P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 44

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 = 44

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	[0.0757819860175, 0.0757819860185]	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344

0x1

已验证构造r = 0.075781985

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

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": [
    [
      "0.075781986",
      "0.075781986"
    ],
    [
      "0.227345958",
      "0.075781986"
    ],
    [
      "0.37890993",
      "0.075781986"
    ],
    [
      "0.530473902",
      "0.075781986"
    ],
    [
      "0.792959764",
      "0.075781986"
    ],
    [
      "0.924218014",
      "0.151563972"
    ],
    [
      "0.661716833",
      "0.151590499"
    ],
    [
      "0.075781986",
      "0.227345958"
    ],
    [
      "0.227345958",
      "0.227345958"
    ],
    [
      "0.37890993",
      "0.227345958"
    ],
    [
      "0.792975083",
      "0.227372485"
    ],
    [
      "0.530473893",
      "0.227398997"
    ],
    [
      "0.661732143",
      "0.303180983"
    ],
    [
      "0.924218014",
      "0.303180999"
    ],
    [
      "0.075781986",
      "0.37890993"
    ],
    [
      "0.227345958",
      "0.37890993"
    ],
    [
      "0.37890993",
      "0.37890993"
    ],
    [
      "0.530473893",
      "0.378962969"
    ],
    [
      "0.792975074",
      "0.378989496"
    ],
    [
      "0.661716824",
      "0.454771482"
    ],
    [
      "0.924218014",
      "0.454797994"
    ],
    [
      "0.227398997",
      "0.530473893"
    ],
    [
      "0.378962969",
      "0.530473893"
    ],
    [
      "0.075781986",
      "0.530473902"
    ],
    [
      "0.792959764",
      "0.53057998"
    ],
    [
      "0.530526812",
      "0.530671538"
    ],
    [
      "0.924218014",
      "0.606361966"
    ],
    [
      "0.642196151",
      "0.640308757"
    ],
    [
      "0.151590499",
      "0.661716833"
    ],
    [
      "0.303180983",
      "0.661732143"
    ],
    [
      "0.459497803",
      "0.664561483"
    ],
    [
      "0.792959764",
      "0.682143952"
    ],
    [
      "0.924211575",
      "0.767599841"
    ],
    [
      "0.686769885",
      "0.79028898"
    ],
    [
      "0.075781986",
      "0.792959764"
    ],
    [
      "0.378989496",
      "0.792975074"
    ],
    [
      "0.227372485",
      "0.792975083"
    ],
    [
      "0.5353063",
      "0.795804423"
    ],
    [
      "0.769985318",
      "0.920203279"
    ],
    [
      "0.924211575",
      "0.924211575"
    ],
    [
      "0.151563972",
      "0.924218014"
    ],
    [
      "0.303180999",
      "0.924218014"
    ],
    [
      "0.454797994",
      "0.924218014"
    ],
    [
      "0.615814606",
      "0.924218014"
    ]
  ],
  "radius": "0.075781985"
}

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": 44
}

起步布局答案

{
  "centers": [
    [
      "0.075781986",
      "0.075781986"
    ],
    [
      "0.227345958",
      "0.075781986"
    ],
    [
      "0.37890993",
      "0.075781986"
    ],
    [
      "0.530473902",
      "0.075781986"
    ],
    [
      "0.792959764",
      "0.075781986"
    ],
    [
      "0.924218014",
      "0.151563972"
    ],
    [
      "0.661716833",
      "0.151590499"
    ],
    [
      "0.075781986",
      "0.227345958"
    ],
    [
      "0.227345958",
      "0.227345958"
    ],
    [
      "0.37890993",
      "0.227345958"
    ],
    [
      "0.792975083",
      "0.227372485"
    ],
    [
      "0.530473893",
      "0.227398997"
    ],
    [
      "0.661732143",
      "0.303180983"
    ],
    [
      "0.924218014",
      "0.303180999"
    ],
    [
      "0.075781986",
      "0.37890993"
    ],
    [
      "0.227345958",
      "0.37890993"
    ],
    [
      "0.37890993",
      "0.37890993"
    ],
    [
      "0.530473893",
      "0.378962969"
    ],
    [
      "0.792975074",
      "0.378989496"
    ],
    [
      "0.661716824",
      "0.454771482"
    ],
    [
      "0.924218014",
      "0.454797994"
    ],
    [
      "0.227398997",
      "0.530473893"
    ],
    [
      "0.378962969",
      "0.530473893"
    ],
    [
      "0.075781986",
      "0.530473902"
    ],
    [
      "0.792959764",
      "0.53057998"
    ],
    [
      "0.530526812",
      "0.530671538"
    ],
    [
      "0.924218014",
      "0.606361966"
    ],
    [
      "0.642196151",
      "0.640308757"
    ],
    [
      "0.151590499",
      "0.661716833"
    ],
    [
      "0.303180983",
      "0.661732143"
    ],
    [
      "0.459497803",
      "0.664561483"
    ],
    [
      "0.792959764",
      "0.682143952"
    ],
    [
      "0.924211575",
      "0.767599841"
    ],
    [
      "0.686769885",
      "0.79028898"
    ],
    [
      "0.075781986",
      "0.792959764"
    ],
    [
      "0.378989496",
      "0.792975074"
    ],
    [
      "0.227372485",
      "0.792975083"
    ],
    [
      "0.5353063",
      "0.795804423"
    ],
    [
      "0.769985318",
      "0.920203279"
    ],
    [
      "0.924211575",
      "0.924211575"
    ],
    [
      "0.151563972",
      "0.924218014"
    ],
    [
      "0.303180999",
      "0.924218014"
    ],
    [
      "0.454797994",
      "0.924218014"
    ],
    [
      "0.615814606",
      "0.924218014"
    ]
  ],
  "radius": "0.075781985"
}

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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