P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 67

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 = 67

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	[0.087245798944625048419650008581, 0.087245798944625048419650008583]	Eckard Specht2010-08-10	Packomania ↗
本站纪录	0.087245798	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667

0x2

已验证构造r = 0.087245798

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
0.085759205→0.087245798
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.066666666→0.085759205
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": [
    [
      "0.250488035",
      "0.087245799"
    ],
    [
      "0.516821418",
      "0.087245799"
    ],
    [
      "0.865804614",
      "0.087245799"
    ],
    [
      "1.214787810",
      "0.087245799"
    ],
    [
      "1.563771005",
      "0.087245799"
    ],
    [
      "1.912754201",
      "0.087245799"
    ],
    [
      "0.691313016",
      "0.087246346"
    ],
    [
      "1.738262603",
      "0.087246709"
    ],
    [
      "1.389279407",
      "0.087246724"
    ],
    [
      "1.040296212",
      "0.087248070"
    ],
    [
      "0.087245799",
      "0.148884180"
    ],
    [
      "0.383654727",
      "0.200001952"
    ],
    [
      "0.778565036",
      "0.238363547"
    ],
    [
      "0.953056634",
      "0.238365818"
    ],
    [
      "1.127548369",
      "0.238366477"
    ],
    [
      "1.302041213",
      "0.238366768"
    ],
    [
      "1.476533919",
      "0.238367797"
    ],
    [
      "1.651026673",
      "0.238368187"
    ],
    [
      "1.825519354",
      "0.238368811"
    ],
    [
      "0.556215660",
      "0.258974895"
    ],
    [
      "0.220412490",
      "0.261640333"
    ],
    [
      "0.389474110",
      "0.374396483"
    ],
    [
      "0.087245799",
      "0.374396486"
    ],
    [
      "0.691313017",
      "0.389474112"
    ],
    [
      "1.040296212",
      "0.389486039"
    ],
    [
      "1.214787810",
      "0.389487638"
    ],
    [
      "1.389279407",
      "0.389488922"
    ],
    [
      "1.563771005",
      "0.389489205"
    ],
    [
      "1.738262603",
      "0.389490839"
    ],
    [
      "1.912754201",
      "0.389491134"
    ],
    [
      "0.865804614",
      "0.389493802"
    ],
    [
      "0.238359955",
      "0.461642285"
    ],
    [
      "0.536040275",
      "0.469083743"
    ],
    [
      "0.087245799",
      "0.548888084"
    ],
    [
      "0.384926120",
      "0.556329545"
    ],
    [
      "0.691313017",
      "0.563965710"
    ],
    [
      "1.040296212",
      "0.563977954"
    ],
    [
      "1.389279407",
      "0.563980520"
    ],
    [
      "1.214787809",
      "0.563981594"
    ],
    [
      "1.563771005",
      "0.563983265"
    ],
    [
      "1.738262603",
      "0.563983586"
    ],
    [
      "1.912754201",
      "0.563983917"
    ],
    [
      "0.865804614",
      "0.563987444"
    ],
    [
      "0.536040277",
      "0.643575343"
    ],
    [
      "0.233811964",
      "0.643575344"
    ],
    [
      "0.384926120",
      "0.730821142"
    ],
    [
      "0.087245799",
      "0.738262603"
    ],
    [
      "0.691313017",
      "0.738457308"
    ],
    [
      "1.040296212",
      "0.738470625"
    ],
    [
      "1.389279407",
      "0.738474357"
    ],
    [
      "1.563771005",
      "0.738474863"
    ],
    [
      "1.214787809",
      "0.738475248"
    ],
    [
      "1.738262603",
      "0.738476571"
    ],
    [
      "1.912754201",
      "0.738476731"
    ],
    [
      "0.865804614",
      "0.738479042"
    ],
    [
      "0.536040277",
      "0.818066941"
    ],
    [
      "0.238359955",
      "0.825508402"
    ],
    [
      "1.031609598",
      "0.912752902"
    ],
    [
      "1.380595613",
      "0.912753101"
    ],
    [
      "1.904074457",
      "0.912753637"
    ],
    [
      "1.206102607",
      "0.912753673"
    ],
    [
      "1.555088617",
      "0.912753673"
    ],
    [
      "0.682623764",
      "0.912753673"
    ],
    [
      "1.729581627",
      "0.912753673"
    ],
    [
      "0.857116308",
      "0.912754201"
    ],
    [
      "0.389474112",
      "0.912754201"
    ],
    [
      "0.087245799",
      "0.912754201"
    ]
  ],
  "radius": "0.087245798"
}

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": 67
}

当前第一名的答案

{
  "centers": [
    [
      "0.250488035",
      "0.087245799"
    ],
    [
      "0.516821418",
      "0.087245799"
    ],
    [
      "0.865804614",
      "0.087245799"
    ],
    [
      "1.214787810",
      "0.087245799"
    ],
    [
      "1.563771005",
      "0.087245799"
    ],
    [
      "1.912754201",
      "0.087245799"
    ],
    [
      "0.691313016",
      "0.087246346"
    ],
    [
      "1.738262603",
      "0.087246709"
    ],
    [
      "1.389279407",
      "0.087246724"
    ],
    [
      "1.040296212",
      "0.087248070"
    ],
    [
      "0.087245799",
      "0.148884180"
    ],
    [
      "0.383654727",
      "0.200001952"
    ],
    [
      "0.778565036",
      "0.238363547"
    ],
    [
      "0.953056634",
      "0.238365818"
    ],
    [
      "1.127548369",
      "0.238366477"
    ],
    [
      "1.302041213",
      "0.238366768"
    ],
    [
      "1.476533919",
      "0.238367797"
    ],
    [
      "1.651026673",
      "0.238368187"
    ],
    [
      "1.825519354",
      "0.238368811"
    ],
    [
      "0.556215660",
      "0.258974895"
    ],
    [
      "0.220412490",
      "0.261640333"
    ],
    [
      "0.389474110",
      "0.374396483"
    ],
    [
      "0.087245799",
      "0.374396486"
    ],
    [
      "0.691313017",
      "0.389474112"
    ],
    [
      "1.040296212",
      "0.389486039"
    ],
    [
      "1.214787810",
      "0.389487638"
    ],
    [
      "1.389279407",
      "0.389488922"
    ],
    [
      "1.563771005",
      "0.389489205"
    ],
    [
      "1.738262603",
      "0.389490839"
    ],
    [
      "1.912754201",
      "0.389491134"
    ],
    [
      "0.865804614",
      "0.389493802"
    ],
    [
      "0.238359955",
      "0.461642285"
    ],
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      "0.536040275",
      "0.469083743"
    ],
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      "0.087245799",
      "0.548888084"
    ],
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      "0.384926120",
      "0.556329545"
    ],
    [
      "0.691313017",
      "0.563965710"
    ],
    [
      "1.040296212",
      "0.563977954"
    ],
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      "0.563980520"
    ],
    [
      "1.214787809",
      "0.563981594"
    ],
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      "0.563983265"
    ],
    [
      "1.738262603",
      "0.563983586"
    ],
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      "1.912754201",
      "0.563983917"
    ],
    [
      "0.865804614",
      "0.563987444"
    ],
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      "0.536040277",
      "0.643575343"
    ],
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      "0.233811964",
      "0.643575344"
    ],
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      "0.384926120",
      "0.730821142"
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      "0.087245799",
      "0.738262603"
    ],
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      "0.691313017",
      "0.738457308"
    ],
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      "1.040296212",
      "0.738470625"
    ],
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      "1.389279407",
      "0.738474357"
    ],
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      "1.563771005",
      "0.738474863"
    ],
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      "1.214787809",
      "0.738475248"
    ],
    [
      "1.738262603",
      "0.738476571"
    ],
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      "1.912754201",
      "0.738476731"
    ],
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      "0.865804614",
      "0.738479042"
    ],
    [
      "0.536040277",
      "0.818066941"
    ],
    [
      "0.238359955",
      "0.825508402"
    ],
    [
      "1.031609598",
      "0.912752902"
    ],
    [
      "1.380595613",
      "0.912753101"
    ],
    [
      "1.904074457",
      "0.912753637"
    ],
    [
      "1.206102607",
      "0.912753673"
    ],
    [
      "1.555088617",
      "0.912753673"
    ],
    [
      "0.682623764",
      "0.912753673"
    ],
    [
      "1.729581627",
      "0.912753673"
    ],
    [
      "0.857116308",
      "0.912754201"
    ],
    [
      "0.389474112",
      "0.912754201"
    ],
    [
      "0.087245799",
      "0.912754201"
    ]
  ],
  "radius": "0.087245798"
}

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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