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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 = 54

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 = 54
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[0.0686455401045, 0.0686455401055]见来源署名packomania.com ↗
本站纪录0.068645540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 = 0.068645540

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 = 54 的当前纪录构型，0.068645540
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
    0.068645539→0.068645540
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": [
    [
      "0.068645540",
      "0.068645540"
    ],
    [
      "0.308757313",
      "0.068645540"
    ],
    [
      "0.446048394",
      "0.068645540"
    ],
    [
      "0.693559334",
      "0.068645540"
    ],
    [
      "0.931354460",
      "0.068645540"
    ],
    [
      "0.569803864",
      "0.128088154"
    ],
    [
      "0.188701426",
      "0.135244526"
    ],
    [
      "0.812456897",
      "0.137291080"
    ],
    [
      "0.068645540",
      "0.205936620"
    ],
    [
      "0.308757313",
      "0.205936620"
    ],
    [
      "0.446048394",
      "0.205936620"
    ],
    [
      "0.693559334",
      "0.205936620"
    ],
    [
      "0.931354460",
      "0.205936620"
    ],
    [
      "0.569803864",
      "0.265379234"
    ],
    [
      "0.188701426",
      "0.272535606"
    ],
    [
      "0.812456897",
      "0.274582160"
    ],
    [
      "0.068645541",
      "0.343227700"
    ],
    [
      "0.308757314",
      "0.343227700"
    ],
    [
      "0.446048394",
      "0.343227701"
    ],
    [
      "0.693559334",
      "0.343227701"
    ],
    [
      "0.931354460",
      "0.343227701"
    ],
    [
      "0.569803864",
      "0.402670315"
    ],
    [
      "0.188701427",
      "0.409826686"
    ],
    [
      "0.812456897",
      "0.411873241"
    ],
    [
      "0.068645541",
      "0.480518780"
    ],
    [
      "0.308757314",
      "0.480518780"
    ],
    [
      "0.446048394",
      "0.480518781"
    ],
    [
      "0.693559334",
      "0.480518781"
    ],
    [
      "0.931354460",
      "0.480518781"
    ],
    [
      "0.569803864",
      "0.539961395"
    ],
    [
      "0.188701427",
      "0.547117766"
    ],
    [
      "0.812456897",
      "0.549164321"
    ],
    [
      "0.068645541",
      "0.617809860"
    ],
    [
      "0.308757314",
      "0.617809861"
    ],
    [
      "0.446048394",
      "0.617809861"
    ],
    [
      "0.931354460",
      "0.617809862"
    ],
    [
      "0.694215673",
      "0.618934326"
    ],
    [
      "0.569926329",
      "0.677252421"
    ],
    [
      "0.188701427",
      "0.684408846"
    ],
    [
      "0.806589532",
      "0.697807368"
    ],
    [
      "0.456563740",
      "0.754697655"
    ],
    [
      "0.068645540",
      "0.755100940"
    ],
    [
      "0.931354460",
      "0.755100942"
    ],
    [
      "0.317105709",
      "0.760538094"
    ],
    [
      "0.593771461",
      "0.812456897"
    ],
    [
      "0.731062541",
      "0.812456897"
    ],
    [
      "0.188701426",
      "0.821699926"
    ],
    [
      "0.398219617",
      "0.878974781"
    ],
    [
      "0.931354460",
      "0.892392022"
    ],
    [
      "0.077272629",
      "0.922727371"
    ],
    [
      "0.271313313",
      "0.931354460"
    ],
    [
      "0.525125921",
      "0.931354460"
    ],
    [
      "0.662417001",
      "0.931354460"
    ],
    [
      "0.799708081",
      "0.931354460"
    ]
  ],
  "radius": "0.068645540"
}
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": 54
}

当前第一名的答案

{
  "centers": [
    [
      "0.068645540",
      "0.068645540"
    ],
    [
      "0.308757313",
      "0.068645540"
    ],
    [
      "0.446048394",
      "0.068645540"
    ],
    [
      "0.693559334",
      "0.068645540"
    ],
    [
      "0.931354460",
      "0.068645540"
    ],
    [
      "0.569803864",
      "0.128088154"
    ],
    [
      "0.188701426",
      "0.135244526"
    ],
    [
      "0.812456897",
      "0.137291080"
    ],
    [
      "0.068645540",
      "0.205936620"
    ],
    [
      "0.308757313",
      "0.205936620"
    ],
    [
      "0.446048394",
      "0.205936620"
    ],
    [
      "0.693559334",
      "0.205936620"
    ],
    [
      "0.931354460",
      "0.205936620"
    ],
    [
      "0.569803864",
      "0.265379234"
    ],
    [
      "0.188701426",
      "0.272535606"
    ],
    [
      "0.812456897",
      "0.274582160"
    ],
    [
      "0.068645541",
      "0.343227700"
    ],
    [
      "0.308757314",
      "0.343227700"
    ],
    [
      "0.446048394",
      "0.343227701"
    ],
    [
      "0.693559334",
      "0.343227701"
    ],
    [
      "0.931354460",
      "0.343227701"
    ],
    [
      "0.569803864",
      "0.402670315"
    ],
    [
      "0.188701427",
      "0.409826686"
    ],
    [
      "0.812456897",
      "0.411873241"
    ],
    [
      "0.068645541",
      "0.480518780"
    ],
    [
      "0.308757314",
      "0.480518780"
    ],
    [
      "0.446048394",
      "0.480518781"
    ],
    [
      "0.693559334",
      "0.480518781"
    ],
    [
      "0.931354460",
      "0.480518781"
    ],
    [
      "0.569803864",
      "0.539961395"
    ],
    [
      "0.188701427",
      "0.547117766"
    ],
    [
      "0.812456897",
      "0.549164321"
    ],
    [
      "0.068645541",
      "0.617809860"
    ],
    [
      "0.308757314",
      "0.617809861"
    ],
    [
      "0.446048394",
      "0.617809861"
    ],
    [
      "0.931354460",
      "0.617809862"
    ],
    [
      "0.694215673",
      "0.618934326"
    ],
    [
      "0.569926329",
      "0.677252421"
    ],
    [
      "0.188701427",
      "0.684408846"
    ],
    [
      "0.806589532",
      "0.697807368"
    ],
    [
      "0.456563740",
      "0.754697655"
    ],
    [
      "0.068645540",
      "0.755100940"
    ],
    [
      "0.931354460",
      "0.755100942"
    ],
    [
      "0.317105709",
      "0.760538094"
    ],
    [
      "0.593771461",
      "0.812456897"
    ],
    [
      "0.731062541",
      "0.812456897"
    ],
    [
      "0.188701426",
      "0.821699926"
    ],
    [
      "0.398219617",
      "0.878974781"
    ],
    [
      "0.931354460",
      "0.892392022"
    ],
    [
      "0.077272629",
      "0.922727371"
    ],
    [
      "0.271313313",
      "0.931354460"
    ],
    [
      "0.525125921",
      "0.931354460"
    ],
    [
      "0.662417001",
      "0.931354460"
    ],
    [
      "0.799708081",
      "0.931354460"
    ]
  ],
  "radius": "0.068645540"
}

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

DISCUSSION

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