P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 48

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 = 48

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	[0.102658308611082398028053376553, 0.102658308611082398028053376555]	David W. Cantrell2010-08-02	Packomania ↗
本站纪录	0.102658308	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748

0x2

已验证构造r = 0.102658308

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
0.100129915→0.102658308
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.08→0.100129915
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": [
    [
      "0.102658309",
      "0.102658309"
    ],
    [
      "0.307974926",
      "0.102658309"
    ],
    [
      "0.513291543",
      "0.102658309"
    ],
    [
      "0.888480663",
      "0.102658309"
    ],
    [
      "1.093797280",
      "0.102658309"
    ],
    [
      "1.299113897",
      "0.102658309"
    ],
    [
      "1.504430514",
      "0.102658309"
    ],
    [
      "1.709747132",
      "0.102658309"
    ],
    [
      "0.700886103",
      "0.186104066"
    ],
    [
      "1.897341691",
      "0.186104066"
    ],
    [
      "0.205316617",
      "0.280467715"
    ],
    [
      "0.410633234",
      "0.280467715"
    ],
    [
      "0.991138971",
      "0.280467715"
    ],
    [
      "1.196455589",
      "0.280467715"
    ],
    [
      "1.401772206",
      "0.280467715"
    ],
    [
      "1.607088823",
      "0.280467715"
    ],
    [
      "0.598227794",
      "0.363913472"
    ],
    [
      "0.803544411",
      "0.363913472"
    ],
    [
      "1.794683383",
      "0.363913472"
    ],
    [
      "0.102658309",
      "0.458277121"
    ],
    [
      "0.307974926",
      "0.458277121"
    ],
    [
      "1.093797280",
      "0.458277121"
    ],
    [
      "1.299113897",
      "0.458277121"
    ],
    [
      "1.504430514",
      "0.458277121"
    ],
    [
      "0.495569486",
      "0.541722879"
    ],
    [
      "0.700886103",
      "0.541722879"
    ],
    [
      "0.906202720",
      "0.541722879"
    ],
    [
      "1.692025074",
      "0.541722879"
    ],
    [
      "1.897341691",
      "0.541722879"
    ],
    [
      "0.205316617",
      "0.636086528"
    ],
    [
      "1.196455589",
      "0.636086528"
    ],
    [
      "1.401772206",
      "0.636086528"
    ],
    [
      "0.392911177",
      "0.719532285"
    ],
    [
      "0.598227794",
      "0.719532285"
    ],
    [
      "0.803544411",
      "0.719532285"
    ],
    [
      "1.008861029",
      "0.719532285"
    ],
    [
      "1.589366766",
      "0.719532285"
    ],
    [
      "1.794683383",
      "0.719532285"
    ],
    [
      "0.102658309",
      "0.813895934"
    ],
    [
      "1.299113897",
      "0.813895934"
    ],
    [
      "0.290252868",
      "0.897341691"
    ],
    [
      "0.495569486",
      "0.897341691"
    ],
    [
      "0.700886103",
      "0.897341691"
    ],
    [
      "0.906202720",
      "0.897341691"
    ],
    [
      "1.111519337",
      "0.897341691"
    ],
    [
      "1.486708457",
      "0.897341691"
    ],
    [
      "1.692025074",
      "0.897341691"
    ],
    [
      "1.897341691",
      "0.897341691"
    ]
  ],
  "radius": "0.102658308"
}

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": 48
}

当前第一名的答案

{
  "centers": [
    [
      "0.102658309",
      "0.102658309"
    ],
    [
      "0.307974926",
      "0.102658309"
    ],
    [
      "0.513291543",
      "0.102658309"
    ],
    [
      "0.888480663",
      "0.102658309"
    ],
    [
      "1.093797280",
      "0.102658309"
    ],
    [
      "1.299113897",
      "0.102658309"
    ],
    [
      "1.504430514",
      "0.102658309"
    ],
    [
      "1.709747132",
      "0.102658309"
    ],
    [
      "0.700886103",
      "0.186104066"
    ],
    [
      "1.897341691",
      "0.186104066"
    ],
    [
      "0.205316617",
      "0.280467715"
    ],
    [
      "0.410633234",
      "0.280467715"
    ],
    [
      "0.991138971",
      "0.280467715"
    ],
    [
      "1.196455589",
      "0.280467715"
    ],
    [
      "1.401772206",
      "0.280467715"
    ],
    [
      "1.607088823",
      "0.280467715"
    ],
    [
      "0.598227794",
      "0.363913472"
    ],
    [
      "0.803544411",
      "0.363913472"
    ],
    [
      "1.794683383",
      "0.363913472"
    ],
    [
      "0.102658309",
      "0.458277121"
    ],
    [
      "0.307974926",
      "0.458277121"
    ],
    [
      "1.093797280",
      "0.458277121"
    ],
    [
      "1.299113897",
      "0.458277121"
    ],
    [
      "1.504430514",
      "0.458277121"
    ],
    [
      "0.495569486",
      "0.541722879"
    ],
    [
      "0.700886103",
      "0.541722879"
    ],
    [
      "0.906202720",
      "0.541722879"
    ],
    [
      "1.692025074",
      "0.541722879"
    ],
    [
      "1.897341691",
      "0.541722879"
    ],
    [
      "0.205316617",
      "0.636086528"
    ],
    [
      "1.196455589",
      "0.636086528"
    ],
    [
      "1.401772206",
      "0.636086528"
    ],
    [
      "0.392911177",
      "0.719532285"
    ],
    [
      "0.598227794",
      "0.719532285"
    ],
    [
      "0.803544411",
      "0.719532285"
    ],
    [
      "1.008861029",
      "0.719532285"
    ],
    [
      "1.589366766",
      "0.719532285"
    ],
    [
      "1.794683383",
      "0.719532285"
    ],
    [
      "0.102658309",
      "0.813895934"
    ],
    [
      "1.299113897",
      "0.813895934"
    ],
    [
      "0.290252868",
      "0.897341691"
    ],
    [
      "0.495569486",
      "0.897341691"
    ],
    [
      "0.700886103",
      "0.897341691"
    ],
    [
      "0.906202720",
      "0.897341691"
    ],
    [
      "1.111519337",
      "0.897341691"
    ],
    [
      "1.486708457",
      "0.897341691"
    ],
    [
      "1.692025074",
      "0.897341691"
    ],
    [
      "1.897341691",
      "0.897341691"
    ]
  ],
  "radius": "0.102658308"
}

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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