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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 = 109

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 = 109
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[0.0490600564535, 0.0490600564545]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 = 0.049060056

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 = 109 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": [
    [
      "0.049133136",
      "0.049060056"
    ],
    [
      "0.147253249",
      "0.049060056"
    ],
    [
      "0.245373362",
      "0.049060056"
    ],
    [
      "0.441600785",
      "0.049060056"
    ],
    [
      "0.611041902",
      "0.049060056"
    ],
    [
      "0.780990923",
      "0.049060056"
    ],
    [
      "0.950939944",
      "0.049060056"
    ],
    [
      "0.343487073",
      "0.050180818"
    ],
    [
      "0.696016412",
      "0.098120113"
    ],
    [
      "0.865965433",
      "0.098120113"
    ],
    [
      "0.526321344",
      "0.098557365"
    ],
    [
      "0.049060056",
      "0.147180142"
    ],
    [
      "0.147180169",
      "0.147180142"
    ],
    [
      "0.245300282",
      "0.147180142"
    ],
    [
      "0.780990923",
      "0.147180169"
    ],
    [
      "0.950939944",
      "0.147180169"
    ],
    [
      "0.611295854",
      "0.147617421"
    ],
    [
      "0.44153344",
      "0.147939224"
    ],
    [
      "0.343413994",
      "0.148300903"
    ],
    [
      "0.865965433",
      "0.196240226"
    ],
    [
      "0.696270364",
      "0.196677478"
    ],
    [
      "0.526507951",
      "0.196999281"
    ],
    [
      "0.049060056",
      "0.245300255"
    ],
    [
      "0.147180169",
      "0.245300255"
    ],
    [
      "0.245300282",
      "0.245300255"
    ],
    [
      "0.950939944",
      "0.245300282"
    ],
    [
      "0.781244875",
      "0.245737534"
    ],
    [
      "0.44153344",
      "0.246059337"
    ],
    [
      "0.611482461",
      "0.246059337"
    ],
    [
      "0.343413994",
      "0.246421016"
    ],
    [
      "0.866219385",
      "0.294797591"
    ],
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      "0.526507951",
      "0.295119394"
    ],
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      "0.696456972",
      "0.295119394"
    ],
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      "0.343420368"
    ],
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    ],
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    ],
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      "0.344294899"
    ],
    [
      "0.343413994",
      "0.344541129"
    ],
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      "0.526507951",
      "0.393239507"
    ],
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      "0.696456972",
      "0.393239507"
    ],
    [
      "0.86615204",
      "0.393676759"
    ],
    [
      "0.049060056",
      "0.441540481"
    ],
    [
      "0.147180169",
      "0.441540481"
    ],
    [
      "0.245300282",
      "0.441540481"
    ],
    [
      "0.44153344",
      "0.442299563"
    ],
    [
      "0.611482461",
      "0.442299563"
    ],
    [
      "0.343413994",
      "0.442661242"
    ],
    [
      "0.78117753",
      "0.442736815"
    ],
    [
      "0.950939944",
      "0.443058618"
    ],
    [
      "0.69620302",
      "0.491796872"
    ],
    [
      "0.865965433",
      "0.492118674"
    ],
    [
      "0.506786134",
      "0.525619505"
    ],
    [
      "0.098120113",
      "0.526514991"
    ],
    [
      "0.196240226",
      "0.526514991"
    ],
    [
      "0.40321488",
      "0.5326281"
    ],
    [
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    ],
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    ],
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      "0.780990923",
      "0.541178731"
    ],
    [
      "0.950939944",
      "0.541178731"
    ],
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      "0.865965433",
      "0.590238787"
    ],
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      "0.695190341",
      "0.591038784"
    ],
    [
      "0.245685889",
      "0.611265702"
    ],
    [
      "0.049060056",
      "0.611489502"
    ],
    [
      "0.147180169",
      "0.611489502"
    ],
    [
      "0.353973047",
      "0.617497402"
    ],
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      "0.61760261"
    ],
    [
      "0.55039505",
      "0.61760261"
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    ],
    [
      "0.659377757",
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      "0.780990923"
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      "0.344191609",
      "0.780990923"
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      "0.781214723"
    ],
    [
      "0.049060056",
      "0.781438522"
    ],
    [
      "0.865057787",
      "0.786474815"
    ],
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      "0.45209316",
      "0.787446422"
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      "0.55039505",
      "0.787551631"
    ],
    [
      "0.950032298",
      "0.835534872"
    ],
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      "0.610317701",
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      "0.708437813",
      "0.865248687"
    ],
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      "0.865248687"
    ],
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      "0.865965433"
    ],
    [
      "0.295131552",
      "0.865965433"
    ],
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      "0.865965433"
    ],
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    ],
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      "0.872420933"
    ],
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      "0.950939944",
      "0.933650786"
    ],
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      "0.049060056",
      "0.950939944"
    ],
    [
      "0.147951383",
      "0.950939944"
    ],
    [
      "0.246071496",
      "0.950939944"
    ],
    [
      "0.344191609",
      "0.950939944"
    ],
    [
      "0.442311722",
      "0.950939944"
    ],
    [
      "0.559994712",
      "0.950939944"
    ],
    [
      "0.658114824",
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    ],
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      "0.756234937",
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    ],
    [
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  ],
  "radius": "0.049060056"
}
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": 109
}

起步布局答案

{
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    [
      "0.049133136",
      "0.049060056"
    ],
    [
      "0.147253249",
      "0.049060056"
    ],
    [
      "0.245373362",
      "0.049060056"
    ],
    [
      "0.441600785",
      "0.049060056"
    ],
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      "0.049060056"
    ],
    [
      "0.780990923",
      "0.049060056"
    ],
    [
      "0.950939944",
      "0.049060056"
    ],
    [
      "0.343487073",
      "0.050180818"
    ],
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      "0.696016412",
      "0.098120113"
    ],
    [
      "0.865965433",
      "0.098120113"
    ],
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      "0.098557365"
    ],
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      "0.147180142"
    ],
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      "0.147180142"
    ],
    [
      "0.245300282",
      "0.147180142"
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      "0.147180169"
    ],
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    ],
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    ],
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    ],
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      "0.950939944"
    ],
    [
      "0.246071496",
      "0.950939944"
    ],
    [
      "0.344191609",
      "0.950939944"
    ],
    [
      "0.442311722",
      "0.950939944"
    ],
    [
      "0.559994712",
      "0.950939944"
    ],
    [
      "0.658114824",
      "0.950939944"
    ],
    [
      "0.756234937",
      "0.950939944"
    ],
    [
      "0.85435505",
      "0.950939944"
    ]
  ],
  "radius": "0.049060056"
}

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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