P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 216

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆,使共同半径尽可能大。

子题n = 216
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[0.0621691262006572210931881492115, 0.0621691262006572210931881492125]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆,圆心在 (1, 1),两个坐标都在 0 到 2 之间
  • 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
挑战这个纪录 ↗

当前展示:起步布局

2y0
0x2
已验证构造r = 0.062169125

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..14 与 19 已证明;其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型,大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 216 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

容器是半径 1 的圆,圆心在 (1, 1),所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位,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 "0.2"。

起步布局答案

{
  "centers": [
    [
      "0.86520448",
      "0.97907554"
    ],
    [
      "0.441512651",
      "0.900780927"
    ],
    [
      "0.570756104",
      "0.166167627"
    ],
    [
      "1.882231955",
      "0.681890702"
    ],
    [
      "1.663709068",
      "0.808833881"
    ],
    [
      "0.866853056",
      "1.553897205"
    ],
    [
      "0.916185285",
      "1.304531939"
    ],
    [
      "1.683822592",
      "0.358195193"
    ],
    [
      "1.349302703",
      "0.248204651"
    ],
    [
      "0.436337783",
      "0.250459038"
    ],
    [
      "0.491162903",
      "1.22615496"
    ],
    [
      "0.069670748",
      "0.882428906"
    ],
    [
      "0.226277991",
      "0.850405032"
    ],
    [
      "1.239207797",
      "0.731888299"
    ],
    [
      "1.189616179",
      "0.406679629"
    ],
    [
      "1.708073491",
      "0.565985915"
    ],
    [
      "1.443940013",
      "0.328850671"
    ],
    [
      "0.813973557",
      "0.080804205"
    ],
    [
      "0.64897498",
      "0.370160046"
    ],
    [
      "0.191908264",
      "0.524064612"
    ],
    [
      "1.284253489",
      "0.487325649"
    ],
    [
      "0.509139415",
      "1.799113655"
    ],
    [
      "1.758346379",
      "0.889479901"
    ],
    [
      "0.306195239",
      "1.631000555"
    ],
    [
      "1.557502892",
      "0.246020652"
    ],
    [
      "1.033345442",
      "1.262896634"
    ],
    [
      "1.830206572",
      "1.09270468"
    ],
    [
      "0.446341476",
      "1.467408704"
    ],
    [
      "1.151618531",
      "1.7959894"
    ],
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      "0.558731442",
      "0.859310986"
    ],
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      "0.910250173",
      "0.73451289"
    ],
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      "0.77056717",
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    ],
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      "0.509081189",
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      "0.313106126",
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      "1.064514438"
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      "1.216684951",
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      "0.748044323",
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    ],
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      "0.34687534",
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      "1.194162104",
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      "0.799025128",
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    [
      "1.618663375",
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    [
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    [
      "1.596462229",
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      "1.117501715",
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      "0.423481799",
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      "0.146355695",
      "1.388353124"
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      "0.581254288",
      "0.737029661"
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      "1.470626594"
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      "1.307397702"
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      "1.781099533",
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      "0.976412571",
      "0.062465797"
    ],
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      "1.832395802",
      "0.56797688"
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      "0.776248011",
      "1.468746004"
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    [
      "0.937318944",
      "1.182002884"
    ],
    [
      "0.764624164",
      "0.324495321"
    ],
    [
      "0.820158788",
      "1.223638189"
    ],
    [
      "1.094978868",
      "0.32603361"
    ],
    [
      "0.603777134",
      "0.614748336"
    ],
    [
      "1.379487521",
      "1.718705516"
    ],
    [
      "1.218333526",
      "1.428991288"
    ]
  ],
  "radius": "0.062169125"
}
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": 216
}

起步布局答案

{
  "centers": [
    [
      "0.86520448",
      "0.97907554"
    ],
    [
      "0.441512651",
      "0.900780927"
    ],
    [
      "0.570756104",
      "0.166167627"
    ],
    [
      "1.882231955",
      "0.681890702"
    ],
    [
      "1.663709068",
      "0.808833881"
    ],
    [
      "0.866853056",
      "1.553897205"
    ],
    [
      "0.916185285",
      "1.304531939"
    ],
    [
      "1.683822592",
      "0.358195193"
    ],
    [
      "1.349302703",
      "0.248204651"
    ],
    [
      "0.436337783",
      "0.250459038"
    ],
    [
      "0.491162903",
      "1.22615496"
    ],
    [
      "0.069670748",
      "0.882428906"
    ],
    [
      "0.226277991",
      "0.850405032"
    ],
    [
      "1.239207797",
      "0.731888299"
    ],
    [
      "1.189616179",
      "0.406679629"
    ],
    [
      "1.708073491",
      "0.565985915"
    ],
    [
      "1.443940013",
      "0.328850671"
    ],
    [
      "0.813973557",
      "0.080804205"
    ],
    [
      "0.64897498",
      "0.370160046"
    ],
    [
      "0.191908264",
      "0.524064612"
    ],
    [
      "1.284253489",
      "0.487325649"
    ],
    [
      "0.509139415",
      "1.799113655"
    ],
    [
      "1.758346379",
      "0.889479901"
    ],
    [
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  "radius": "0.062169125"
}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。 · 验证器 v1.0.0

DISCUSSION

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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。

还没有帖子。第一个聊聊这道题的思路?