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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 = 111

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 = 111
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[0.0486439363445, 0.0486439363455]见来源署名packomania.com ↗
本站纪录0.048643936Ἀθηνᾶ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 = 0.048643936

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 = 111 的当前纪录构型，0.048643936
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Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.048643935→0.048643936
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

{
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      "0.048644116",
      "0.048643936"
    ],
    [
      "0.217291836",
      "0.048643936"
    ],
    [
      "0.389133646",
      "0.048643936"
    ],
    [
      "0.570890015",
      "0.048643936"
    ],
    [
      "0.757011997",
      "0.048643936"
    ],
    [
      "0.951356064",
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    ],
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      "0.854184030",
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    [
      "0.663951006",
      "0.077009094"
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      "0.083372682"
    ],
    [
      "0.303212741",
      "0.094278662"
    ],
    [
      "0.132967976",
      "0.097166270"
    ],
    [
      "0.906880347",
      "0.135170468"
    ],
    [
      "0.555526899",
      "0.144711126"
    ],
    [
      "0.733321804",
      "0.145219229"
    ],
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      "0.385694026",
      "0.145870986"
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      "0.145931708"
    ],
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      "0.048643936",
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    ],
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      "0.048643936",
      "0.243219682"
    ],
    [
      "0.867102294",
      "0.270340937"
    ],
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      "0.639115739",
      "0.290640655"
    ],
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      "0.470608200",
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    ],
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      "0.301440320",
      "0.291802813"
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    ],
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    ],
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      "0.385694100",
      "0.437734604"
    ],
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      "0.048643936",
      "0.437795427"
    ],
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      "0.867102294",
      "0.464916682"
    ],
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      "0.698594751",
      "0.464916691"
    ],
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      "0.530087175",
      "0.464916756"
    ],
    [
      "0.301440321",
      "0.486378558"
    ],
    [
      "0.132897737",
      "0.486439309"
    ],
    [
      "0.951356064",
      "0.513560619"
    ],
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      "0.782848525",
      "0.513560619"
    ],
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      "0.614340971",
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      "0.782848530",
      "0.805424246"
    ],
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      "0.951356064",
      "0.805424255"
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      "0.614341002",
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      "0.445214236",
      "0.806586403"
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      "0.276671706",
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  ],
  "radius": "0.048643936"
}
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": 111
}

当前第一名的答案

{
  "centers": [
    [
      "0.048644116",
      "0.048643936"
    ],
    [
      "0.217291836",
      "0.048643936"
    ],
    [
      "0.389133646",
      "0.048643936"
    ],
    [
      "0.570890015",
      "0.048643936"
    ],
    [
      "0.757011997",
      "0.048643936"
    ],
    [
      "0.951356064",
      "0.048643936"
    ],
    [
      "0.854184030",
      "0.053390102"
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    [
      "0.663951006",
      "0.077009094"
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      "0.083372682"
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    [
      "0.867102294",
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  ],
  "radius": "0.048643936"
}

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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