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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 = 77

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 = 77
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[0.082307226525211902738439125965, 0.082307226525211902738439125967]Eckard Specht1999–2010Packomania ↗
本站纪录0.082307226NUE_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 = 0.082307226

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 = 77 的当前纪录构型，0.082307226
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）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.079102316→0.082307226
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.057142856→0.079102316
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": [
    [
      "0.082307227",
      "0.082307227"
    ],
    [
      "0.246921680",
      "0.082307227"
    ],
    [
      "0.442088644",
      "0.082307227"
    ],
    [
      "0.606703097",
      "0.082307227"
    ],
    [
      "0.771317550",
      "0.082307227"
    ],
    [
      "0.966484515",
      "0.082307227"
    ],
    [
      "1.131098968",
      "0.082307227"
    ],
    [
      "1.295713421",
      "0.082307227"
    ],
    [
      "1.490880385",
      "0.082307227"
    ],
    [
      "1.655494838",
      "0.082307227"
    ],
    [
      "1.820109291",
      "0.082307227"
    ],
    [
      "0.344505162",
      "0.214879404"
    ],
    [
      "0.868901032",
      "0.214879404"
    ],
    [
      "1.393296903",
      "0.214879404"
    ],
    [
      "1.917692773",
      "0.214879404"
    ],
    [
      "0.164614453",
      "0.224867525"
    ],
    [
      "0.524395871",
      "0.224867525"
    ],
    [
      "0.689010324",
      "0.224867525"
    ],
    [
      "1.048791741",
      "0.224867525"
    ],
    [
      "1.213406194",
      "0.224867525"
    ],
    [
      "1.573187612",
      "0.224867525"
    ],
    [
      "1.737802065",
      "0.224867525"
    ],
    [
      "0.262197935",
      "0.357439702"
    ],
    [
      "0.426812388",
      "0.357439702"
    ],
    [
      "0.786593806",
      "0.357439702"
    ],
    [
      "0.951208259",
      "0.357439702"
    ],
    [
      "1.310989676",
      "0.357439702"
    ],
    [
      "1.475604129",
      "0.357439702"
    ],
    [
      "1.835385547",
      "0.357439702"
    ],
    [
      "0.082307227",
      "0.367427823"
    ],
    [
      "0.606703097",
      "0.367427823"
    ],
    [
      "1.131098968",
      "0.367427823"
    ],
    [
      "1.655494838",
      "0.367427823"
    ],
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      "0.179890709",
      "0.500000000"
    ],
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      "0.344505162",
      "0.500000000"
    ],
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      "0.509119615",
      "0.500000000"
    ],
    [
      "0.704286579",
      "0.500000000"
    ],
    [
      "0.868901032",
      "0.500000000"
    ],
    [
      "1.033515485",
      "0.500000000"
    ],
    [
      "1.228682450",
      "0.500000000"
    ],
    [
      "1.393296903",
      "0.500000000"
    ],
    [
      "1.557911356",
      "0.500000000"
    ],
    [
      "1.753078320",
      "0.500000000"
    ],
    [
      "1.917692773",
      "0.500000000"
    ],
    [
      "0.082307227",
      "0.632572177"
    ],
    [
      "0.606703097",
      "0.632572177"
    ],
    [
      "1.131098968",
      "0.632572177"
    ],
    [
      "1.655494838",
      "0.632572177"
    ],
    [
      "0.262197935",
      "0.642560298"
    ],
    [
      "0.426812388",
      "0.642560298"
    ],
    [
      "0.786593806",
      "0.642560298"
    ],
    [
      "0.951208259",
      "0.642560298"
    ],
    [
      "1.310989676",
      "0.642560298"
    ],
    [
      "1.475604129",
      "0.642560298"
    ],
    [
      "1.835385547",
      "0.642560298"
    ],
    [
      "0.164614453",
      "0.775132475"
    ],
    [
      "0.524395871",
      "0.775132475"
    ],
    [
      "0.689010324",
      "0.775132475"
    ],
    [
      "1.048791741",
      "0.775132475"
    ],
    [
      "1.213406194",
      "0.775132475"
    ],
    [
      "1.573187612",
      "0.775132475"
    ],
    [
      "1.737802065",
      "0.775132475"
    ],
    [
      "0.344505162",
      "0.785120596"
    ],
    [
      "0.868901032",
      "0.785120596"
    ],
    [
      "1.393296903",
      "0.785120596"
    ],
    [
      "1.917692773",
      "0.785120596"
    ],
    [
      "0.082307227",
      "0.917692773"
    ],
    [
      "0.246921680",
      "0.917692773"
    ],
    [
      "0.442088644",
      "0.917692773"
    ],
    [
      "0.606703097",
      "0.917692773"
    ],
    [
      "0.771317550",
      "0.917692773"
    ],
    [
      "0.966484515",
      "0.917692773"
    ],
    [
      "1.131098968",
      "0.917692773"
    ],
    [
      "1.295713421",
      "0.917692773"
    ],
    [
      "1.490880385",
      "0.917692773"
    ],
    [
      "1.655494838",
      "0.917692773"
    ],
    [
      "1.820109291",
      "0.917692773"
    ]
  ],
  "radius": "0.082307226"
}
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": 77
}

当前第一名的答案

{
  "centers": [
    [
      "0.082307227",
      "0.082307227"
    ],
    [
      "0.246921680",
      "0.082307227"
    ],
    [
      "0.442088644",
      "0.082307227"
    ],
    [
      "0.606703097",
      "0.082307227"
    ],
    [
      "0.771317550",
      "0.082307227"
    ],
    [
      "0.966484515",
      "0.082307227"
    ],
    [
      "1.131098968",
      "0.082307227"
    ],
    [
      "1.295713421",
      "0.082307227"
    ],
    [
      "1.490880385",
      "0.082307227"
    ],
    [
      "1.655494838",
      "0.082307227"
    ],
    [
      "1.820109291",
      "0.082307227"
    ],
    [
      "0.344505162",
      "0.214879404"
    ],
    [
      "0.868901032",
      "0.214879404"
    ],
    [
      "1.393296903",
      "0.214879404"
    ],
    [
      "1.917692773",
      "0.214879404"
    ],
    [
      "0.164614453",
      "0.224867525"
    ],
    [
      "0.524395871",
      "0.224867525"
    ],
    [
      "0.689010324",
      "0.224867525"
    ],
    [
      "1.048791741",
      "0.224867525"
    ],
    [
      "1.213406194",
      "0.224867525"
    ],
    [
      "1.573187612",
      "0.224867525"
    ],
    [
      "1.737802065",
      "0.224867525"
    ],
    [
      "0.262197935",
      "0.357439702"
    ],
    [
      "0.426812388",
      "0.357439702"
    ],
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      "0.357439702"
    ],
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      "0.951208259",
      "0.357439702"
    ],
    [
      "1.310989676",
      "0.357439702"
    ],
    [
      "1.475604129",
      "0.357439702"
    ],
    [
      "1.835385547",
      "0.357439702"
    ],
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      "0.082307227",
      "0.367427823"
    ],
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      "0.606703097",
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    ],
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    ],
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      "0.179890709",
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      "0.509119615",
      "0.500000000"
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      "0.868901032",
      "0.500000000"
    ],
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      "1.033515485",
      "0.500000000"
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    [
      "1.228682450",
      "0.500000000"
    ],
    [
      "1.393296903",
      "0.500000000"
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      "0.632572177"
    ],
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      "0.606703097",
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      "0.262197935",
      "0.642560298"
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      "0.642560298"
    ],
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      "0.642560298"
    ],
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      "0.642560298"
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      "0.642560298"
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      "0.642560298"
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      "0.524395871",
      "0.775132475"
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      "0.689010324",
      "0.775132475"
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    [
      "1.048791741",
      "0.775132475"
    ],
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      "1.213406194",
      "0.775132475"
    ],
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      "1.573187612",
      "0.775132475"
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      "1.737802065",
      "0.775132475"
    ],
    [
      "0.344505162",
      "0.785120596"
    ],
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      "0.868901032",
      "0.785120596"
    ],
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      "1.393296903",
      "0.785120596"
    ],
    [
      "1.917692773",
      "0.785120596"
    ],
    [
      "0.082307227",
      "0.917692773"
    ],
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      "0.246921680",
      "0.917692773"
    ],
    [
      "0.442088644",
      "0.917692773"
    ],
    [
      "0.606703097",
      "0.917692773"
    ],
    [
      "0.771317550",
      "0.917692773"
    ],
    [
      "0.966484515",
      "0.917692773"
    ],
    [
      "1.131098968",
      "0.917692773"
    ],
    [
      "1.295713421",
      "0.917692773"
    ],
    [
      "1.490880385",
      "0.917692773"
    ],
    [
      "1.655494838",
      "0.917692773"
    ],
    [
      "1.820109291",
      "0.917692773"
    ]
  ],
  "radius": "0.082307226"
}

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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