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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 = 92

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 = 92
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[0.0533170851755, 0.0533170851765]见来源署名packomania.com ↗
本站纪录0.053317085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 = 0.053317085

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 = 92 的当前纪录构型，0.053317085
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
    0.053317084→0.053317085
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": [
    [
      "0.946682915",
      "0.053317085"
    ],
    [
      "0.053321051",
      "0.053317085"
    ],
    [
      "0.238638421",
      "0.053317086"
    ],
    [
      "0.761982874",
      "0.053317085"
    ],
    [
      "0.470030289",
      "0.053317086"
    ],
    [
      "0.345272591",
      "0.053317086"
    ],
    [
      "0.576664459",
      "0.053317085"
    ],
    [
      "0.669323666",
      "0.106091299"
    ],
    [
      "0.145979736",
      "0.106092218"
    ],
    [
      "0.854332895",
      "0.106630499"
    ],
    [
      "0.407651440",
      "0.139802492"
    ],
    [
      "0.577290252",
      "0.159949419"
    ],
    [
      "0.238013684",
      "0.159949426"
    ],
    [
      "0.761987114",
      "0.159951255"
    ],
    [
      "0.053317086",
      "0.159951255"
    ],
    [
      "0.946682915",
      "0.159951255"
    ],
    [
      "0.484942352",
      "0.213266505"
    ],
    [
      "0.669638154",
      "0.213266505"
    ],
    [
      "0.145665913",
      "0.213266735"
    ],
    [
      "0.330361188",
      "0.213267199"
    ],
    [
      "0.854335014",
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    ],
    [
      "0.577290253",
      "0.266583589"
    ],
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      "0.238013416",
      "0.266584508"
    ],
    [
      "0.761987113",
      "0.266585425"
    ],
    [
      "0.053317085",
      "0.266585425"
    ],
    [
      "0.946682915",
      "0.266585425"
    ],
    [
      "0.407652100",
      "0.286731212"
    ],
    [
      "0.669638154",
      "0.319900676"
    ],
    [
      "0.145665515",
      "0.319901593"
    ],
    [
      "0.854335014",
      "0.319902511"
    ],
    [
      "0.500000001",
      "0.340048296"
    ],
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      "0.315304329",
      "0.340048521"
    ],
    [
      "0.761987114",
      "0.373219597"
    ],
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      "0.053317085",
      "0.373219595"
    ],
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    ],
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      "0.592347901",
      "0.393365382"
    ],
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      "0.222956428",
      "0.393365605"
    ],
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      "0.407652230",
      "0.393365605"
    ],
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      "0.854335015",
      "0.426536681"
    ],
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      "0.500000130",
      "0.446682691"
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      "0.315304328",
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      "0.407652229",
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    ],
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      "0.500000000"
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      "0.777043443",
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      "0.315304458",
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    ],
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    ],
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    ],
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      "0.553317309"
    ],
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      "0.145664985",
      "0.573463319"
    ],
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    ],
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      "0.499999999",
      "0.659951704"
    ],
    [
      "0.145664986",
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    ],
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      "0.854334485",
      "0.680098407"
    ],
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    ],
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      "0.592347899",
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      "0.053317085",
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    ],
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    ],
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    ],
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      "0.733416411"
    ],
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      "0.840048745"
    ],
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      "0.840048745"
    ],
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      "0.761986316",
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    ],
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      "0.592348561",
      "0.860197508"
    ],
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      "0.145667105",
      "0.893369501"
    ],
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      "0.854020264",
      "0.893907783"
    ],
    [
      "0.330676334",
      "0.893908703"
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      "0.761361580",
      "0.946682915"
    ],
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      "0.946682915"
    ],
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      "0.053317085",
      "0.946682915"
    ],
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      "0.529969712",
      "0.946682914"
    ],
    [
      "0.423335542",
      "0.946682915"
    ],
    [
      "0.946678948",
      "0.946682915"
    ],
    [
      "0.238017126",
      "0.946682915"
    ]
  ],
  "radius": "0.053317085"
}
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": 92
}

当前第一名的答案

{
  "centers": [
    [
      "0.946682915",
      "0.053317085"
    ],
    [
      "0.053321051",
      "0.053317085"
    ],
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  "radius": "0.053317085"
}

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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