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 = 109

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 = 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[0.0864893358953348412274819939045, 0.0864893358953348412274819939055]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 = 0.086489335

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 = 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": [
    [
      "0.999999793",
      "0.09619321"
    ],
    [
      "0.822451558",
      "0.103909444"
    ],
    [
      "1.177548442",
      "0.103909444"
    ],
    [
      "0.543244668",
      "0.208876558"
    ],
    [
      "1.456755332",
      "0.208876558"
    ],
    [
      "0.71368309",
      "0.238412386"
    ],
    [
      "1.28631691",
      "0.238412386"
    ],
    [
      "0.913510664",
      "0.25098038"
    ],
    [
      "1.086489336",
      "0.25098038"
    ],
    [
      "0.483604234",
      "0.371248476"
    ],
    [
      "1.516395766",
      "0.371248476"
    ],
    [
      "0.312737036",
      "0.398193261"
    ],
    [
      "1.687262964",
      "0.398193261"
    ],
    [
      "0.654042656",
      "0.400784304"
    ],
    [
      "0.827021328",
      "0.400784304"
    ],
    [
      "1",
      "0.400784304"
    ],
    [
      "1.172978672",
      "0.400784304"
    ],
    [
      "1.345957344",
      "0.400784304"
    ],
    [
      "1.782719749",
      "0.548096779"
    ],
    [
      "0.217285482",
      "0.548099907"
    ],
    [
      "0.394574649",
      "0.550588228"
    ],
    [
      "0.567553321",
      "0.550588228"
    ],
    [
      "0.740531992",
      "0.550588228"
    ],
    [
      "0.913510664",
      "0.550588228"
    ],
    [
      "1.086489336",
      "0.550588228"
    ],
    [
      "1.259468008",
      "0.550588228"
    ],
    [
      "1.432446679",
      "0.550588228"
    ],
    [
      "1.605425351",
      "0.550588228"
    ],
    [
      "0.308085313",
      "0.700392152"
    ],
    [
      "0.481063985",
      "0.700392152"
    ],
    [
      "0.654042656",
      "0.700392152"
    ],
    [
      "0.827021328",
      "0.700392152"
    ],
    [
      "1",
      "0.700392152"
    ],
    [
      "1.172978672",
      "0.700392152"
    ],
    [
      "1.345957344",
      "0.700392152"
    ],
    [
      "1.518936015",
      "0.700392152"
    ],
    [
      "1.691914687",
      "0.700392152"
    ],
    [
      "0.135188593",
      "0.705716184"
    ],
    [
      "1.864811407",
      "0.705716184"
    ],
    [
      "0.394574649",
      "0.850196076"
    ],
    [
      "0.567553321",
      "0.850196076"
    ],
    [
      "0.740531992",
      "0.850196076"
    ],
    [
      "0.913510664",
      "0.850196076"
    ],
    [
      "1.086489336",
      "0.850196076"
    ],
    [
      "1.259468008",
      "0.850196076"
    ],
    [
      "1.432446679",
      "0.850196076"
    ],
    [
      "1.605425351",
      "0.850196076"
    ],
    [
      "0.197287324",
      "0.867163911"
    ],
    [
      "1.802712676",
      "0.867163911"
    ],
    [
      "0.086489336",
      "1"
    ],
    [
      "0.308085313",
      "1"
    ],
    [
      "0.481063985",
      "1"
    ],
    [
      "0.654042656",
      "1"
    ],
    [
      "0.827021328",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.172978672",
      "1"
    ],
    [
      "1.345957344",
      "1"
    ],
    [
      "1.518936015",
      "1"
    ],
    [
      "1.691914687",
      "1"
    ],
    [
      "1.913510664",
      "1"
    ],
    [
      "0.197287324",
      "1.132836089"
    ],
    [
      "1.802712676",
      "1.132836089"
    ],
    [
      "0.394574649",
      "1.149803924"
    ],
    [
      "0.567553321",
      "1.149803924"
    ],
    [
      "0.740531992",
      "1.149803924"
    ],
    [
      "0.913510664",
      "1.149803924"
    ],
    [
      "1.086489336",
      "1.149803924"
    ],
    [
      "1.259468008",
      "1.149803924"
    ],
    [
      "1.432446679",
      "1.149803924"
    ],
    [
      "1.605425351",
      "1.149803924"
    ],
    [
      "0.135188593",
      "1.294283816"
    ],
    [
      "1.864811407",
      "1.294283816"
    ],
    [
      "0.308085313",
      "1.299607848"
    ],
    [
      "0.481063985",
      "1.299607848"
    ],
    [
      "0.654042656",
      "1.299607848"
    ],
    [
      "0.827021328",
      "1.299607848"
    ],
    [
      "1",
      "1.299607848"
    ],
    [
      "1.172978672",
      "1.299607848"
    ],
    [
      "1.345957344",
      "1.299607848"
    ],
    [
      "1.518936015",
      "1.299607848"
    ],
    [
      "1.691914687",
      "1.299607848"
    ],
    [
      "0.394574649",
      "1.449411772"
    ],
    [
      "0.567553321",
      "1.449411772"
    ],
    [
      "0.740531992",
      "1.449411772"
    ],
    [
      "0.913510664",
      "1.449411772"
    ],
    [
      "1.086489336",
      "1.449411772"
    ],
    [
      "1.259468008",
      "1.449411772"
    ],
    [
      "1.432446679",
      "1.449411772"
    ],
    [
      "1.605425351",
      "1.449411772"
    ],
    [
      "0.217285296",
      "1.45190012"
    ],
    [
      "1.782714224",
      "1.451900612"
    ],
    [
      "0.654042656",
      "1.599215696"
    ],
    [
      "0.827021328",
      "1.599215696"
    ],
    [
      "1",
      "1.599215696"
    ],
    [
      "1.172978672",
      "1.599215696"
    ],
    [
      "1.345957344",
      "1.599215696"
    ],
    [
      "0.312737036",
      "1.601806739"
    ],
    [
      "1.687262964",
      "1.601806739"
    ],
    [
      "0.483604234",
      "1.628751524"
    ],
    [
      "1.516395766",
      "1.628751524"
    ],
    [
      "0.913510664",
      "1.74901962"
    ],
    [
      "1.086489336",
      "1.74901962"
    ],
    [
      "0.71368309",
      "1.761587614"
    ],
    [
      "1.28631691",
      "1.761587614"
    ],
    [
      "0.543244668",
      "1.791123442"
    ],
    [
      "1.456755332",
      "1.791123442"
    ],
    [
      "0.822451558",
      "1.896090556"
    ],
    [
      "1.177548442",
      "1.896090556"
    ],
    [
      "1.000000365",
      "1.903800883"
    ]
  ],
  "radius": "0.086489335"
}
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

{
  "centers": [
    [
      "0.999999793",
      "0.09619321"
    ],
    [
      "0.822451558",
      "0.103909444"
    ],
    [
      "1.177548442",
      "0.103909444"
    ],
    [
      "0.543244668",
      "0.208876558"
    ],
    [
      "1.456755332",
      "0.208876558"
    ],
    [
      "0.71368309",
      "0.238412386"
    ],
    [
      "1.28631691",
      "0.238412386"
    ],
    [
      "0.913510664",
      "0.25098038"
    ],
    [
      "1.086489336",
      "0.25098038"
    ],
    [
      "0.483604234",
      "0.371248476"
    ],
    [
      "1.516395766",
      "0.371248476"
    ],
    [
      "0.312737036",
      "0.398193261"
    ],
    [
      "1.687262964",
      "0.398193261"
    ],
    [
      "0.654042656",
      "0.400784304"
    ],
    [
      "0.827021328",
      "0.400784304"
    ],
    [
      "1",
      "0.400784304"
    ],
    [
      "1.172978672",
      "0.400784304"
    ],
    [
      "1.345957344",
      "0.400784304"
    ],
    [
      "1.782719749",
      "0.548096779"
    ],
    [
      "0.217285482",
      "0.548099907"
    ],
    [
      "0.394574649",
      "0.550588228"
    ],
    [
      "0.567553321",
      "0.550588228"
    ],
    [
      "0.740531992",
      "0.550588228"
    ],
    [
      "0.913510664",
      "0.550588228"
    ],
    [
      "1.086489336",
      "0.550588228"
    ],
    [
      "1.259468008",
      "0.550588228"
    ],
    [
      "1.432446679",
      "0.550588228"
    ],
    [
      "1.605425351",
      "0.550588228"
    ],
    [
      "0.308085313",
      "0.700392152"
    ],
    [
      "0.481063985",
      "0.700392152"
    ],
    [
      "0.654042656",
      "0.700392152"
    ],
    [
      "0.827021328",
      "0.700392152"
    ],
    [
      "1",
      "0.700392152"
    ],
    [
      "1.172978672",
      "0.700392152"
    ],
    [
      "1.345957344",
      "0.700392152"
    ],
    [
      "1.518936015",
      "0.700392152"
    ],
    [
      "1.691914687",
      "0.700392152"
    ],
    [
      "0.135188593",
      "0.705716184"
    ],
    [
      "1.864811407",
      "0.705716184"
    ],
    [
      "0.394574649",
      "0.850196076"
    ],
    [
      "0.567553321",
      "0.850196076"
    ],
    [
      "0.740531992",
      "0.850196076"
    ],
    [
      "0.913510664",
      "0.850196076"
    ],
    [
      "1.086489336",
      "0.850196076"
    ],
    [
      "1.259468008",
      "0.850196076"
    ],
    [
      "1.432446679",
      "0.850196076"
    ],
    [
      "1.605425351",
      "0.850196076"
    ],
    [
      "0.197287324",
      "0.867163911"
    ],
    [
      "1.802712676",
      "0.867163911"
    ],
    [
      "0.086489336",
      "1"
    ],
    [
      "0.308085313",
      "1"
    ],
    [
      "0.481063985",
      "1"
    ],
    [
      "0.654042656",
      "1"
    ],
    [
      "0.827021328",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.172978672",
      "1"
    ],
    [
      "1.345957344",
      "1"
    ],
    [
      "1.518936015",
      "1"
    ],
    [
      "1.691914687",
      "1"
    ],
    [
      "1.913510664",
      "1"
    ],
    [
      "0.197287324",
      "1.132836089"
    ],
    [
      "1.802712676",
      "1.132836089"
    ],
    [
      "0.394574649",
      "1.149803924"
    ],
    [
      "0.567553321",
      "1.149803924"
    ],
    [
      "0.740531992",
      "1.149803924"
    ],
    [
      "0.913510664",
      "1.149803924"
    ],
    [
      "1.086489336",
      "1.149803924"
    ],
    [
      "1.259468008",
      "1.149803924"
    ],
    [
      "1.432446679",
      "1.149803924"
    ],
    [
      "1.605425351",
      "1.149803924"
    ],
    [
      "0.135188593",
      "1.294283816"
    ],
    [
      "1.864811407",
      "1.294283816"
    ],
    [
      "0.308085313",
      "1.299607848"
    ],
    [
      "0.481063985",
      "1.299607848"
    ],
    [
      "0.654042656",
      "1.299607848"
    ],
    [
      "0.827021328",
      "1.299607848"
    ],
    [
      "1",
      "1.299607848"
    ],
    [
      "1.172978672",
      "1.299607848"
    ],
    [
      "1.345957344",
      "1.299607848"
    ],
    [
      "1.518936015",
      "1.299607848"
    ],
    [
      "1.691914687",
      "1.299607848"
    ],
    [
      "0.394574649",
      "1.449411772"
    ],
    [
      "0.567553321",
      "1.449411772"
    ],
    [
      "0.740531992",
      "1.449411772"
    ],
    [
      "0.913510664",
      "1.449411772"
    ],
    [
      "1.086489336",
      "1.449411772"
    ],
    [
      "1.259468008",
      "1.449411772"
    ],
    [
      "1.432446679",
      "1.449411772"
    ],
    [
      "1.605425351",
      "1.449411772"
    ],
    [
      "0.217285296",
      "1.45190012"
    ],
    [
      "1.782714224",
      "1.451900612"
    ],
    [
      "0.654042656",
      "1.599215696"
    ],
    [
      "0.827021328",
      "1.599215696"
    ],
    [
      "1",
      "1.599215696"
    ],
    [
      "1.172978672",
      "1.599215696"
    ],
    [
      "1.345957344",
      "1.599215696"
    ],
    [
      "0.312737036",
      "1.601806739"
    ],
    [
      "1.687262964",
      "1.601806739"
    ],
    [
      "0.483604234",
      "1.628751524"
    ],
    [
      "1.516395766",
      "1.628751524"
    ],
    [
      "0.913510664",
      "1.74901962"
    ],
    [
      "1.086489336",
      "1.74901962"
    ],
    [
      "0.71368309",
      "1.761587614"
    ],
    [
      "1.28631691",
      "1.761587614"
    ],
    [
      "0.543244668",
      "1.791123442"
    ],
    [
      "1.456755332",
      "1.791123442"
    ],
    [
      "0.822451558",
      "1.896090556"
    ],
    [
      "1.177548442",
      "1.896090556"
    ],
    [
      "1.000000365",
      "1.903800883"
    ]
  ],
  "radius": "0.086489335"
}

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

讨论区

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?