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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 = 73

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 = 73
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[0.083864941158572615084386635525, 0.083864941158572615084386635527]Eckard Specht1999–2010Packomania ↗
本站纪录0.08386548Joost de Winter纪录详情 ↓

已超越外部纪录 · 审核通过。 审核依据 ↗

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 = 0.083865480

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 = 73 的当前纪录构型，0.08386548
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Joost de Winter
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）
  1. Joost de WinterAI · OpenAI: GPT-6.1 Sol
    0.083864941→0.08386548
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.08386494→0.083864941
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.080945672→0.08386494
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.057142856→0.080945672
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": [
    [
      "0.083865481",
      "0.083865481"
    ],
    [
      "0.251596443",
      "0.083865481"
    ],
    [
      "0.419327405",
      "0.083865481"
    ],
    [
      "0.679354088",
      "0.083865481"
    ],
    [
      "0.847085050",
      "0.083865481"
    ],
    [
      "1.014816012",
      "0.083865481"
    ],
    [
      "1.182546974",
      "0.083865481"
    ],
    [
      "1.350277936",
      "0.083865481"
    ],
    [
      "1.518008898",
      "0.083865481"
    ],
    [
      "1.778035582",
      "0.083865481"
    ],
    [
      "1.916134519",
      "0.179061904"
    ],
    [
      "0.549340747",
      "0.189838149"
    ],
    [
      "1.648022240",
      "0.189838149"
    ],
    [
      "0.167730962",
      "0.229124755"
    ],
    [
      "0.335461924",
      "0.229124755"
    ],
    [
      "0.763219569",
      "0.229124755"
    ],
    [
      "0.930950531",
      "0.229124755"
    ],
    [
      "1.098681493",
      "0.229124755"
    ],
    [
      "1.266412455",
      "0.229124755"
    ],
    [
      "1.434143417",
      "0.229124755"
    ],
    [
      "1.745542917",
      "0.328723707"
    ],
    [
      "0.465475266",
      "0.335097423"
    ],
    [
      "0.633206227",
      "0.335097423"
    ],
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      "1.564156759",
      "0.335097423"
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      "1.916134519",
      "0.346792866"
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      "0.083865481",
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    ],
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      "0.847085050",
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    ],
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      "0.480356697"
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      "0.465475266",
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      "0.633206227",
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    ],
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    ],
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      "1.564156759",
      "0.625615971"
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    ],
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      "1.014816012",
      "0.664902577"
    ],
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      "1.182546974",
      "0.664902577"
    ],
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      "0.765839553"
    ],
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      "0.770875245"
    ],
    [
      "0.549340747",
      "0.770875245"
    ],
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      "0.717071708",
      "0.770875245"
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      "0.884802670",
      "0.770875245"
    ],
    [
      "1.312560316",
      "0.770875245"
    ],
    [
      "1.648022240",
      "0.770875245"
    ],
    [
      "1.098681493",
      "0.810161851"
    ],
    [
      "0.098567385",
      "0.896401908"
    ],
    [
      "0.297744304",
      "0.916134519"
    ],
    [
      "0.465475266",
      "0.916134519"
    ],
    [
      "0.633206227",
      "0.916134519"
    ],
    [
      "0.800937189",
      "0.916134519"
    ],
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      "0.968668151",
      "0.916134519"
    ],
    [
      "1.228694835",
      "0.916134519"
    ],
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      "1.396425797",
      "0.916134519"
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      "1.916134519",
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  ],
  "radius": "0.083865480"
}
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": 73
}

当前第一名的答案

{
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    [
      "0.083865481",
      "0.083865481"
    ],
    [
      "0.251596443",
      "0.083865481"
    ],
    [
      "0.419327405",
      "0.083865481"
    ],
    [
      "0.679354088",
      "0.083865481"
    ],
    [
      "0.847085050",
      "0.083865481"
    ],
    [
      "1.014816012",
      "0.083865481"
    ],
    [
      "1.182546974",
      "0.083865481"
    ],
    [
      "1.350277936",
      "0.083865481"
    ],
    [
      "1.518008898",
      "0.083865481"
    ],
    [
      "1.778035582",
      "0.083865481"
    ],
    [
      "1.916134519",
      "0.179061904"
    ],
    [
      "0.549340747",
      "0.189838149"
    ],
    [
      "1.648022240",
      "0.189838149"
    ],
    [
      "0.167730962",
      "0.229124755"
    ],
    [
      "0.335461924",
      "0.229124755"
    ],
    [
      "0.763219569",
      "0.229124755"
    ],
    [
      "0.930950531",
      "0.229124755"
    ],
    [
      "1.098681493",
      "0.229124755"
    ],
    [
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    ],
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      "0.229124755"
    ],
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      "1.745542917",
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      "0.465475266",
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}

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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