P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 56

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 = 56

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	[0.095707426403254254068587403303, 0.095707426403254254068587403305]	Eckard Specht1999–2010	Packomania ↗
本站纪录	0.095707714	condor	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556

0x2

已验证构造r = 0.095707714

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（4 次易主）

condorAI · OpenAI: GPT-6.1 Sol
0.09570761→0.095707714
2026-10-27
Joost de WinterAI · OpenAI: GPT-6.1 Sol
0.095707426→0.09570761
2026-10-27
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.093479303→0.095707426
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.066666666→0.093479303
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": [
    [
      "0.287115713",
      "0.095707715"
    ],
    [
      "0.478531142",
      "0.095707715"
    ],
    [
      "0.669946572",
      "0.095707715"
    ],
    [
      "0.861362001",
      "0.095707715"
    ],
    [
      "1.109917560",
      "0.095707715"
    ],
    [
      "1.301332989",
      "0.095707715"
    ],
    [
      "1.492748419",
      "0.095707715"
    ],
    [
      "1.684163848",
      "0.095707715"
    ],
    [
      "0.095707715",
      "0.097394324"
    ],
    [
      "1.903328902",
      "0.096614654"
    ],
    [
      "0.985639781",
      "0.241292398"
    ],
    [
      "1.397040704",
      "0.261478339"
    ],
    [
      "0.764928622",
      "0.261895189"
    ],
    [
      "1.205625275",
      "0.261478339"
    ],
    [
      "1.588456133",
      "0.261478339"
    ],
    [
      "1.780014506",
      "0.261395729"
    ],
    [
      "0.573513192",
      "0.261895189"
    ],
    [
      "0.382097763",
      "0.261895189"
    ],
    [
      "0.176554551",
      "0.270898374"
    ],
    [
      "0.889789123",
      "0.406980412"
    ],
    [
      "1.081347495",
      "0.407063023"
    ],
    [
      "1.904292285",
      "0.406980412"
    ],
    [
      "1.301332989",
      "0.427248964"
    ],
    [
      "1.492748419",
      "0.427248964"
    ],
    [
      "0.669220907",
      "0.427665814"
    ],
    [
      "1.684306791",
      "0.427166353"
    ],
    [
      "0.477805478",
      "0.427665814"
    ],
    [
      "0.286390048",
      "0.427665814"
    ],
    [
      "0.095707715",
      "0.444402425"
    ],
    [
      "1.177055210",
      "0.572833647"
    ],
    [
      "0.794081408",
      "0.572751036"
    ],
    [
      "0.985496837",
      "0.572751036"
    ],
    [
      "1.808584571",
      "0.572751036"
    ],
    [
      "1.397040704",
      "0.593019588"
    ],
    [
      "1.588599077",
      "0.592936977"
    ],
    [
      "0.573513192",
      "0.593436438"
    ],
    [
      "0.382097763",
      "0.593436438"
    ],
    [
      "0.095707715",
      "0.635817854"
    ],
    [
      "0.257237262",
      "0.738521661"
    ],
    [
      "0.698373693",
      "0.738521661"
    ],
    [
      "1.081204552",
      "0.738521661"
    ],
    [
      "1.272762924",
      "0.738604271"
    ],
    [
      "1.712876856",
      "0.738521661"
    ],
    [
      "0.889789123",
      "0.738521661"
    ],
    [
      "1.904292285",
      "0.738521661"
    ],
    [
      "1.492891362",
      "0.758707602"
    ],
    [
      "0.477805478",
      "0.759207063"
    ],
    [
      "0.139994357",
      "0.902981641"
    ],
    [
      "0.352944977",
      "0.904292285"
    ],
    [
      "0.602665979",
      "0.904292285"
    ],
    [
      "0.794081408",
      "0.904292285"
    ],
    [
      "0.985496837",
      "0.904292285"
    ],
    [
      "1.176912267",
      "0.904292285"
    ],
    [
      "1.368613582",
      "0.904292285"
    ],
    [
      "1.617169141",
      "0.904292285"
    ],
    [
      "1.808584571",
      "0.904292285"
    ]
  ],
  "radius": "0.095707714"
}

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": 56
}

当前第一名的答案

{
  "centers": [
    [
      "0.287115713",
      "0.095707715"
    ],
    [
      "0.478531142",
      "0.095707715"
    ],
    [
      "0.669946572",
      "0.095707715"
    ],
    [
      "0.861362001",
      "0.095707715"
    ],
    [
      "1.109917560",
      "0.095707715"
    ],
    [
      "1.301332989",
      "0.095707715"
    ],
    [
      "1.492748419",
      "0.095707715"
    ],
    [
      "1.684163848",
      "0.095707715"
    ],
    [
      "0.095707715",
      "0.097394324"
    ],
    [
      "1.903328902",
      "0.096614654"
    ],
    [
      "0.985639781",
      "0.241292398"
    ],
    [
      "1.397040704",
      "0.261478339"
    ],
    [
      "0.764928622",
      "0.261895189"
    ],
    [
      "1.205625275",
      "0.261478339"
    ],
    [
      "1.588456133",
      "0.261478339"
    ],
    [
      "1.780014506",
      "0.261395729"
    ],
    [
      "0.573513192",
      "0.261895189"
    ],
    [
      "0.382097763",
      "0.261895189"
    ],
    [
      "0.176554551",
      "0.270898374"
    ],
    [
      "0.889789123",
      "0.406980412"
    ],
    [
      "1.081347495",
      "0.407063023"
    ],
    [
      "1.904292285",
      "0.406980412"
    ],
    [
      "1.301332989",
      "0.427248964"
    ],
    [
      "1.492748419",
      "0.427248964"
    ],
    [
      "0.669220907",
      "0.427665814"
    ],
    [
      "1.684306791",
      "0.427166353"
    ],
    [
      "0.477805478",
      "0.427665814"
    ],
    [
      "0.286390048",
      "0.427665814"
    ],
    [
      "0.095707715",
      "0.444402425"
    ],
    [
      "1.177055210",
      "0.572833647"
    ],
    [
      "0.794081408",
      "0.572751036"
    ],
    [
      "0.985496837",
      "0.572751036"
    ],
    [
      "1.808584571",
      "0.572751036"
    ],
    [
      "1.397040704",
      "0.593019588"
    ],
    [
      "1.588599077",
      "0.592936977"
    ],
    [
      "0.573513192",
      "0.593436438"
    ],
    [
      "0.382097763",
      "0.593436438"
    ],
    [
      "0.095707715",
      "0.635817854"
    ],
    [
      "0.257237262",
      "0.738521661"
    ],
    [
      "0.698373693",
      "0.738521661"
    ],
    [
      "1.081204552",
      "0.738521661"
    ],
    [
      "1.272762924",
      "0.738604271"
    ],
    [
      "1.712876856",
      "0.738521661"
    ],
    [
      "0.889789123",
      "0.738521661"
    ],
    [
      "1.904292285",
      "0.738521661"
    ],
    [
      "1.492891362",
      "0.758707602"
    ],
    [
      "0.477805478",
      "0.759207063"
    ],
    [
      "0.139994357",
      "0.902981641"
    ],
    [
      "0.352944977",
      "0.904292285"
    ],
    [
      "0.602665979",
      "0.904292285"
    ],
    [
      "0.794081408",
      "0.904292285"
    ],
    [
      "0.985496837",
      "0.904292285"
    ],
    [
      "1.176912267",
      "0.904292285"
    ],
    [
      "1.368613582",
      "0.904292285"
    ],
    [
      "1.617169141",
      "0.904292285"
    ],
    [
      "1.808584571",
      "0.904292285"
    ]
  ],
  "radius": "0.095707714"
}

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