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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 = 97

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 = 97
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[0.0521147955015, 0.0521147955025]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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：起步布局

1y0
0x1
已验证构造r = 0.052114795

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 = 97 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。
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"。

起步布局答案

{
  "centers": [
    [
      "0.052114796",
      "0.052114796"
    ],
    [
      "0.231457146",
      "0.052114796"
    ],
    [
      "0.411988093",
      "0.052114796"
    ],
    [
      "0.592519041",
      "0.052114796"
    ],
    [
      "0.773049988",
      "0.052114796"
    ],
    [
      "0.947576283",
      "0.052453074"
    ],
    [
      "0.32172262",
      "0.104229591"
    ],
    [
      "0.502253567",
      "0.104229591"
    ],
    [
      "0.682784514",
      "0.104229591"
    ],
    [
      "0.141785971",
      "0.105245653"
    ],
    [
      "0.857619731",
      "0.113039062"
    ],
    [
      "0.411988093",
      "0.156344387"
    ],
    [
      "0.592519041",
      "0.156344387"
    ],
    [
      "0.232051444",
      "0.157360448"
    ],
    [
      "0.052114796",
      "0.15837651"
    ],
    [
      "0.767354257",
      "0.165153857"
    ],
    [
      "0.947885204",
      "0.165153857"
    ],
    [
      "0.502253567",
      "0.208459182"
    ],
    [
      "0.322316918",
      "0.209475244"
    ],
    [
      "0.142380269",
      "0.210491305"
    ],
    [
      "0.677088784",
      "0.217268653"
    ],
    [
      "0.857619731",
      "0.217268653"
    ],
    [
      "0.412582392",
      "0.261590039"
    ],
    [
      "0.052114796",
      "0.262606101"
    ],
    [
      "0.232645743",
      "0.262606101"
    ],
    [
      "0.58682331",
      "0.269383448"
    ],
    [
      "0.767354257",
      "0.269383448"
    ],
    [
      "0.947885204",
      "0.269383448"
    ],
    [
      "0.142380269",
      "0.314720896"
    ],
    [
      "0.322911216",
      "0.314720896"
    ],
    [
      "0.677088784",
      "0.321498244"
    ],
    [
      "0.857619731",
      "0.321498244"
    ],
    [
      "0.497152135",
      "0.322514305"
    ],
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      "0.052114796",
      "0.366835692"
    ],
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      "0.232645743",
      "0.366835692"
    ],
    [
      "0.767354257",
      "0.373613039"
    ],
    [
      "0.947885204",
      "0.373613039"
    ],
    [
      "0.587417608",
      "0.374629101"
    ],
    [
      "0.401040789",
      "0.383710588"
    ],
    [
      "0.142380269",
      "0.418950487"
    ],
    [
      "0.857619731",
      "0.425727835"
    ],
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    ],
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    ],
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    ],
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      "0.052114796",
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    ],
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    ],
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    ],
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      "0.220509842",
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    ],
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      "0.401040789",
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    ],
    [
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    ],
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      "0.530973487"
    ],
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    ],
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      "0.310775315",
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    ],
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      "0.586240524",
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    ],
    [
      "0.766771471",
      "0.583081636"
    ],
    [
      "0.947885204",
      "0.584104344"
    ],
    [
      "0.220509842",
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      "0.401040789",
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    ],
    [
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    ],
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    ],
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    ],
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    ],
    [
      "0.310775315",
      "0.644284565"
    ],
    [
      "0.142380269",
      "0.661159461"
    ],
    [
      "0.586240524",
      "0.687311227"
    ],
    [
      "0.766771471",
      "0.687311227"
    ],
    [
      "0.947885204",
      "0.688333935"
    ],
    [
      "0.401040789",
      "0.696399361"
    ],
    [
      "0.052114796",
      "0.713274257"
    ],
    [
      "0.232645743",
      "0.713274257"
    ],
    [
      "0.49597505",
      "0.739426022"
    ],
    [
      "0.676505998",
      "0.739426022"
    ],
    [
      "0.857036945",
      "0.739426022"
    ],
    [
      "0.142380269",
      "0.765389052"
    ],
    [
      "0.322911216",
      "0.765389052"
    ],
    [
      "0.766771471",
      "0.791540818"
    ],
    [
      "0.586240524",
      "0.791540818"
    ],
    [
      "0.947885204",
      "0.792563526"
    ],
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      "0.052114796",
      "0.817503848"
    ],
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      "0.232645743",
      "0.817503848"
    ],
    [
      "0.391745459",
      "0.843655613"
    ],
    [
      "0.49597505",
      "0.843655613"
    ],
    [
      "0.676505998",
      "0.843655613"
    ],
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    ],
    [
      "0.142380269",
      "0.869618643"
    ],
    [
      "0.301479986",
      "0.895770409"
    ],
    [
      "0.586240524",
      "0.895770409"
    ],
    [
      "0.766771471",
      "0.895770409"
    ],
    [
      "0.947885204",
      "0.896793117"
    ],
    [
      "0.052315792",
      "0.92315597"
    ],
    [
      "0.211214512",
      "0.947885204"
    ],
    [
      "0.391745459",
      "0.947885204"
    ],
    [
      "0.49597505",
      "0.947885204"
    ],
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      "0.676505998",
      "0.947885204"
    ],
    [
      "0.857036945",
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    ]
  ],
  "radius": "0.052114795"
}
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": 97
}

起步布局答案

{
  "centers": [
    [
      "0.052114796",
      "0.052114796"
    ],
    [
      "0.231457146",
      "0.052114796"
    ],
    [
      "0.411988093",
      "0.052114796"
    ],
    [
      "0.592519041",
      "0.052114796"
    ],
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    ],
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      "0.052453074"
    ],
    [
      "0.32172262",
      "0.104229591"
    ],
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      "0.104229591"
    ],
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      "0.156344387"
    ],
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    ],
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    ],
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      "0.210491305"
    ],
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      "0.586240524",
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    ],
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      "0.895770409"
    ],
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      "0.586240524",
      "0.895770409"
    ],
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      "0.766771471",
      "0.895770409"
    ],
    [
      "0.947885204",
      "0.896793117"
    ],
    [
      "0.052315792",
      "0.92315597"
    ],
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      "0.211214512",
      "0.947885204"
    ],
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      "0.391745459",
      "0.947885204"
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    [
      "0.49597505",
      "0.947885204"
    ],
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      "0.676505998",
      "0.947885204"
    ],
    [
      "0.857036945",
      "0.947885204"
    ]
  ],
  "radius": "0.052114795"
}

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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