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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 72

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 72
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[0.0598010021265, 0.0598010021275]见来源署名packomania.com ↗
本站纪录0.059801002游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
放大来摆，然后提交 ↗

当前展示：本站纪录

1y0
0x1
已验证构造r = 0.059801002

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 72 的当前纪录构型，0.059801002
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.059801001→0.059801002
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。

当前第一名的答案

{
  "centers": [
    [
      "0.059801002",
      "0.059801002"
    ],
    [
      "0.266957750",
      "0.059801002"
    ],
    [
      "0.474114498",
      "0.059801002"
    ],
    [
      "0.681271246",
      "0.059801002"
    ],
    [
      "0.918907165",
      "0.059801002"
    ],
    [
      "0.800089206",
      "0.073473310"
    ],
    [
      "0.163379376",
      "0.119602004"
    ],
    [
      "0.370536124",
      "0.119602004"
    ],
    [
      "0.577692872",
      "0.119602004"
    ],
    [
      "0.728839660",
      "0.169536527"
    ],
    [
      "0.940198998",
      "0.177492537"
    ],
    [
      "0.059801002",
      "0.179403006"
    ],
    [
      "0.266957750",
      "0.179403006"
    ],
    [
      "0.474114498",
      "0.179403006"
    ],
    [
      "0.625261286",
      "0.229337529"
    ],
    [
      "0.832418034",
      "0.229337529"
    ],
    [
      "0.163379376",
      "0.239204009"
    ],
    [
      "0.370536124",
      "0.239204009"
    ],
    [
      "0.521682912",
      "0.289138531"
    ],
    [
      "0.728839660",
      "0.289138531"
    ],
    [
      "0.940198998",
      "0.297094541"
    ],
    [
      "0.059801002",
      "0.299005011"
    ],
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      "0.266957750",
      "0.299005011"
    ],
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      "0.418104538",
      "0.348939534"
    ],
    [
      "0.625261286",
      "0.348939534"
    ],
    [
      "0.832418034",
      "0.348939534"
    ],
    [
      "0.163379376",
      "0.358806013"
    ],
    [
      "0.314526164",
      "0.408740536"
    ],
    [
      "0.521682912",
      "0.408740536"
    ],
    [
      "0.728839660",
      "0.408740536"
    ],
    [
      "0.940198998",
      "0.416696545"
    ],
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      "0.059801002",
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      "0.521682912",
      "0.647944544"
    ],
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      "0.728839660",
      "0.647944544"
    ],
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      "0.940198998",
      "0.655900554"
    ],
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      "0.059801002",
      "0.657811024"
    ],
    [
      "0.171334784",
      "0.700994990"
    ],
    [
      "0.378491532",
      "0.700994990"
    ],
    [
      "0.625261286",
      "0.707745546"
    ],
    [
      "0.832418034",
      "0.707745546"
    ],
    [
      "0.274913158",
      "0.760795992"
    ],
    [
      "0.482069906",
      "0.760795992"
    ],
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      "0.728839660",
      "0.767546548"
    ],
    [
      "0.940198998",
      "0.775502558"
    ],
    [
      "0.059801002",
      "0.777413028"
    ],
    [
      "0.171334784",
      "0.820596994"
    ],
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      "0.378491532",
      "0.820596994"
    ],
    [
      "0.585648280",
      "0.820596994"
    ],
    [
      "0.832418034",
      "0.827347550"
    ],
    [
      "0.274913158",
      "0.880397996"
    ],
    [
      "0.482069906",
      "0.880397996"
    ],
    [
      "0.689226654",
      "0.880397996"
    ],
    [
      "0.059801002",
      "0.897015032"
    ],
    [
      "0.922591038",
      "0.922591038"
    ],
    [
      "0.171334784",
      "0.940198998"
    ],
    [
      "0.378491532",
      "0.940198998"
    ],
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      "0.585648280",
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    ],
    [
      "0.792805028",
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    ]
  ],
  "radius": "0.059801002"
}
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": 72
}

当前第一名的答案

{
  "centers": [
    [
      "0.059801002",
      "0.059801002"
    ],
    [
      "0.266957750",
      "0.059801002"
    ],
    [
      "0.474114498",
      "0.059801002"
    ],
    [
      "0.681271246",
      "0.059801002"
    ],
    [
      "0.918907165",
      "0.059801002"
    ],
    [
      "0.800089206",
      "0.073473310"
    ],
    [
      "0.163379376",
      "0.119602004"
    ],
    [
      "0.370536124",
      "0.119602004"
    ],
    [
      "0.577692872",
      "0.119602004"
    ],
    [
      "0.728839660",
      "0.169536527"
    ],
    [
      "0.940198998",
      "0.177492537"
    ],
    [
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      "0.179403006"
    ],
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      "0.179403006"
    ],
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    ],
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      "0.625261286",
      "0.229337529"
    ],
    [
      "0.832418034",
      "0.229337529"
    ],
    [
      "0.163379376",
      "0.239204009"
    ],
    [
      "0.370536124",
      "0.239204009"
    ],
    [
      "0.521682912",
      "0.289138531"
    ],
    [
      "0.728839660",
      "0.289138531"
    ],
    [
      "0.940198998",
      "0.297094541"
    ],
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      "0.299005011"
    ],
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      "0.358806013"
    ],
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      "0.314526164",
      "0.408740536"
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      "0.922591038"
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      "0.940198998"
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      "0.940198998"
    ],
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      "0.585648280",
      "0.940198998"
    ],
    [
      "0.792805028",
      "0.940198998"
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  ],
  "radius": "0.059801002"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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