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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 = 118

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 = 118
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[0.0475723626795, 0.0475723626805]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 = 0.047572362

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 = 118 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": [
    [
      "0.047572363",
      "0.047572363"
    ],
    [
      "0.212367861",
      "0.047572363"
    ],
    [
      "0.376731442",
      "0.047572363"
    ],
    [
      "0.540644094",
      "0.047572363"
    ],
    [
      "0.705439324",
      "0.047572363"
    ],
    [
      "0.870234822",
      "0.047572363"
    ],
    [
      "0.129970112",
      "0.095144725"
    ],
    [
      "0.787837073",
      "0.095144725"
    ],
    [
      "0.623041709",
      "0.095144958"
    ],
    [
      "0.952427637",
      "0.095497929"
    ],
    [
      "0.294549652",
      "0.095516831"
    ],
    [
      "0.458687768",
      "0.09590123"
    ],
    [
      "0.047572363",
      "0.142717088"
    ],
    [
      "0.705439458",
      "0.14271732"
    ],
    [
      "0.870029888",
      "0.143070291"
    ],
    [
      "0.212151902",
      "0.143089194"
    ],
    [
      "0.541085383",
      "0.143473825"
    ],
    [
      "0.376505977",
      "0.143845699"
    ],
    [
      "0.952427637",
      "0.190642654"
    ],
    [
      "0.787632273",
      "0.190642887"
    ],
    [
      "0.129754153",
      "0.190661557"
    ],
    [
      "0.623483132",
      "0.191046188"
    ],
    [
      "0.294108228",
      "0.191418061"
    ],
    [
      "0.458903592",
      "0.191418294"
    ],
    [
      "0.870030022",
      "0.238215249"
    ],
    [
      "0.047572363",
      "0.238606025"
    ],
    [
      "0.705675947",
      "0.238971754"
    ],
    [
      "0.211710479",
      "0.238990424"
    ],
    [
      "0.376505843",
      "0.238990657"
    ],
    [
      "0.541301342",
      "0.238990657"
    ],
    [
      "0.952427637",
      "0.285787844"
    ],
    [
      "0.788073697",
      "0.286544117"
    ],
    [
      "0.294108094",
      "0.286563019"
    ],
    [
      "0.458903592",
      "0.286563019"
    ],
    [
      "0.623494157",
      "0.286916223"
    ],
    [
      "0.129528688",
      "0.286934893"
    ],
    [
      "0.870471312",
      "0.334116712"
    ],
    [
      "0.376505843",
      "0.334135382"
    ],
    [
      "0.541096408",
      "0.334488585"
    ],
    [
      "0.705891906",
      "0.334488585"
    ],
    [
      "0.211926303",
      "0.334507488"
    ],
    [
      "0.047572363",
      "0.33526376"
    ],
    [
      "0.458698658",
      "0.382060948"
    ],
    [
      "0.623494157",
      "0.382060948"
    ],
    [
      "0.788289521",
      "0.38206118"
    ],
    [
      "0.294324053",
      "0.382079851"
    ],
    [
      "0.952427637",
      "0.382445579"
    ],
    [
      "0.129969978",
      "0.382836355"
    ],
    [
      "0.541096408",
      "0.429633311"
    ],
    [
      "0.705891772",
      "0.429633543"
    ],
    [
      "0.376516868",
      "0.430005417"
    ],
    [
      "0.870245847",
      "0.430390048"
    ],
    [
      "0.212367727",
      "0.430408718"
    ],
    [
      "0.047572363",
      "0.43040895"
    ],
    [
      "0.623494023",
      "0.477205906"
    ],
    [
      "0.458914617",
      "0.477577779"
    ],
    [
      "0.787848098",
      "0.477962411"
    ],
    [
      "0.129970112",
      "0.477981313"
    ],
    [
      "0.294560542",
      "0.478334284"
    ],
    [
      "0.952427637",
      "0.478334517"
    ],
    [
      "0.541312232",
      "0.525150374"
    ],
    [
      "0.705450348",
      "0.525534773"
    ],
    [
      "0.047572363",
      "0.525553676"
    ],
    [
      "0.376958291",
      "0.525906647"
    ],
    [
      "0.212162927",
      "0.525906879"
    ],
    [
      "0.870029888",
      "0.525906879"
    ],
    [
      "0.129765178",
      "0.573479242"
    ],
    [
      "0.459355906",
      "0.573479242"
    ],
    [
      "0.623268558",
      "0.573479242"
    ],
    [
      "0.787632139",
      "0.573479242"
    ],
    [
      "0.952427637",
      "0.573479242"
    ],
    [
      "0.294560676",
      "0.573479242"
    ],
    [
      "0.870029888",
      "0.621051605"
    ],
    [
      "0.212162927",
      "0.621051605"
    ],
    [
      "0.376958291",
      "0.621051837"
    ],
    [
      "0.047572363",
      "0.621404808"
    ],
    [
      "0.705450348",
      "0.621423711"
    ],
    [
      "0.541312232",
      "0.621808109"
    ],
    [
      "0.952427637",
      "0.668623967"
    ],
    [
      "0.294560542",
      "0.6686242"
    ],
    [
      "0.129970112",
      "0.668977171"
    ],
    [
      "0.787848098",
      "0.668996073"
    ],
    [
      "0.458914617",
      "0.669380704"
    ],
    [
      "0.623494023",
      "0.669752578"
    ],
    [
      "0.047572363",
      "0.716549533"
    ],
    [
      "0.212367727",
      "0.716549766"
    ],
    [
      "0.870245847",
      "0.716568436"
    ],
    [
      "0.376516868",
      "0.716953067"
    ],
    [
      "0.705891772",
      "0.717324941"
    ],
    [
      "0.541096408",
      "0.717325173"
    ],
    [
      "0.130044398",
      "0.764250796"
    ],
    [
      "0.952427637",
      "0.764512905"
    ],
    [
      "0.294324053",
      "0.764878633"
    ],
    [
      "0.788289521",
      "0.764897303"
    ],
    [
      "0.458698658",
      "0.764897536"
    ],
    [
      "0.623494157",
      "0.764897536"
    ],
    [
      "0.047572363",
      "0.811694259"
    ],
    [
      "0.541096408",
      "0.812469898"
    ],
    [
      "0.705891906",
      "0.812469898"
    ],
    [
      "0.376505843",
      "0.812823102"
    ],
    [
      "0.215969643",
      "0.818851823"
    ],
    [
      "0.86610702",
      "0.819641761"
    ],
    [
      "0.129895624",
      "0.859395405"
    ],
    [
      "0.952427637",
      "0.85965763"
    ],
    [
      "0.623494157",
      "0.860042261"
    ],
    [
      "0.458903592",
      "0.860395465"
    ],
    [
      "0.298151433",
      "0.866796292"
    ],
    [
      "0.783709405",
      "0.867214356"
    ],
    [
      "0.531693725",
      "0.922000267"
    ],
    [
      "0.050827208",
      "0.929589774"
    ],
    [
      "0.244727697",
      "0.945526357"
    ],
    [
      "0.836344895",
      "0.946473576"
    ],
    [
      "0.149833592",
      "0.952427637"
    ],
    [
      "0.339621801",
      "0.952427637"
    ],
    [
      "0.434766527",
      "0.952427637"
    ],
    [
      "0.646241926",
      "0.952427637"
    ],
    [
      "0.741386652",
      "0.952427637"
    ],
    [
      "0.931303138",
      "0.952427637"
    ]
  ],
  "radius": "0.047572362"
}
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": 118
}

起步布局答案

{
  "centers": [
    [
      "0.047572363",
      "0.047572363"
    ],
    [
      "0.212367861",
      "0.047572363"
    ],
    [
      "0.376731442",
      "0.047572363"
    ],
    [
      "0.540644094",
      "0.047572363"
    ],
    [
      "0.705439324",
      "0.047572363"
    ],
    [
      "0.870234822",
      "0.047572363"
    ],
    [
      "0.129970112",
      "0.095144725"
    ],
    [
      "0.787837073",
      "0.095144725"
    ],
    [
      "0.623041709",
      "0.095144958"
    ],
    [
      "0.952427637",
      "0.095497929"
    ],
    [
      "0.294549652",
      "0.095516831"
    ],
    [
      "0.458687768",
      "0.09590123"
    ],
    [
      "0.047572363",
      "0.142717088"
    ],
    [
      "0.705439458",
      "0.14271732"
    ],
    [
      "0.870029888",
      "0.143070291"
    ],
    [
      "0.212151902",
      "0.143089194"
    ],
    [
      "0.541085383",
      "0.143473825"
    ],
    [
      "0.376505977",
      "0.143845699"
    ],
    [
      "0.952427637",
      "0.190642654"
    ],
    [
      "0.787632273",
      "0.190642887"
    ],
    [
      "0.129754153",
      "0.190661557"
    ],
    [
      "0.623483132",
      "0.191046188"
    ],
    [
      "0.294108228",
      "0.191418061"
    ],
    [
      "0.458903592",
      "0.191418294"
    ],
    [
      "0.870030022",
      "0.238215249"
    ],
    [
      "0.047572363",
      "0.238606025"
    ],
    [
      "0.705675947",
      "0.238971754"
    ],
    [
      "0.211710479",
      "0.238990424"
    ],
    [
      "0.376505843",
      "0.238990657"
    ],
    [
      "0.541301342",
      "0.238990657"
    ],
    [
      "0.952427637",
      "0.285787844"
    ],
    [
      "0.788073697",
      "0.286544117"
    ],
    [
      "0.294108094",
      "0.286563019"
    ],
    [
      "0.458903592",
      "0.286563019"
    ],
    [
      "0.623494157",
      "0.286916223"
    ],
    [
      "0.129528688",
      "0.286934893"
    ],
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      "0.870471312",
      "0.334116712"
    ],
    [
      "0.376505843",
      "0.334135382"
    ],
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      "0.334488585"
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      "0.334488585"
    ],
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      "0.334507488"
    ],
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      "0.047572363",
      "0.33526376"
    ],
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      "0.382060948"
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      "0.382060948"
    ],
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      "0.429633311"
    ],
    [
      "0.705891772",
      "0.429633543"
    ],
    [
      "0.376516868",
      "0.430005417"
    ],
    [
      "0.870245847",
      "0.430390048"
    ],
    [
      "0.212367727",
      "0.430408718"
    ],
    [
      "0.047572363",
      "0.43040895"
    ],
    [
      "0.623494023",
      "0.477205906"
    ],
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      "0.477577779"
    ],
    [
      "0.787848098",
      "0.477962411"
    ],
    [
      "0.129970112",
      "0.477981313"
    ],
    [
      "0.294560542",
      "0.478334284"
    ],
    [
      "0.952427637",
      "0.478334517"
    ],
    [
      "0.541312232",
      "0.525150374"
    ],
    [
      "0.705450348",
      "0.525534773"
    ],
    [
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      "0.525553676"
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      "0.668623967"
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      "0.6686242"
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    [
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      "0.716953067"
    ],
    [
      "0.705891772",
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    [
      "0.541096408",
      "0.717325173"
    ],
    [
      "0.130044398",
      "0.764250796"
    ],
    [
      "0.952427637",
      "0.764512905"
    ],
    [
      "0.294324053",
      "0.764878633"
    ],
    [
      "0.788289521",
      "0.764897303"
    ],
    [
      "0.458698658",
      "0.764897536"
    ],
    [
      "0.623494157",
      "0.764897536"
    ],
    [
      "0.047572363",
      "0.811694259"
    ],
    [
      "0.541096408",
      "0.812469898"
    ],
    [
      "0.705891906",
      "0.812469898"
    ],
    [
      "0.376505843",
      "0.812823102"
    ],
    [
      "0.215969643",
      "0.818851823"
    ],
    [
      "0.86610702",
      "0.819641761"
    ],
    [
      "0.129895624",
      "0.859395405"
    ],
    [
      "0.952427637",
      "0.85965763"
    ],
    [
      "0.623494157",
      "0.860042261"
    ],
    [
      "0.458903592",
      "0.860395465"
    ],
    [
      "0.298151433",
      "0.866796292"
    ],
    [
      "0.783709405",
      "0.867214356"
    ],
    [
      "0.531693725",
      "0.922000267"
    ],
    [
      "0.050827208",
      "0.929589774"
    ],
    [
      "0.244727697",
      "0.945526357"
    ],
    [
      "0.836344895",
      "0.946473576"
    ],
    [
      "0.149833592",
      "0.952427637"
    ],
    [
      "0.339621801",
      "0.952427637"
    ],
    [
      "0.434766527",
      "0.952427637"
    ],
    [
      "0.646241926",
      "0.952427637"
    ],
    [
      "0.741386652",
      "0.952427637"
    ],
    [
      "0.931303138",
      "0.952427637"
    ]
  ],
  "radius": "0.047572362"
}

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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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复