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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 = 94

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 = 94
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[0.0527953627265, 0.0527953627275]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 = 0.052795362

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 = 94 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": [
    [
      "0.052795363",
      "0.052795363"
    ],
    [
      "0.235683864",
      "0.052795363"
    ],
    [
      "0.418572365",
      "0.052795363"
    ],
    [
      "0.601460867",
      "0.052795363"
    ],
    [
      "0.74757673",
      "0.052795363"
    ],
    [
      "0.947204637",
      "0.052795363"
    ],
    [
      "0.847390683",
      "0.087242074"
    ],
    [
      "0.144239613",
      "0.105590725"
    ],
    [
      "0.327128115",
      "0.105590725"
    ],
    [
      "0.510016616",
      "0.105590725"
    ],
    [
      "0.674518798",
      "0.129031443"
    ],
    [
      "0.052795363",
      "0.158386088"
    ],
    [
      "0.235683864",
      "0.158386088"
    ],
    [
      "0.418572365",
      "0.158386088"
    ],
    [
      "0.947204637",
      "0.158386088"
    ],
    [
      "0.774332752",
      "0.163478155"
    ],
    [
      "0.583074548",
      "0.181826806"
    ],
    [
      "0.144239613",
      "0.211181451"
    ],
    [
      "0.327128115",
      "0.211181451"
    ],
    [
      "0.862552789",
      "0.221500792"
    ],
    [
      "0.491630297",
      "0.234622168"
    ],
    [
      "0.674518798",
      "0.234622168"
    ],
    [
      "0.052795363",
      "0.263976814"
    ],
    [
      "0.235683864",
      "0.263976814"
    ],
    [
      "0.771850382",
      "0.275560709"
    ],
    [
      "0.947204637",
      "0.284615495"
    ],
    [
      "0.400186046",
      "0.287417531"
    ],
    [
      "0.583074548",
      "0.287417531"
    ],
    [
      "0.144239613",
      "0.316772176"
    ],
    [
      "0.85650223",
      "0.338675413"
    ],
    [
      "0.308741796",
      "0.340212894"
    ],
    [
      "0.491630297",
      "0.340212894"
    ],
    [
      "0.674518798",
      "0.340212894"
    ],
    [
      "0.052795363",
      "0.369567539"
    ],
    [
      "0.947204637",
      "0.392735331"
    ],
    [
      "0.217297545",
      "0.393008257"
    ],
    [
      "0.400186046",
      "0.393008257"
    ],
    [
      "0.583074548",
      "0.393008257"
    ],
    [
      "0.765963049",
      "0.393008257"
    ],
    [
      "0.125853294",
      "0.445803619"
    ],
    [
      "0.308741796",
      "0.445803619"
    ],
    [
      "0.491630297",
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    ],
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    ],
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    ],
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      "0.498598982"
    ],
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      "0.498598982"
    ],
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      "0.583074548",
      "0.498598982"
    ],
    [
      "0.765221205",
      "0.499863537"
    ],
    [
      "0.947204637",
      "0.501401018"
    ],
    [
      "0.052795363",
      "0.5220397"
    ],
    [
      "0.308741796",
      "0.551394345"
    ],
    [
      "0.491630297",
      "0.551394345"
    ],
    [
      "0.673776954",
      "0.5526589"
    ],
    [
      "0.855760387",
      "0.554196381"
    ],
    [
      "0.144239613",
      "0.574835062"
    ],
    [
      "0.400186046",
      "0.604189708"
    ],
    [
      "0.582332704",
      "0.605454263"
    ],
    [
      "0.764316136",
      "0.606991743"
    ],
    [
      "0.947204637",
      "0.606991743"
    ],
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      "0.052795363",
      "0.627630425"
    ],
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      "0.235683864",
      "0.627630425"
    ],
    [
      "0.490888453",
      "0.658249625"
    ],
    [
      "0.672871885",
      "0.659787106"
    ],
    [
      "0.855760387",
      "0.659787106"
    ],
    [
      "0.144239613",
      "0.680425788"
    ],
    [
      "0.327128115",
      "0.680425788"
    ],
    [
      "0.581427635",
      "0.712582469"
    ],
    [
      "0.764316136",
      "0.712582469"
    ],
    [
      "0.947204637",
      "0.712582469"
    ],
    [
      "0.052795363",
      "0.733221151"
    ],
    [
      "0.235683864",
      "0.733221151"
    ],
    [
      "0.417830521",
      "0.734485706"
    ],
    [
      "0.672871885",
      "0.765377832"
    ],
    [
      "0.855760387",
      "0.765377832"
    ],
    [
      "0.144239613",
      "0.786016513"
    ],
    [
      "0.326386271",
      "0.787281068"
    ],
    [
      "0.508369703",
      "0.788818549"
    ],
    [
      "0.764316136",
      "0.818173194"
    ],
    [
      "0.947204637",
      "0.818173194"
    ],
    [
      "0.052795363",
      "0.838811876"
    ],
    [
      "0.23494202",
      "0.840076431"
    ],
    [
      "0.416925452",
      "0.841613912"
    ],
    [
      "0.599813954",
      "0.841613912"
    ],
    [
      "0.855760387",
      "0.870968557"
    ],
    [
      "0.14349777",
      "0.892871794"
    ],
    [
      "0.325481202",
      "0.894409275"
    ],
    [
      "0.508369703",
      "0.894409275"
    ],
    [
      "0.691258204",
      "0.894409275"
    ],
    [
      "0.946972674",
      "0.946972673"
    ],
    [
      "0.052798275",
      "0.947201646"
    ],
    [
      "0.234036951",
      "0.947204637"
    ],
    [
      "0.416925452",
      "0.947204637"
    ],
    [
      "0.599813954",
      "0.947204637"
    ],
    [
      "0.782702455",
      "0.947204637"
    ]
  ],
  "radius": "0.052795362"
}
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": 94
}

起步布局答案

{
  "centers": [
    [
      "0.052795363",
      "0.052795363"
    ],
    [
      "0.235683864",
      "0.052795363"
    ],
    [
      "0.418572365",
      "0.052795363"
    ],
    [
      "0.601460867",
      "0.052795363"
    ],
    [
      "0.74757673",
      "0.052795363"
    ],
    [
      "0.947204637",
      "0.052795363"
    ],
    [
      "0.847390683",
      "0.087242074"
    ],
    [
      "0.144239613",
      "0.105590725"
    ],
    [
      "0.327128115",
      "0.105590725"
    ],
    [
      "0.510016616",
      "0.105590725"
    ],
    [
      "0.674518798",
      "0.129031443"
    ],
    [
      "0.052795363",
      "0.158386088"
    ],
    [
      "0.235683864",
      "0.158386088"
    ],
    [
      "0.418572365",
      "0.158386088"
    ],
    [
      "0.947204637",
      "0.158386088"
    ],
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      "0.774332752",
      "0.163478155"
    ],
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    ],
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      "0.144239613",
      "0.211181451"
    ],
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      "0.327128115",
      "0.211181451"
    ],
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      "0.862552789",
      "0.221500792"
    ],
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      "0.491630297",
      "0.234622168"
    ],
    [
      "0.674518798",
      "0.234622168"
    ],
    [
      "0.052795363",
      "0.263976814"
    ],
    [
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      "0.263976814"
    ],
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      "0.275560709"
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      "0.284615495"
    ],
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      "0.400186046",
      "0.287417531"
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      "0.583074548",
      "0.287417531"
    ],
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      "0.316772176"
    ],
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      "0.85650223",
      "0.338675413"
    ],
    [
      "0.308741796",
      "0.340212894"
    ],
    [
      "0.491630297",
      "0.340212894"
    ],
    [
      "0.674518798",
      "0.340212894"
    ],
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      "0.052795363",
      "0.369567539"
    ],
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      "0.947204637",
      "0.392735331"
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      "0.217297545",
      "0.393008257"
    ],
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      "0.393008257"
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    ],
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    ],
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      "0.445803619"
    ],
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    ],
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      "0.498598982"
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      "0.499863537"
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      "0.947204637",
      "0.501401018"
    ],
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      "0.052795363",
      "0.5220397"
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      "0.308741796",
      "0.551394345"
    ],
    [
      "0.491630297",
      "0.551394345"
    ],
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      "0.673776954",
      "0.5526589"
    ],
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      "0.855760387",
      "0.554196381"
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    ],
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      "0.582332704",
      "0.605454263"
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      "0.606991743"
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      "0.627630425"
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      "0.627630425"
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      "0.490888453",
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      "0.855760387",
      "0.659787106"
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      "0.144239613",
      "0.680425788"
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      "0.327128115",
      "0.680425788"
    ],
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      "0.764316136",
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      "0.947204637",
      "0.712582469"
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    [
      "0.052795363",
      "0.733221151"
    ],
    [
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      "0.733221151"
    ],
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      "0.417830521",
      "0.734485706"
    ],
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      "0.672871885",
      "0.765377832"
    ],
    [
      "0.855760387",
      "0.765377832"
    ],
    [
      "0.144239613",
      "0.786016513"
    ],
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      "0.326386271",
      "0.787281068"
    ],
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      "0.788818549"
    ],
    [
      "0.764316136",
      "0.818173194"
    ],
    [
      "0.947204637",
      "0.818173194"
    ],
    [
      "0.052795363",
      "0.838811876"
    ],
    [
      "0.23494202",
      "0.840076431"
    ],
    [
      "0.416925452",
      "0.841613912"
    ],
    [
      "0.599813954",
      "0.841613912"
    ],
    [
      "0.855760387",
      "0.870968557"
    ],
    [
      "0.14349777",
      "0.892871794"
    ],
    [
      "0.325481202",
      "0.894409275"
    ],
    [
      "0.508369703",
      "0.894409275"
    ],
    [
      "0.691258204",
      "0.894409275"
    ],
    [
      "0.946972674",
      "0.946972673"
    ],
    [
      "0.052798275",
      "0.947201646"
    ],
    [
      "0.234036951",
      "0.947204637"
    ],
    [
      "0.416925452",
      "0.947204637"
    ],
    [
      "0.599813954",
      "0.947204637"
    ],
    [
      "0.782702455",
      "0.947204637"
    ]
  ],
  "radius": "0.052795362"
}

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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