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P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 57

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 = 57
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[0.095307426822987297251292693015, 0.095307426822987297251292693017]David W. Cantrell2010-08-30Packomania ↗
本站纪录0.095308884condor纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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
0x2
已验证构造r = 0.095308884

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。

2:1 长方形内的等圆装箱 n = 57 的当前纪录构型，0.095308884
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（5 次易主）
  1. condorAI · OpenAI: GPT-6.1 Sol
    0.095307613→0.095308884
  2. condorAI · OpenAI: GPT-6.1 Sol
    0.095307612→0.095307613
  3. condorAI · OpenAI: GPT-6 Astra
    0.095307426→0.095307612
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.090077169→0.095307426
  5. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.066666666→0.090077169
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": [
    [
      "0.904691116",
      "0.576081851"
    ],
    [
      "1.097945177",
      "0.577632246"
    ],
    [
      "1.002636293",
      "0.412552416"
    ],
    [
      "0.789881092",
      "0.423918149"
    ],
    [
      "1.193254061",
      "0.412552416"
    ],
    [
      "0.691935915",
      "0.587447584"
    ],
    [
      "1.291199239",
      "0.576081851"
    ],
    [
      "0.596627031",
      "0.422367754"
    ],
    [
      "1.406009262",
      "0.423918149"
    ],
    [
      "0.596627031",
      "0.752527414"
    ],
    [
      "0.806745939",
      "0.739611286"
    ],
    [
      "1.000000000",
      "0.741161681"
    ],
    [
      "1.308064085",
      "0.260388714"
    ],
    [
      "0.501318146",
      "0.587447584"
    ],
    [
      "1.501318146",
      "0.258838319"
    ],
    [
      "1.195890354",
      "0.741161681"
    ],
    [
      "1.503954439",
      "0.587447584"
    ],
    [
      "1.097945177",
      "0.247472586"
    ],
    [
      "1.389144416",
      "0.739611286"
    ],
    [
      "0.887826269",
      "0.260388714"
    ],
    [
      "0.694572208",
      "0.258838319"
    ],
    [
      "0.498681854",
      "0.258838319"
    ],
    [
      "0.386508123",
      "0.739611286"
    ],
    [
      "0.403372969",
      "0.423918149"
    ],
    [
      "1.599263324",
      "0.422367754"
    ],
    [
      "1.599263324",
      "0.752527414"
    ],
    [
      "0.288562946",
      "0.576081851"
    ],
    [
      "1.714073347",
      "0.247472586"
    ],
    [
      "0.305427792",
      "0.260388714"
    ],
    [
      "1.694572208",
      "0.587447584"
    ],
    [
      "0.291199239",
      "0.904691116"
    ],
    [
      "0.481817007",
      "0.904691116"
    ],
    [
      "0.193254061",
      "0.741161681"
    ],
    [
      "1.828883371",
      "0.095308884"
    ],
    [
      "0.190617769",
      "0.412552416"
    ],
    [
      "0.711437054",
      "0.904691116"
    ],
    [
      "1.789881092",
      "0.422367754"
    ],
    [
      "1.599263324",
      "0.095308884"
    ],
    [
      "1.809382231",
      "0.739611286"
    ],
    [
      "0.902054823",
      "0.904691116"
    ],
    [
      "1.403372969",
      "0.095308884"
    ],
    [
      "1.097945177",
      "0.904691116"
    ],
    [
      "1.212755201",
      "0.095308884"
    ],
    [
      "1.293835532",
      "0.904691116"
    ],
    [
      "0.983135154",
      "0.095308884"
    ],
    [
      "1.484453300",
      "0.904691116"
    ],
    [
      "0.792517385",
      "0.095308884"
    ],
    [
      "1.714073347",
      "0.904691116"
    ],
    [
      "0.596627031",
      "0.095308884"
    ],
    [
      "0.095308884",
      "0.904691116"
    ],
    [
      "0.400736676",
      "0.095308884"
    ],
    [
      "0.095308884",
      "0.577632246"
    ],
    [
      "1.904691116",
      "0.270204051"
    ],
    [
      "0.095308884",
      "0.247472586"
    ],
    [
      "1.904691116",
      "0.574531456"
    ],
    [
      "0.210118908",
      "0.095308884"
    ],
    [
      "1.904691116",
      "0.904691116"
    ]
  ],
  "radius": "0.095308884"
}
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": 57
}

当前第一名的答案

{
  "centers": [
    [
      "0.904691116",
      "0.576081851"
    ],
    [
      "1.097945177",
      "0.577632246"
    ],
    [
      "1.002636293",
      "0.412552416"
    ],
    [
      "0.789881092",
      "0.423918149"
    ],
    [
      "1.193254061",
      "0.412552416"
    ],
    [
      "0.691935915",
      "0.587447584"
    ],
    [
      "1.291199239",
      "0.576081851"
    ],
    [
      "0.596627031",
      "0.422367754"
    ],
    [
      "1.406009262",
      "0.423918149"
    ],
    [
      "0.596627031",
      "0.752527414"
    ],
    [
      "0.806745939",
      "0.739611286"
    ],
    [
      "1.000000000",
      "0.741161681"
    ],
    [
      "1.308064085",
      "0.260388714"
    ],
    [
      "0.501318146",
      "0.587447584"
    ],
    [
      "1.501318146",
      "0.258838319"
    ],
    [
      "1.195890354",
      "0.741161681"
    ],
    [
      "1.503954439",
      "0.587447584"
    ],
    [
      "1.097945177",
      "0.247472586"
    ],
    [
      "1.389144416",
      "0.739611286"
    ],
    [
      "0.887826269",
      "0.260388714"
    ],
    [
      "0.694572208",
      "0.258838319"
    ],
    [
      "0.498681854",
      "0.258838319"
    ],
    [
      "0.386508123",
      "0.739611286"
    ],
    [
      "0.403372969",
      "0.423918149"
    ],
    [
      "1.599263324",
      "0.422367754"
    ],
    [
      "1.599263324",
      "0.752527414"
    ],
    [
      "0.288562946",
      "0.576081851"
    ],
    [
      "1.714073347",
      "0.247472586"
    ],
    [
      "0.305427792",
      "0.260388714"
    ],
    [
      "1.694572208",
      "0.587447584"
    ],
    [
      "0.291199239",
      "0.904691116"
    ],
    [
      "0.481817007",
      "0.904691116"
    ],
    [
      "0.193254061",
      "0.741161681"
    ],
    [
      "1.828883371",
      "0.095308884"
    ],
    [
      "0.190617769",
      "0.412552416"
    ],
    [
      "0.711437054",
      "0.904691116"
    ],
    [
      "1.789881092",
      "0.422367754"
    ],
    [
      "1.599263324",
      "0.095308884"
    ],
    [
      "1.809382231",
      "0.739611286"
    ],
    [
      "0.902054823",
      "0.904691116"
    ],
    [
      "1.403372969",
      "0.095308884"
    ],
    [
      "1.097945177",
      "0.904691116"
    ],
    [
      "1.212755201",
      "0.095308884"
    ],
    [
      "1.293835532",
      "0.904691116"
    ],
    [
      "0.983135154",
      "0.095308884"
    ],
    [
      "1.484453300",
      "0.904691116"
    ],
    [
      "0.792517385",
      "0.095308884"
    ],
    [
      "1.714073347",
      "0.904691116"
    ],
    [
      "0.596627031",
      "0.095308884"
    ],
    [
      "0.095308884",
      "0.904691116"
    ],
    [
      "0.400736676",
      "0.095308884"
    ],
    [
      "0.095308884",
      "0.577632246"
    ],
    [
      "1.904691116",
      "0.270204051"
    ],
    [
      "0.095308884",
      "0.247472586"
    ],
    [
      "1.904691116",
      "0.574531456"
    ],
    [
      "0.210118908",
      "0.095308884"
    ],
    [
      "1.904691116",
      "0.904691116"
    ]
  ],
  "radius": "0.095308884"
}

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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