P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 85

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 = 85

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	[0.078713466124604292987915660891, 0.078713466124604292987915660893]	Eckard Specht2010-08-04	Packomania ↗
本站纪录	0.078713465	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485

0x2

已验证构造r = 0.078713465

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
0.074417333→0.078713465
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.057142856→0.074417333
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": [
    [
      "0.236140398",
      "0.078713466"
    ],
    [
      "0.393567331",
      "0.078713466"
    ],
    [
      "0.550994263",
      "0.078713466"
    ],
    [
      "0.708421195",
      "0.078713466"
    ],
    [
      "1.023275060",
      "0.078713466"
    ],
    [
      "1.338128924",
      "0.078713466"
    ],
    [
      "1.495555856",
      "0.078713466"
    ],
    [
      "1.652982789",
      "0.078713466"
    ],
    [
      "1.810409721",
      "0.078713466"
    ],
    [
      "0.078713466",
      "0.078713466"
    ],
    [
      "0.865848127",
      "0.078713469"
    ],
    [
      "1.180701992",
      "0.078713473"
    ],
    [
      "1.921286534",
      "0.190470217"
    ],
    [
      "0.314853864",
      "0.215049189"
    ],
    [
      "0.472280797",
      "0.215049189"
    ],
    [
      "1.574269322",
      "0.215049189"
    ],
    [
      "0.787134659",
      "0.215049190"
    ],
    [
      "1.416842361",
      "0.215049206"
    ],
    [
      "1.101988443",
      "0.215049237"
    ],
    [
      "1.259415375",
      "0.215049252"
    ],
    [
      "0.629707722",
      "0.215119171"
    ],
    [
      "0.944561537",
      "0.215179571"
    ],
    [
      "0.155296593",
      "0.216257145"
    ],
    [
      "1.759292541",
      "0.232312446"
    ],
    [
      "1.921286534",
      "0.347897149"
    ],
    [
      "0.236102645",
      "0.351363107"
    ],
    [
      "0.550884692",
      "0.351448113"
    ],
    [
      "0.865671031",
      "0.351487004"
    ],
    [
      "1.652741277",
      "0.351524063"
    ],
    [
      "1.495314344",
      "0.351524063"
    ],
    [
      "1.180460480",
      "0.351524063"
    ],
    [
      "1.337887412",
      "0.351524098"
    ],
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      "1.023065381",
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    ],
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      "0.508951030"
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      "0.512116709"
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      "0.515835723"
    ],
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      "1.758426974",
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    ],
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      "0.645407534"
    ],
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      "1.259383526",
      "0.645407551"
    ],
    [
      "0.787134591",
      "0.648419415"
    ],
    [
      "0.472280788",
      "0.648498742"
    ],
    [
      "1.101988425",
      "0.648573171"
    ],
    [
      "0.944561493",
      "0.648573217"
    ],
    [
      "0.629707716",
      "0.648573268"
    ],
    [
      "0.157426932",
      "0.648573276"
    ],
    [
      "0.314853864",
      "0.648573276"
    ],
    [
      "1.574204904",
      "0.648610442"
    ],
    [
      "1.921286534",
      "0.673262656"
    ],
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      "1.338097091",
      "0.781743284"
    ],
    [
      "1.023275060",
      "0.784908980"
    ],
    [
      "0.708421195",
      "0.784908983"
    ],
    [
      "0.078713466",
      "0.784908998"
    ],
    [
      "0.236140398",
      "0.784908998"
    ],
    [
      "0.393567331",
      "0.784908998"
    ],
    [
      "0.550994263",
      "0.784908998"
    ],
    [
      "0.865848127",
      "0.784908998"
    ],
    [
      "1.180701992",
      "0.784908998"
    ],
    [
      "1.495491438",
      "0.784946165"
    ],
    [
      "1.652918371",
      "0.784946165"
    ],
    [
      "1.810345303",
      "0.784955459"
    ],
    [
      "1.920538806",
      "0.920538806"
    ],
    [
      "1.416769923",
      "0.921285645"
    ],
    [
      "1.101916059",
      "0.921286527"
    ],
    [
      "0.787062194",
      "0.921286529"
    ],
    [
      "0.157354465",
      "0.921286534"
    ],
    [
      "0.314781398",
      "0.921286534"
    ],
    [
      "0.472208330",
      "0.921286534"
    ],
    [
      "0.629635262",
      "0.921286534"
    ],
    [
      "0.944489127",
      "0.921286534"
    ],
    [
      "1.259342991",
      "0.921286534"
    ],
    [
      "1.574196856",
      "0.921286534"
    ],
    [
      "1.731623788",
      "0.921286534"
    ]
  ],
  "radius": "0.078713465"
}

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": 85
}

当前第一名的答案

{
  "centers": [
    [
      "0.236140398",
      "0.078713466"
    ],
    [
      "0.393567331",
      "0.078713466"
    ],
    [
      "0.550994263",
      "0.078713466"
    ],
    [
      "0.708421195",
      "0.078713466"
    ],
    [
      "1.023275060",
      "0.078713466"
    ],
    [
      "1.338128924",
      "0.078713466"
    ],
    [
      "1.495555856",
      "0.078713466"
    ],
    [
      "1.652982789",
      "0.078713466"
    ],
    [
      "1.810409721",
      "0.078713466"
    ],
    [
      "0.078713466",
      "0.078713466"
    ],
    [
      "0.865848127",
      "0.078713469"
    ],
    [
      "1.180701992",
      "0.078713473"
    ],
    [
      "1.921286534",
      "0.190470217"
    ],
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    ],
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    ],
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      "0.215049190"
    ],
    [
      "1.416842361",
      "0.215049206"
    ],
    [
      "1.101988443",
      "0.215049237"
    ],
    [
      "1.259415375",
      "0.215049252"
    ],
    [
      "0.629707722",
      "0.215119171"
    ],
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      "0.944561537",
      "0.215179571"
    ],
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      "0.155296593",
      "0.216257145"
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    ],
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      "1.921286534",
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    ],
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      "1.652741277",
      "0.351524063"
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      "1.495314344",
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    ],
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    ],
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      "0.784946165"
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      "0.784955459"
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      "0.920538806"
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      "1.416769923",
      "0.921285645"
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      "1.101916059",
      "0.921286527"
    ],
    [
      "0.787062194",
      "0.921286529"
    ],
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      "0.157354465",
      "0.921286534"
    ],
    [
      "0.314781398",
      "0.921286534"
    ],
    [
      "0.472208330",
      "0.921286534"
    ],
    [
      "0.629635262",
      "0.921286534"
    ],
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      "0.944489127",
      "0.921286534"
    ],
    [
      "1.259342991",
      "0.921286534"
    ],
    [
      "1.574196856",
      "0.921286534"
    ],
    [
      "1.731623788",
      "0.921286534"
    ]
  ],
  "radius": "0.078713465"
}

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