极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 69

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 = 69
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[0.0613835716855, 0.0613835716865]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 = 0.061383571

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 = 69 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": [
    [
      "0.061383572",
      "0.061383572"
    ],
    [
      "0.27301862",
      "0.061383572"
    ],
    [
      "0.484271786",
      "0.061383572"
    ],
    [
      "0.695695124",
      "0.061383572"
    ],
    [
      "0.938616428",
      "0.061383572"
    ],
    [
      "0.817155776",
      "0.079246425"
    ],
    [
      "0.167201096",
      "0.123628436"
    ],
    [
      "0.589983455",
      "0.123808042"
    ],
    [
      "0.378645203",
      "0.123951906"
    ],
    [
      "0.938616428",
      "0.184150715"
    ],
    [
      "0.061383572",
      "0.185873299"
    ],
    [
      "0.272827679",
      "0.18619677"
    ],
    [
      "0.484356872",
      "0.186376376"
    ],
    [
      "0.695328866",
      "0.186848631"
    ],
    [
      "0.817155776",
      "0.202013568"
    ],
    [
      "0.167010154",
      "0.248441634"
    ],
    [
      "0.378539348",
      "0.24862124"
    ],
    [
      "0.589702283",
      "0.249416966"
    ],
    [
      "0.938616428",
      "0.306917858"
    ],
    [
      "0.272721824",
      "0.310866104"
    ],
    [
      "0.061383572",
      "0.311009969"
    ],
    [
      "0.483884759",
      "0.31166183"
    ],
    [
      "0.695000816",
      "0.312535824"
    ],
    [
      "0.817155776",
      "0.324780711"
    ],
    [
      "0.167095241",
      "0.373434439"
    ],
    [
      "0.378067235",
      "0.373906694"
    ],
    [
      "0.589183292",
      "0.374780688"
    ],
    [
      "0.938616428",
      "0.429685002"
    ],
    [
      "0.061383572",
      "0.435858908"
    ],
    [
      "0.272440652",
      "0.436475028"
    ],
    [
      "0.483365768",
      "0.437025552"
    ],
    [
      "0.69481351",
      "0.437342886"
    ],
    [
      "0.817155776",
      "0.447547855"
    ],
    [
      "0.166728982",
      "0.498899498"
    ],
    [
      "0.588995986",
      "0.49958775"
    ],
    [
      "0.377739185",
      "0.499593887"
    ],
    [
      "0.938616428",
      "0.552452145"
    ],
    [
      "0.061383572",
      "0.561940088"
    ],
    [
      "0.272027516",
      "0.562018357"
    ],
    [
      "0.694664823",
      "0.562084697"
    ],
    [
      "0.483369403",
      "0.562156084"
    ],
    [
      "0.817155776",
      "0.570314998"
    ],
    [
      "0.377657734",
      "0.624580554"
    ],
    [
      "0.58903824",
      "0.624653031"
    ],
    [
      "0.166682105",
      "0.625058946"
    ],
    [
      "0.938616428",
      "0.675219289"
    ],
    [
      "0.483326571",
      "0.687077501"
    ],
    [
      "0.694516962",
      "0.687470313"
    ],
    [
      "0.272312323",
      "0.687621144"
    ],
    [
      "0.061383572",
      "0.688177805"
    ],
    [
      "0.817155776",
      "0.693082142"
    ],
    [
      "0.588805292",
      "0.749894782"
    ],
    [
      "0.37798116",
      "0.750118091"
    ],
    [
      "0.16701379",
      "0.750740003"
    ],
    [
      "0.938616428",
      "0.797986432"
    ],
    [
      "0.69443551",
      "0.81245698"
    ],
    [
      "0.483459882",
      "0.812935372"
    ],
    [
      "0.272682627",
      "0.813236949"
    ],
    [
      "0.061383572",
      "0.8133022"
    ],
    [
      "0.817155776",
      "0.815849285"
    ],
    [
      "0.589090099",
      "0.87549757"
    ],
    [
      "0.167052409",
      "0.875799147"
    ],
    [
      "0.378161348",
      "0.876054231"
    ],
    [
      "0.938616428",
      "0.920753575"
    ],
    [
      "0.061384166",
      "0.938615834"
    ],
    [
      "0.27253113",
      "0.938616428"
    ],
    [
      "0.483791566",
      "0.938616428"
    ],
    [
      "0.694388633",
      "0.938616428"
    ],
    [
      "0.817155776",
      "0.938616428"
    ]
  ],
  "radius": "0.061383571"
}
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": 69
}

起步布局答案

{
  "centers": [
    [
      "0.061383572",
      "0.061383572"
    ],
    [
      "0.27301862",
      "0.061383572"
    ],
    [
      "0.484271786",
      "0.061383572"
    ],
    [
      "0.695695124",
      "0.061383572"
    ],
    [
      "0.938616428",
      "0.061383572"
    ],
    [
      "0.817155776",
      "0.079246425"
    ],
    [
      "0.167201096",
      "0.123628436"
    ],
    [
      "0.589983455",
      "0.123808042"
    ],
    [
      "0.378645203",
      "0.123951906"
    ],
    [
      "0.938616428",
      "0.184150715"
    ],
    [
      "0.061383572",
      "0.185873299"
    ],
    [
      "0.272827679",
      "0.18619677"
    ],
    [
      "0.484356872",
      "0.186376376"
    ],
    [
      "0.695328866",
      "0.186848631"
    ],
    [
      "0.817155776",
      "0.202013568"
    ],
    [
      "0.167010154",
      "0.248441634"
    ],
    [
      "0.378539348",
      "0.24862124"
    ],
    [
      "0.589702283",
      "0.249416966"
    ],
    [
      "0.938616428",
      "0.306917858"
    ],
    [
      "0.272721824",
      "0.310866104"
    ],
    [
      "0.061383572",
      "0.311009969"
    ],
    [
      "0.483884759",
      "0.31166183"
    ],
    [
      "0.695000816",
      "0.312535824"
    ],
    [
      "0.817155776",
      "0.324780711"
    ],
    [
      "0.167095241",
      "0.373434439"
    ],
    [
      "0.378067235",
      "0.373906694"
    ],
    [
      "0.589183292",
      "0.374780688"
    ],
    [
      "0.938616428",
      "0.429685002"
    ],
    [
      "0.061383572",
      "0.435858908"
    ],
    [
      "0.272440652",
      "0.436475028"
    ],
    [
      "0.483365768",
      "0.437025552"
    ],
    [
      "0.69481351",
      "0.437342886"
    ],
    [
      "0.817155776",
      "0.447547855"
    ],
    [
      "0.166728982",
      "0.498899498"
    ],
    [
      "0.588995986",
      "0.49958775"
    ],
    [
      "0.377739185",
      "0.499593887"
    ],
    [
      "0.938616428",
      "0.552452145"
    ],
    [
      "0.061383572",
      "0.561940088"
    ],
    [
      "0.272027516",
      "0.562018357"
    ],
    [
      "0.694664823",
      "0.562084697"
    ],
    [
      "0.483369403",
      "0.562156084"
    ],
    [
      "0.817155776",
      "0.570314998"
    ],
    [
      "0.377657734",
      "0.624580554"
    ],
    [
      "0.58903824",
      "0.624653031"
    ],
    [
      "0.166682105",
      "0.625058946"
    ],
    [
      "0.938616428",
      "0.675219289"
    ],
    [
      "0.483326571",
      "0.687077501"
    ],
    [
      "0.694516962",
      "0.687470313"
    ],
    [
      "0.272312323",
      "0.687621144"
    ],
    [
      "0.061383572",
      "0.688177805"
    ],
    [
      "0.817155776",
      "0.693082142"
    ],
    [
      "0.588805292",
      "0.749894782"
    ],
    [
      "0.37798116",
      "0.750118091"
    ],
    [
      "0.16701379",
      "0.750740003"
    ],
    [
      "0.938616428",
      "0.797986432"
    ],
    [
      "0.69443551",
      "0.81245698"
    ],
    [
      "0.483459882",
      "0.812935372"
    ],
    [
      "0.272682627",
      "0.813236949"
    ],
    [
      "0.061383572",
      "0.8133022"
    ],
    [
      "0.817155776",
      "0.815849285"
    ],
    [
      "0.589090099",
      "0.87549757"
    ],
    [
      "0.167052409",
      "0.875799147"
    ],
    [
      "0.378161348",
      "0.876054231"
    ],
    [
      "0.938616428",
      "0.920753575"
    ],
    [
      "0.061384166",
      "0.938615834"
    ],
    [
      "0.27253113",
      "0.938616428"
    ],
    [
      "0.483791566",
      "0.938616428"
    ],
    [
      "0.694388633",
      "0.938616428"
    ],
    [
      "0.817155776",
      "0.938616428"
    ]
  ],
  "radius": "0.061383571"
}

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

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。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复