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 = 214

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 = 214
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[0.0624297828229502429122238324905, 0.0624297828229502429122238324915]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 = 0.062429782

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 = 214 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": [
    [
      "0.737558045",
      "1.320842598"
    ],
    [
      "0.616064519",
      "1.855354581"
    ],
    [
      "0.645593697",
      "0.630432401"
    ],
    [
      "1.053699271",
      "1.773142962"
    ],
    [
      "1.303047406",
      "0.112756977"
    ],
    [
      "1.603022028",
      "1.717162"
    ],
    [
      "0.950994548",
      "1.450474783"
    ],
    [
      "0.956541063",
      "1.200817257"
    ],
    [
      "0.068218745",
      "1.104026942"
    ],
    [
      "0.530121045",
      "0.940812857"
    ],
    [
      "0.844276297",
      "1.385658691"
    ],
    [
      "1.271232998",
      "1.897480124"
    ],
    [
      "0.526894799",
      "1.066381649"
    ],
    [
      "0.064175546",
      "0.94281169"
    ],
    [
      "0.42661068",
      "0.750457742"
    ],
    [
      "1.492905578",
      "1.400068957"
    ],
    [
      "0.414215644",
      "1.250460769"
    ],
    [
      "1.393816241",
      "1.728752855"
    ],
    [
      "1.313531772",
      "0.237175582"
    ],
    [
      "1.085008031",
      "0.384875858"
    ],
    [
      "0.96988948",
      "0.185285791"
    ],
    [
      "0.745877817",
      "0.946356307"
    ],
    [
      "1.922008301",
      "1.170113507"
    ],
    [
      "1.619112844",
      "0.577614106"
    ],
    [
      "0.521348285",
      "1.316039175"
    ],
    [
      "1.285015589",
      "1.020779245"
    ],
    [
      "0.432126774",
      "0.500744681"
    ],
    [
      "0.420629536",
      "1.000825527"
    ],
    [
      "1.282242332",
      "1.145608008"
    ],
    [
      "0.536102188",
      "0.690445071"
    ],
    [
      "1.613116386",
      "0.45208736"
    ],
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      "0.648366954",
      "0.505603637"
    ],
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      "0.622520022",
      "1.630512795"
    ],
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      "1.340056287"
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      "1.188875429",
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    ],
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      "1.275859661",
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    ],
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      "1.428739549",
      "0.28531163"
    ],
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    ],
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      "1.711888596",
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    ],
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      "1.503138605",
      "0.899210853"
    ],
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      "0.84669312",
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    ],
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    [
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      "1.275240194"
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      "0.444047916"
    ],
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      "0.119459302",
      "0.677996909"
    ],
    [
      "0.743104559",
      "1.071185071"
    ],
    [
      "0.313911284",
      "0.936009435"
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      "0.630839793",
      "1.256026505"
    ],
    [
      "0.345689292",
      "1.671502353"
    ]
  ],
  "radius": "0.062429782"
}
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": 214
}

起步布局答案

{
  "centers": [
    [
      "0.737558045",
      "1.320842598"
    ],
    [
      "0.616064519",
      "1.855354581"
    ],
    [
      "0.645593697",
      "0.630432401"
    ],
    [
      "1.053699271",
      "1.773142962"
    ],
    [
      "1.303047406",
      "0.112756977"
    ],
    [
      "1.603022028",
      "1.717162"
    ],
    [
      "0.950994548",
      "1.450474783"
    ],
    [
      "0.956541063",
      "1.200817257"
    ],
    [
      "0.068218745",
      "1.104026942"
    ],
    [
      "0.530121045",
      "0.940812857"
    ],
    [
      "0.844276297",
      "1.385658691"
    ],
    [
      "1.271232998",
      "1.897480124"
    ],
    [
      "0.526894799",
      "1.066381649"
    ],
    [
      "0.064175546",
      "0.94281169"
    ],
    [
      "0.42661068",
      "0.750457742"
    ],
    [
      "1.492905578",
      "1.400068957"
    ],
    [
      "0.414215644",
      "1.250460769"
    ],
    [
      "1.393816241",
      "1.728752855"
    ],
    [
      "1.313531772",
      "0.237175582"
    ],
    [
      "1.085008031",
      "0.384875858"
    ],
    [
      "0.96988948",
      "0.185285791"
    ],
    [
      "0.745877817",
      "0.946356307"
    ],
    [
      "1.922008301",
      "1.170113507"
    ],
    [
      "1.619112844",
      "0.577614106"
    ],
    [
      "0.521348285",
      "1.316039175"
    ],
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  "radius": "0.062429782"
}

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?