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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 = 79

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 = 79
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[0.081552742428921051072617734001, 0.081552742428921051072617734003]Eckard Specht2010-08-10Packomania ↗
本站纪录0.081552742NUE_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
0x2
已验证构造r = 0.081552742

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 = 79 的当前纪录构型，0.081552742
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（3 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.081552741→0.081552742
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.07936081→0.081552741
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.057142856→0.07936081
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": [
    [
      "0.244168144",
      "0.081552742"
    ],
    [
      "0.416213312",
      "0.081552742"
    ],
    [
      "0.579318796",
      "0.081552742"
    ],
    [
      "0.744584267",
      "0.081552742"
    ],
    [
      "0.923609207",
      "0.081552742"
    ],
    [
      "1.086714692",
      "0.081552742"
    ],
    [
      "1.249863955",
      "0.081552742"
    ],
    [
      "1.429420228",
      "0.081552742"
    ],
    [
      "1.592525712",
      "0.081552742"
    ],
    [
      "1.755631197",
      "0.081552742"
    ],
    [
      "1.918447258",
      "0.091265091"
    ],
    [
      "0.081552742",
      "0.094187229"
    ],
    [
      "1.339642091",
      "0.217726475"
    ],
    [
      "0.834096737",
      "0.217901255"
    ],
    [
      "0.330190728",
      "0.220129488"
    ],
    [
      "0.661951531",
      "0.222177172"
    ],
    [
      "1.168289323",
      "0.222793596"
    ],
    [
      "1.005161949",
      "0.222806236"
    ],
    [
      "1.510972970",
      "0.222806236"
    ],
    [
      "1.674078455",
      "0.222806236"
    ],
    [
      "0.493216715",
      "0.225340084"
    ],
    [
      "1.886320617",
      "0.251175310"
    ],
    [
      "0.091157753",
      "0.257009657"
    ],
    [
      "0.227363606",
      "0.346739054"
    ],
    [
      "0.751464002",
      "0.358525685"
    ],
    [
      "1.763897625",
      "0.358952906"
    ],
    [
      "1.258067460",
      "0.358967328"
    ],
    [
      "1.421194834",
      "0.358979968"
    ],
    [
      "0.915649479",
      "0.359154748"
    ],
    [
      "0.580398789",
      "0.363430666"
    ],
    [
      "1.086736581",
      "0.364047089"
    ],
    [
      "1.601081565",
      "0.368665254"
    ],
    [
      "0.388424268",
      "0.372485170"
    ],
    [
      "1.918447258",
      "0.411085529"
    ],
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      "0.081552742",
      "0.419832085"
    ],
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      "0.285629374",
      "0.499197137"
    ],
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      "0.490620653",
      "0.499604398"
    ],
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      "1.503827569",
      "0.499604398"
    ],
    [
      "0.669911259",
      "0.499779178"
    ],
    [
      "0.833016744",
      "0.499779178"
    ],
    [
      "1.176514717",
      "0.500220822"
    ],
    [
      "1.339620202",
      "0.500220822"
    ],
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      "0.997224111",
      "0.500395602"
    ],
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      "1.690900735",
      "0.504811925"
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      "1.918447258",
      "0.574191013"
    ],
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      "0.081552742",
      "0.582937569"
    ],
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      "0.390421405",
      "0.628303699"
    ],
    [
      "1.593646738",
      "0.635751069"
    ],
    [
      "0.580133123",
      "0.635952911"
    ],
    [
      "1.087002247",
      "0.636569334"
    ],
    [
      "1.422252937",
      "0.640845252"
    ],
    [
      "0.914591376",
      "0.641020032"
    ],
    [
      "0.751464002",
      "0.641032672"
    ],
    [
      "1.258067460",
      "0.641474315"
    ],
    [
      "1.772124888",
      "0.646254620"
    ],
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      "0.229110978",
      "0.652435529"
    ],
    [
      "1.907694987",
      "0.736941706"
    ],
    [
      "0.095144862",
      "0.745475729"
    ],
    [
      "0.493679411",
      "0.774482596"
    ],
    [
      "1.004369512",
      "0.777193764"
    ],
    [
      "1.511765407",
      "0.777193764"
    ],
    [
      "1.674870892",
      "0.777193764"
    ],
    [
      "0.661685865",
      "0.777206404"
    ],
    [
      "1.168554990",
      "0.777822828"
    ],
    [
      "0.330664969",
      "0.780068542"
    ],
    [
      "1.340700195",
      "0.782098745"
    ],
    [
      "0.833038633",
      "0.782273525"
    ],
    [
      "1.918447258",
      "0.899692398"
    ],
    [
      "0.081552742",
      "0.908013890"
    ],
    [
      "0.244324190",
      "0.918447258"
    ],
    [
      "0.417005749",
      "0.918447258"
    ],
    [
      "0.580111234",
      "0.918447258"
    ],
    [
      "0.743260497",
      "0.918447258"
    ],
    [
      "0.922816770",
      "0.918447258"
    ],
    [
      "1.085922254",
      "0.918447258"
    ],
    [
      "1.251187725",
      "0.918447258"
    ],
    [
      "1.430212665",
      "0.918447258"
    ],
    [
      "1.593318150",
      "0.918447258"
    ],
    [
      "1.756423634",
      "0.918447258"
    ]
  ],
  "radius": "0.081552742"
}
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": 79
}

当前第一名的答案

{
  "centers": [
    [
      "0.244168144",
      "0.081552742"
    ],
    [
      "0.416213312",
      "0.081552742"
    ],
    [
      "0.579318796",
      "0.081552742"
    ],
    [
      "0.744584267",
      "0.081552742"
    ],
    [
      "0.923609207",
      "0.081552742"
    ],
    [
      "1.086714692",
      "0.081552742"
    ],
    [
      "1.249863955",
      "0.081552742"
    ],
    [
      "1.429420228",
      "0.081552742"
    ],
    [
      "1.592525712",
      "0.081552742"
    ],
    [
      "1.755631197",
      "0.081552742"
    ],
    [
      "1.918447258",
      "0.091265091"
    ],
    [
      "0.081552742",
      "0.094187229"
    ],
    [
      "1.339642091",
      "0.217726475"
    ],
    [
      "0.834096737",
      "0.217901255"
    ],
    [
      "0.330190728",
      "0.220129488"
    ],
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      "0.661951531",
      "0.222177172"
    ],
    [
      "1.168289323",
      "0.222793596"
    ],
    [
      "1.005161949",
      "0.222806236"
    ],
    [
      "1.510972970",
      "0.222806236"
    ],
    [
      "1.674078455",
      "0.222806236"
    ],
    [
      "0.493216715",
      "0.225340084"
    ],
    [
      "1.886320617",
      "0.251175310"
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      "0.915649479",
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      "0.285629374",
      "0.499197137"
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      "0.499604398"
    ],
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      "0.499604398"
    ],
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      "0.669911259",
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      "0.745475729"
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      "1.674870892",
      "0.777193764"
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      "0.782273525"
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      "0.899692398"
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      "0.918447258"
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      "0.918447258"
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      "0.918447258"
    ],
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      "0.743260497",
      "0.918447258"
    ],
    [
      "0.922816770",
      "0.918447258"
    ],
    [
      "1.085922254",
      "0.918447258"
    ],
    [
      "1.251187725",
      "0.918447258"
    ],
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      "1.430212665",
      "0.918447258"
    ],
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      "1.593318150",
      "0.918447258"
    ],
    [
      "1.756423634",
      "0.918447258"
    ]
  ],
  "radius": "0.081552742"
}

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 条回复好！！！NUE_13#310 条回复