P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 49

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 = 49

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	[0.101004283938801899632894910957, 0.101004283938801899632894910959]	David W. Cantrell2010-08-11	Packomania ↗
本站纪录	0.101004283	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849

0x2

已验证构造r = 0.101004283

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
0.099060402→0.101004283
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.08→0.099060402
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": [
    [
      "0.101004284",
      "0.101004284"
    ],
    [
      "0.499543137",
      "0.101004284"
    ],
    [
      "0.701551704",
      "0.101004284"
    ],
    [
      "0.903560272",
      "0.101004284"
    ],
    [
      "1.297839144",
      "0.101004284"
    ],
    [
      "1.499847712",
      "0.101004284"
    ],
    [
      "1.701856280",
      "0.101004284"
    ],
    [
      "0.300273710",
      "0.134157824"
    ],
    [
      "1.100699708",
      "0.145089477"
    ],
    [
      "1.898995716",
      "0.145089477"
    ],
    [
      "0.101004284",
      "0.303012852"
    ],
    [
      "0.499543137",
      "0.303012852"
    ],
    [
      "0.701551704",
      "0.303012852"
    ],
    [
      "0.903560272",
      "0.303012852"
    ],
    [
      "1.297839144",
      "0.303012852"
    ],
    [
      "1.499847712",
      "0.303012852"
    ],
    [
      "1.701856280",
      "0.303012852"
    ],
    [
      "0.300273710",
      "0.336166392"
    ],
    [
      "1.100699708",
      "0.347098045"
    ],
    [
      "1.898995716",
      "0.347098045"
    ],
    [
      "0.101004284",
      "0.505021420"
    ],
    [
      "0.499543137",
      "0.505021420"
    ],
    [
      "0.701551704",
      "0.505021420"
    ],
    [
      "0.903560272",
      "0.505021420"
    ],
    [
      "1.297839144",
      "0.505021420"
    ],
    [
      "1.499847712",
      "0.505021420"
    ],
    [
      "1.701856280",
      "0.505021420"
    ],
    [
      "0.300273710",
      "0.538174960"
    ],
    [
      "1.100699708",
      "0.549106613"
    ],
    [
      "1.898995716",
      "0.549106613"
    ],
    [
      "0.802555988",
      "0.679965971"
    ],
    [
      "1.398843428",
      "0.679965971"
    ],
    [
      "1.600851996",
      "0.679965971"
    ],
    [
      "0.600244408",
      "0.680140916"
    ],
    [
      "0.171927189",
      "0.694170575"
    ],
    [
      "0.399257908",
      "0.718730861"
    ],
    [
      "0.999695424",
      "0.724051164"
    ],
    [
      "1.201703992",
      "0.724051164"
    ],
    [
      "1.797991432",
      "0.724051164"
    ],
    [
      "0.701551704",
      "0.854910523"
    ],
    [
      "1.499847712",
      "0.854910523"
    ],
    [
      "0.101004284",
      "0.883319731"
    ],
    [
      "0.302403700",
      "0.898995716"
    ],
    [
      "0.504412268",
      "0.898995716"
    ],
    [
      "0.898691140",
      "0.898995716"
    ],
    [
      "1.100699708",
      "0.898995716"
    ],
    [
      "1.302708276",
      "0.898995716"
    ],
    [
      "1.696987148",
      "0.898995716"
    ],
    [
      "1.898995716",
      "0.898995716"
    ]
  ],
  "radius": "0.101004283"
}

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": 49
}

当前第一名的答案

{
  "centers": [
    [
      "0.101004284",
      "0.101004284"
    ],
    [
      "0.499543137",
      "0.101004284"
    ],
    [
      "0.701551704",
      "0.101004284"
    ],
    [
      "0.903560272",
      "0.101004284"
    ],
    [
      "1.297839144",
      "0.101004284"
    ],
    [
      "1.499847712",
      "0.101004284"
    ],
    [
      "1.701856280",
      "0.101004284"
    ],
    [
      "0.300273710",
      "0.134157824"
    ],
    [
      "1.100699708",
      "0.145089477"
    ],
    [
      "1.898995716",
      "0.145089477"
    ],
    [
      "0.101004284",
      "0.303012852"
    ],
    [
      "0.499543137",
      "0.303012852"
    ],
    [
      "0.701551704",
      "0.303012852"
    ],
    [
      "0.903560272",
      "0.303012852"
    ],
    [
      "1.297839144",
      "0.303012852"
    ],
    [
      "1.499847712",
      "0.303012852"
    ],
    [
      "1.701856280",
      "0.303012852"
    ],
    [
      "0.300273710",
      "0.336166392"
    ],
    [
      "1.100699708",
      "0.347098045"
    ],
    [
      "1.898995716",
      "0.347098045"
    ],
    [
      "0.101004284",
      "0.505021420"
    ],
    [
      "0.499543137",
      "0.505021420"
    ],
    [
      "0.701551704",
      "0.505021420"
    ],
    [
      "0.903560272",
      "0.505021420"
    ],
    [
      "1.297839144",
      "0.505021420"
    ],
    [
      "1.499847712",
      "0.505021420"
    ],
    [
      "1.701856280",
      "0.505021420"
    ],
    [
      "0.300273710",
      "0.538174960"
    ],
    [
      "1.100699708",
      "0.549106613"
    ],
    [
      "1.898995716",
      "0.549106613"
    ],
    [
      "0.802555988",
      "0.679965971"
    ],
    [
      "1.398843428",
      "0.679965971"
    ],
    [
      "1.600851996",
      "0.679965971"
    ],
    [
      "0.600244408",
      "0.680140916"
    ],
    [
      "0.171927189",
      "0.694170575"
    ],
    [
      "0.399257908",
      "0.718730861"
    ],
    [
      "0.999695424",
      "0.724051164"
    ],
    [
      "1.201703992",
      "0.724051164"
    ],
    [
      "1.797991432",
      "0.724051164"
    ],
    [
      "0.701551704",
      "0.854910523"
    ],
    [
      "1.499847712",
      "0.854910523"
    ],
    [
      "0.101004284",
      "0.883319731"
    ],
    [
      "0.302403700",
      "0.898995716"
    ],
    [
      "0.504412268",
      "0.898995716"
    ],
    [
      "0.898691140",
      "0.898995716"
    ],
    [
      "1.100699708",
      "0.898995716"
    ],
    [
      "1.302708276",
      "0.898995716"
    ],
    [
      "1.696987148",
      "0.898995716"
    ],
    [
      "1.898995716",
      "0.898995716"
    ]
  ],
  "radius": "0.101004283"
}

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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