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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 = 68

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 = 68
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[0.086402531473385728267242717167, 0.086402531473385728267242717169]Eckard Specht2010-08-10Packomania ↗
本站纪录0.086402531NUE_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 = 0.086402531

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 = 68 的当前纪录构型，0.086402531
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
    0.08333248→0.086402531
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.066666666→0.08333248
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": [
    [
      "0.086402531",
      "0.086402531"
    ],
    [
      "0.259207594",
      "0.086402531"
    ],
    [
      "0.432012657",
      "0.086402531"
    ],
    [
      "0.617606558",
      "0.086402531"
    ],
    [
      "0.790411621",
      "0.086402531"
    ],
    [
      "0.963216684",
      "0.086402531"
    ],
    [
      "1.136021747",
      "0.086402531"
    ],
    [
      "1.308826809",
      "0.086402531"
    ],
    [
      "1.481631872",
      "0.086402531"
    ],
    [
      "1.654436935",
      "0.086402531"
    ],
    [
      "1.828138570",
      "0.086523420"
    ],
    [
      "0.524809608",
      "0.232177412"
    ],
    [
      "0.876814152",
      "0.236056106"
    ],
    [
      "1.049619215",
      "0.236056106"
    ],
    [
      "1.222424278",
      "0.236056106"
    ],
    [
      "1.395229341",
      "0.236056106"
    ],
    [
      "1.568034404",
      "0.236056106"
    ],
    [
      "1.740839467",
      "0.236056106"
    ],
    [
      "0.170850599",
      "0.237167626"
    ],
    [
      "0.343655662",
      "0.237167626"
    ],
    [
      "0.697452832",
      "0.239654488"
    ],
    [
      "1.913597469",
      "0.240088801"
    ],
    [
      "0.438284429",
      "0.381760109"
    ],
    [
      "0.790096113",
      "0.385527078"
    ],
    [
      "1.136021747",
      "0.385709680"
    ],
    [
      "1.308826809",
      "0.385709680"
    ],
    [
      "1.481631872",
      "0.385709680"
    ],
    [
      "1.654436935",
      "0.385709680"
    ],
    [
      "0.962900791",
      "0.385891617"
    ],
    [
      "0.086402531",
      "0.387932720"
    ],
    [
      "0.612619656",
      "0.391148413"
    ],
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      "0.262480695",
      "0.393098521"
    ],
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      "1.858987704",
      "0.404038072"
    ],
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      "1.744308353",
      "0.533306151"
    ],
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      "1.049618447",
      "0.535362811"
    ],
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      "1.395229341",
      "0.535363255"
    ],
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      "0.709925987",
      "0.538609868"
    ],
    [
      "0.369348018",
      "0.540219445"
    ],
    [
      "0.086402531",
      "0.560737783"
    ],
    [
      "1.567437327",
      "0.565351764"
    ],
    [
      "1.220955413",
      "0.565617218"
    ],
    [
      "1.913597469",
      "0.567987343"
    ],
    [
      "0.539775805",
      "0.568784448"
    ],
    [
      "0.880076169",
      "0.568784448"
    ],
    [
      "0.235013992",
      "0.648920682"
    ],
    [
      "1.744308353",
      "0.706111214"
    ],
    [
      "1.066197115",
      "0.707370769"
    ],
    [
      "1.405379315",
      "0.707869973"
    ],
    [
      "0.726504655",
      "0.710617826"
    ],
    [
      "0.086402531",
      "0.737103582"
    ],
    [
      "0.383587430",
      "0.737167626"
    ],
    [
      "0.556354472",
      "0.740792406"
    ],
    [
      "0.896654837",
      "0.740792406"
    ],
    [
      "1.235739393",
      "0.740792406"
    ],
    [
      "1.575019237",
      "0.740792406"
    ],
    [
      "1.913597469",
      "0.740792406"
    ],
    [
      "0.234975970",
      "0.825350525"
    ],
    [
      "1.744308353",
      "0.878916277"
    ],
    [
      "1.066197115",
      "0.880175832"
    ],
    [
      "1.405379315",
      "0.880675036"
    ],
    [
      "0.726504655",
      "0.883422889"
    ],
    [
      "0.086402531",
      "0.913597469"
    ],
    [
      "0.383549409",
      "0.913597469"
    ],
    [
      "0.556354472",
      "0.913597469"
    ],
    [
      "0.896654837",
      "0.913597469"
    ],
    [
      "1.235739393",
      "0.913597469"
    ],
    [
      "1.575019237",
      "0.913597469"
    ],
    [
      "1.913597469",
      "0.913597469"
    ]
  ],
  "radius": "0.086402531"
}
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": 68
}

当前第一名的答案

{
  "centers": [
    [
      "0.086402531",
      "0.086402531"
    ],
    [
      "0.259207594",
      "0.086402531"
    ],
    [
      "0.432012657",
      "0.086402531"
    ],
    [
      "0.617606558",
      "0.086402531"
    ],
    [
      "0.790411621",
      "0.086402531"
    ],
    [
      "0.963216684",
      "0.086402531"
    ],
    [
      "1.136021747",
      "0.086402531"
    ],
    [
      "1.308826809",
      "0.086402531"
    ],
    [
      "1.481631872",
      "0.086402531"
    ],
    [
      "1.654436935",
      "0.086402531"
    ],
    [
      "1.828138570",
      "0.086523420"
    ],
    [
      "0.524809608",
      "0.232177412"
    ],
    [
      "0.876814152",
      "0.236056106"
    ],
    [
      "1.049619215",
      "0.236056106"
    ],
    [
      "1.222424278",
      "0.236056106"
    ],
    [
      "1.395229341",
      "0.236056106"
    ],
    [
      "1.568034404",
      "0.236056106"
    ],
    [
      "1.740839467",
      "0.236056106"
    ],
    [
      "0.170850599",
      "0.237167626"
    ],
    [
      "0.343655662",
      "0.237167626"
    ],
    [
      "0.697452832",
      "0.239654488"
    ],
    [
      "1.913597469",
      "0.240088801"
    ],
    [
      "0.438284429",
      "0.381760109"
    ],
    [
      "0.790096113",
      "0.385527078"
    ],
    [
      "1.136021747",
      "0.385709680"
    ],
    [
      "1.308826809",
      "0.385709680"
    ],
    [
      "1.481631872",
      "0.385709680"
    ],
    [
      "1.654436935",
      "0.385709680"
    ],
    [
      "0.962900791",
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      "0.086402531",
      "0.387932720"
    ],
    [
      "0.612619656",
      "0.391148413"
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      "0.533306151"
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      "0.535362811"
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      "0.896654837",
      "0.740792406"
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      "0.740792406"
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      "1.575019237",
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      "1.913597469",
      "0.740792406"
    ],
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      "0.234975970",
      "0.825350525"
    ],
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      "1.744308353",
      "0.878916277"
    ],
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      "1.066197115",
      "0.880175832"
    ],
    [
      "1.405379315",
      "0.880675036"
    ],
    [
      "0.726504655",
      "0.883422889"
    ],
    [
      "0.086402531",
      "0.913597469"
    ],
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      "0.383549409",
      "0.913597469"
    ],
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      "0.556354472",
      "0.913597469"
    ],
    [
      "0.896654837",
      "0.913597469"
    ],
    [
      "1.235739393",
      "0.913597469"
    ],
    [
      "1.575019237",
      "0.913597469"
    ],
    [
      "1.913597469",
      "0.913597469"
    ]
  ],
  "radius": "0.086402531"
}

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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