559909"
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      "0.639559909"
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      "0.779119818"
    ],
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      "0.918679727"
    ],
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      "0.248352950",
      "0.918679727"
    ],
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      "0.415385628",
      "0.918679727"
    ],
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      "0.918679727"
    ],
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      "0.918679727"
    ],
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      "0.916483661",
      "0.918679727"
    ],
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      "0.918679727"
    ],
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      "1.250549017",
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    ],
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      "1.417581694",
      "0.918679727"
    ],
    [
      "1.584614372",
      "0.918679727"
    ],
    [
      "1.751647050",
      "0.918679727"
    ],
    [
      "1.918679727",
      "0.918679727"
    ]
  ],
  "radius": "0.081320272"
}

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