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 = 100

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 = 100
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[0.0902352002884737157275787807815, 0.0902352002884737157275787807825]见来源署名packomania.com ↗
本站纪录0.090235200游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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:本站纪录

2y0
0x2
已验证构造r = 0.090235200

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 = 100 的当前纪录构型,0.090235200
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.090235199→0.090235200
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"。

当前第一名的答案

{
  "centers": [
    [
      "1.028242566",
      "0.090673685"
    ],
    [
      "0.848192936",
      "0.102990186"
    ],
    [
      "0.674117037",
      "0.150604753"
    ],
    [
      "1.196417529",
      "0.156146943"
    ],
    [
      "1.376219117",
      "0.171669651"
    ],
    [
      "0.512864898",
      "0.231643714"
    ],
    [
      "1.055628542",
      "0.269054104"
    ],
    [
      "1.556330778",
      "0.280165588"
    ],
    [
      "0.875550120",
      "0.281391814"
    ],
    [
      "0.699442010",
      "0.329289423"
    ],
    [
      "0.370781921",
      "0.342918118"
    ],
    [
      "1.390576806",
      "0.351568017"
    ],
    [
      "1.210400786",
      "0.361871763"
    ],
    [
      "0.538189870",
      "0.410328384"
    ],
    [
      "1.699642242",
      "0.418473798"
    ],
    [
      "0.997631337",
      "0.439951411"
    ],
    [
      "0.818852945",
      "0.464606067"
    ],
    [
      "1.526700981",
      "0.470057571"
    ],
    [
      "0.253459186",
      "0.480049229"
    ],
    [
      "1.309412102",
      "0.512756901"
    ],
    [
      "0.657600806",
      "0.545645027"
    ],
    [
      "1.136036208",
      "0.564321675"
    ],
    [
      "1.800665249",
      "0.568019503"
    ],
    [
      "0.480977180",
      "0.584266420"
    ],
    [
      "0.962884411",
      "0.617045218"
    ],
    [
      "1.627723987",
      "0.619603275"
    ],
    [
      "1.445561268",
      "0.631307102"
    ],
    [
      "0.313038557",
      "0.650401405"
    ],
    [
      "0.789878897",
      "0.668418946"
    ],
    [
      "1.267253982",
      "0.688234131"
    ],
    [
      "0.140252072",
      "0.702501275"
    ],
    [
      "0.617413903",
      "0.721584159"
    ],
    [
      "1.094518728",
      "0.740503600"
    ],
    [
      "1.758867474",
      "0.743582915"
    ],
    [
      "0.444678649",
      "0.773853629"
    ],
    [
      "0.921783474",
      "0.792773071"
    ],
    [
      "1.578409284",
      "0.793205192"
    ],
    [
      "1.398888298",
      "0.811692515"
    ],
    [
      "0.271765125",
      "0.826088824"
    ],
    [
      "0.749048220",
      "0.845042541"
    ],
    [
      "1.898584966",
      "0.857813330"
    ],
    [
      "1.226153044",
      "0.863961984"
    ],
    [
      "0.098275323",
      "0.879357597"
    ],
    [
      "0.576312965",
      "0.897312012"
    ],
    [
      "1.053417790",
      "0.916231454"
    ],
    [
      "1.710260016",
      "0.917384182"
    ],
    [
      "0.403577711",
      "0.949581482"
    ],
    [
      "0.880682536",
      "0.968500924"
    ],
    [
      "1.537308346",
      "0.968933045"
    ],
    [
      "1.357787360",
      "0.987420368"
    ],
    [
      "0.230844842",
      "1.001858833"
    ],
    [
      "0.707947282",
      "1.020770394"
    ],
    [
      "1.908971485",
      "1.037984598"
    ],
    [
      "1.185052106",
      "1.039689837"
    ],
    [
      "0.535212027",
      "1.073039865"
    ],
    [
      "1.012316852",
      "1.091959307"
    ],
    [
      "1.737983031",
      "1.095712531"
    ],
    [
      "0.098854731",
      "1.124936760"
    ],
    [
      "0.361701893",
      "1.126140762"
    ],
    [
      "0.839581598",
      "1.144228777"
    ],
    [
      "1.563317730",
      "1.147519375"
    ],
    [
      "1.316686422",
      "1.163148221"
    ],
    [
      "0.666846344",
      "1.196498247"
    ],
    [
      "1.143951168",
      "1.215417690"
    ],
    [
      "1.882207861",
      "1.222188367"
    ],
    [
      "0.494111090",
      "1.248767718"
    ],
    [
      "0.971215914",
      "1.267687160"
    ],
    [
      "1.694952046",
      "1.270977758"
    ],
    [
      "1.448320738",
      "1.286606604"
    ],
    [
      "0.141246676",
      "1.300357649"
    ],
    [
      "0.321708433",
      "1.302123967"
    ],
    [
      "0.798480660",
      "1.319956630"
    ],
    [
      "1.275585484",
      "1.338876074"
    ],
    [
      "0.625745406",
      "1.372226101"
    ],
    [
      "1.102850230",
      "1.391145543"
    ],
    [
      "1.813717192",
      "1.406861552"
    ],
    [
      "1.579955054",
      "1.410064987"
    ],
    [
      "0.453342749",
      "1.425582350"
    ],
    [
      "0.930114976",
      "1.443415013"
    ],
    [
      "1.407219800",
      "1.462334457"
    ],
    [
      "0.217431261",
      "1.463959221"
    ],
    [
      "0.757379722",
      "1.495684484"
    ],
    [
      "1.234484545",
      "1.514603927"
    ],
    [
      "1.717395668",
      "1.559477834"
    ],
    [
      "1.061749291",
      "1.566873397"
    ],
    [
      "0.552245668",
      "1.576538562"
    ],
    [
      "0.357100497",
      "1.578248634"
    ],
    [
      "1.538854116",
      "1.585792840"
    ],
    [
      "0.883102872",
      "1.625242327"
    ],
    [
      "1.366118861",
      "1.638062310"
    ],
    [
      "0.707206908",
      "1.669040344"
    ],
    [
      "1.206007100",
      "1.721331688"
    ],
    [
      "0.456003416",
      "1.729204846"
    ],
    [
      "1.026960827",
      "1.743959049"
    ],
    [
      "1.497753177",
      "1.761520693"
    ],
    [
      "0.610646138",
      "1.822238141"
    ],
    [
      "0.790994215",
      "1.828881680"
    ],
    [
      "1.337641416",
      "1.844790071"
    ],
    [
      "1.157204921",
      "1.896067638"
    ],
    [
      "0.952929738",
      "1.908546300"
    ]
  ],
  "radius": "0.090235200"
}
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": 100
}

当前第一名的答案

{
  "centers": [
    [
      "1.028242566",
      "0.090673685"
    ],
    [
      "0.848192936",
      "0.102990186"
    ],
    [
      "0.674117037",
      "0.150604753"
    ],
    [
      "1.196417529",
      "0.156146943"
    ],
    [
      "1.376219117",
      "0.171669651"
    ],
    [
      "0.512864898",
      "0.231643714"
    ],
    [
      "1.055628542",
      "0.269054104"
    ],
    [
      "1.556330778",
      "0.280165588"
    ],
    [
      "0.875550120",
      "0.281391814"
    ],
    [
      "0.699442010",
      "0.329289423"
    ],
    [
      "0.370781921",
      "0.342918118"
    ],
    [
      "1.390576806",
      "0.351568017"
    ],
    [
      "1.210400786",
      "0.361871763"
    ],
    [
      "0.538189870",
      "0.410328384"
    ],
    [
      "1.699642242",
      "0.418473798"
    ],
    [
      "0.997631337",
      "0.439951411"
    ],
    [
      "0.818852945",
      "0.464606067"
    ],
    [
      "1.526700981",
      "0.470057571"
    ],
    [
      "0.253459186",
      "0.480049229"
    ],
    [
      "1.309412102",
      "0.512756901"
    ],
    [
      "0.657600806",
      "0.545645027"
    ],
    [
      "1.136036208",
      "0.564321675"
    ],
    [
      "1.800665249",
      "0.568019503"
    ],
    [
      "0.480977180",
      "0.584266420"
    ],
    [
      "0.962884411",
      "0.617045218"
    ],
    [
      "1.627723987",
      "0.619603275"
    ],
    [
      "1.445561268",
      "0.631307102"
    ],
    [
      "0.313038557",
      "0.650401405"
    ],
    [
      "0.789878897",
      "0.668418946"
    ],
    [
      "1.267253982",
      "0.688234131"
    ],
    [
      "0.140252072",
      "0.702501275"
    ],
    [
      "0.617413903",
      "0.721584159"
    ],
    [
      "1.094518728",
      "0.740503600"
    ],
    [
      "1.758867474",
      "0.743582915"
    ],
    [
      "0.444678649",
      "0.773853629"
    ],
    [
      "0.921783474",
      "0.792773071"
    ],
    [
      "1.578409284",
      "0.793205192"
    ],
    [
      "1.398888298",
      "0.811692515"
    ],
    [
      "0.271765125",
      "0.826088824"
    ],
    [
      "0.749048220",
      "0.845042541"
    ],
    [
      "1.898584966",
      "0.857813330"
    ],
    [
      "1.226153044",
      "0.863961984"
    ],
    [
      "0.098275323",
      "0.879357597"
    ],
    [
      "0.576312965",
      "0.897312012"
    ],
    [
      "1.053417790",
      "0.916231454"
    ],
    [
      "1.710260016",
      "0.917384182"
    ],
    [
      "0.403577711",
      "0.949581482"
    ],
    [
      "0.880682536",
      "0.968500924"
    ],
    [
      "1.537308346",
      "0.968933045"
    ],
    [
      "1.357787360",
      "0.987420368"
    ],
    [
      "0.230844842",
      "1.001858833"
    ],
    [
      "0.707947282",
      "1.020770394"
    ],
    [
      "1.908971485",
      "1.037984598"
    ],
    [
      "1.185052106",
      "1.039689837"
    ],
    [
      "0.535212027",
      "1.073039865"
    ],
    [
      "1.012316852",
      "1.091959307"
    ],
    [
      "1.737983031",
      "1.095712531"
    ],
    [
      "0.098854731",
      "1.124936760"
    ],
    [
      "0.361701893",
      "1.126140762"
    ],
    [
      "0.839581598",
      "1.144228777"
    ],
    [
      "1.563317730",
      "1.147519375"
    ],
    [
      "1.316686422",
      "1.163148221"
    ],
    [
      "0.666846344",
      "1.196498247"
    ],
    [
      "1.143951168",
      "1.215417690"
    ],
    [
      "1.882207861",
      "1.222188367"
    ],
    [
      "0.494111090",
      "1.248767718"
    ],
    [
      "0.971215914",
      "1.267687160"
    ],
    [
      "1.694952046",
      "1.270977758"
    ],
    [
      "1.448320738",
      "1.286606604"
    ],
    [
      "0.141246676",
      "1.300357649"
    ],
    [
      "0.321708433",
      "1.302123967"
    ],
    [
      "0.798480660",
      "1.319956630"
    ],
    [
      "1.275585484",
      "1.338876074"
    ],
    [
      "0.625745406",
      "1.372226101"
    ],
    [
      "1.102850230",
      "1.391145543"
    ],
    [
      "1.813717192",
      "1.406861552"
    ],
    [
      "1.579955054",
      "1.410064987"
    ],
    [
      "0.453342749",
      "1.425582350"
    ],
    [
      "0.930114976",
      "1.443415013"
    ],
    [
      "1.407219800",
      "1.462334457"
    ],
    [
      "0.217431261",
      "1.463959221"
    ],
    [
      "0.757379722",
      "1.495684484"
    ],
    [
      "1.234484545",
      "1.514603927"
    ],
    [
      "1.717395668",
      "1.559477834"
    ],
    [
      "1.061749291",
      "1.566873397"
    ],
    [
      "0.552245668",
      "1.576538562"
    ],
    [
      "0.357100497",
      "1.578248634"
    ],
    [
      "1.538854116",
      "1.585792840"
    ],
    [
      "0.883102872",
      "1.625242327"
    ],
    [
      "1.366118861",
      "1.638062310"
    ],
    [
      "0.707206908",
      "1.669040344"
    ],
    [
      "1.206007100",
      "1.721331688"
    ],
    [
      "0.456003416",
      "1.729204846"
    ],
    [
      "1.026960827",
      "1.743959049"
    ],
    [
      "1.497753177",
      "1.761520693"
    ],
    [
      "0.610646138",
      "1.822238141"
    ],
    [
      "0.790994215",
      "1.828881680"
    ],
    [
      "1.337641416",
      "1.844790071"
    ],
    [
      "1.157204921",
      "1.896067638"
    ],
    [
      "0.952929738",
      "1.908546300"
    ]
  ],
  "radius": "0.090235200"
}

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?