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 = 220

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 = 220
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[0.0616367206187595306826943379885, 0.0616367206187595306826943379895]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 = 0.061636720

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 = 220 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": [
    [
      "1.586945641",
      "0.842280409"
    ],
    [
      "0.606787542",
      "0.80656018"
    ],
    [
      "0.660639451",
      "1.172813256"
    ],
    [
      "1.522997852",
      "1.779101231"
    ],
    [
      "1.554478704",
      "1.42532451"
    ],
    [
      "0.173641092",
      "1.131038478"
    ],
    [
      "0.914602019",
      "0.69918863"
    ],
    [
      "1.664910578",
      "0.605851579"
    ],
    [
      "0.948644004",
      "1.63553484"
    ],
    [
      "0.341991767",
      "0.471096939"
    ],
    [
      "0.993402658",
      "0.500504282"
    ],
    [
      "1.101063036",
      "1.232642204"
    ],
    [
      "0.43144285",
      "1.081896954"
    ],
    [
      "0.474205193",
      "0.638765905"
    ],
    [
      "1.505487852",
      "1.048388449"
    ],
    [
      "1.622676069",
      "1.701997263"
    ],
    [
      "0.739230617",
      "0.974178194"
    ],
    [
      "1.669068639",
      "1.470772151"
    ],
    [
      "1.508898997",
      "0.21161719"
    ],
    [
      "0.624640389",
      "0.928730183"
    ],
    [
      "1.06200924",
      "1.93631218"
    ],
    [
      "0.219026104",
      "0.47980252"
    ],
    [
      "1.604881797",
      "0.964242024"
    ],
    [
      "0.546049516",
      "1.127365615"
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      "1.408834072",
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      "0.912045527"
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      "1.709325624",
      "1.614314905"
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      "0.518086419",
      "0.19483856"
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      "0.837417386"
    ],
    [
      "0.835884396",
      "0.897664219"
    ],
    [
      "1.215652971",
      "1.278089845"
    ],
    [
      "0.958625727",
      "0.254307155"
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    [
      "1.401160911",
      "0.15171021"
    ],
    [
      "0.73743585",
      "1.666646072"
    ]
  ],
  "radius": "0.06163672"
}
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": 220
}

起步布局答案

{
  "centers": [
    [
      "1.586945641",
      "0.842280409"
    ],
    [
      "0.606787542",
      "0.80656018"
    ],
    [
      "0.660639451",
      "1.172813256"
    ],
    [
      "1.522997852",
      "1.779101231"
    ],
    [
      "1.554478704",
      "1.42532451"
    ],
    [
      "0.173641092",
      "1.131038478"
    ],
    [
      "0.914602019",
      "0.69918863"
    ],
    [
      "1.664910578",
      "0.605851579"
    ],
    [
      "0.948644004",
      "1.63553484"
    ],
    [
      "0.341991767",
      "0.471096939"
    ],
    [
      "0.993402658",
      "0.500504282"
    ],
    [
      "1.101063036",
      "1.232642204"
    ],
    [
      "0.43144285",
      "1.081896954"
    ],
    [
      "0.474205193",
      "0.638765905"
    ],
    [
      "1.505487852",
      "1.048388449"
    ],
    [
      "1.622676069",
      "1.701997263"
    ],
    [
      "0.739230617",
      "0.974178194"
    ],
    [
      "1.669068639",
      "1.470772151"
    ],
    [
      "1.508898997",
      "0.21161719"
    ],
    [
      "0.624640389",
      "0.928730183"
    ],
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  "radius": "0.06163672"
}

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?