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 = 205

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 = 205
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[0.0637308487440471958845732012245, 0.0637308487440471958845732012255]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 = 0.063730848

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 = 205 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": [
    [
      "0.389163094",
      "1.243433265"
    ],
    [
      "1.42985832",
      "1.831758227"
    ],
    [
      "1.064705524",
      "1.894169749"
    ],
    [
      "0.744133112",
      "0.951541877"
    ],
    [
      "1.92396693",
      "0.84872199"
    ],
    [
      "1.046981456",
      "1.65379277"
    ],
    [
      "1.687263611",
      "1.426389848"
    ],
    [
      "1.275352927",
      "1.347092867"
    ],
    [
      "1.148754379",
      "1.332284352"
    ],
    [
      "1.099386689",
      "0.072313171"
    ],
    [
      "0.490112542",
      "1.477516896"
    ],
    [
      "1.048628406",
      "0.542608673"
    ],
    [
      "0.744143333",
      "0.160216253"
    ],
    [
      "0.541410741",
      "1.038966664"
    ],
    [
      "0.834955875",
      "1.868286558"
    ],
    [
      "0.480555966",
      "1.778959446"
    ],
    [
      "0.946855483",
      "0.864117091"
    ],
    [
      "1.375341753",
      "1.587852765"
    ],
    [
      "0.895557284",
      "1.302667323"
    ],
    [
      "1.098279655",
      "1.215242537"
    ],
    [
      "0.28739449",
      "1.60728355"
    ],
    [
      "0.59270894",
      "0.600416431"
    ],
    [
      "1.326651126",
      "0.908542634"
    ],
    [
      "0.190734333",
      "1.100431495"
    ],
    [
      "1.75945541",
      "1.299091726"
    ],
    [
      "0.439637818",
      "1.360475081"
    ],
    [
      "0.642360189",
      "1.273050294"
    ],
    [
      "0.070659919",
      "0.886303312"
    ],
    [
      "0.466112665",
      "0.585588498"
    ],
    [
      "0.228692944",
      "1.242693137"
    ],
    [
      "1.197290731",
      "0.335567691"
    ],
    [
      "0.516569437",
      "0.702638055"
    ],
    [
      "1.402774949",
      "0.806309333"
    ],
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      "0.287524666",
      "1.01749709"
    ],
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      "1.249703827",
      "1.566367983"
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      "0.557417188"
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      "1.379223074"
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      "1.301825501",
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      "1.897719477",
      "1.265894086"
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      "0.290405514",
      "0.444812245"
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      "0.377562431"
    ],
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      "0.769782212",
      "0.732266761"
    ],
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      "0.385327338",
      "0.68417965"
    ],
    [
      "0.192401033",
      "0.526308294"
    ],
    [
      "0.87073166",
      "0.966350391"
    ],
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      "0.949213703",
      "0.307875045"
    ],
    [
      "0.668832764",
      "0.49818313"
    ],
    [
      "1.200052578",
      "0.893734119"
    ],
    [
      "1.312903756",
      "1.882434792"
    ],
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      "1.174141587",
      "1.669224978"
    ],
    [
      "0.920382909",
      "1.638984255"
    ],
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      "0.743309637",
      "1.507133925"
    ],
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      "1.251350777",
      "0.455183887"
    ],
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      "1.682622078",
      "0.602874762"
    ],
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      "1.072630556",
      "1.434517653"
    ],
    [
      "0.693658388",
      "0.834500062"
    ],
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      "0.313039271",
      "1.345666566"
    ],
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      "1.07612214",
      "0.296011949"
    ],
    [
      "1.5275894",
      "1.383386163"
    ],
    [
      "1.504709995",
      "0.483342187"
    ],
    [
      "1.803888662",
      "1.179625537"
    ],
    [
      "0.258188918",
      "0.675107359"
    ],
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      "0.795431311",
      "0.512991644"
    ],
    [
      "0.077232482",
      "1.158429894"
    ],
    [
      "0.996506732",
      "1.536750954"
    ],
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      "0.590711184",
      "1.714832134"
    ],
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      "1.301002026",
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    ],
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      "0.491659223",
      "0.365529957"
    ],
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      "0.845906035",
      "0.63003346"
    ],
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      "0.692834913",
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      "1.276176402",
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      "1.149577854",
      "0.776692304"
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      "0.719307488",
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      "1.317475838"
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      "0.196088417",
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    ],
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      "1.428424048",
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    ],
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      "0.869908185",
      "1.52194244"
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      "1.401951474",
      "1.361901381"
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    [
      "0.998153682",
      "0.425566858"
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      "1.780189571",
      "1.054386416"
    ],
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      "0.921206384",
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      "0.537353636",
      "1.597324933"
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      "0.61671109",
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      "1.676992185",
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    [
      "1.481413536",
      "1.2622351"
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    [
      "0.643183664",
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      "1.37712585",
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      "0.329801319",
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      "1.504868404"
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      "0.145407739"
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      "0.310952802",
      "1.14278718"
    ],
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      "1.783203233",
      "0.821648978"
    ],
    [
      "1.894857848",
      "0.724628696"
    ],
    [
      "0.967770575",
      "0.064285734"
    ],
    [
      "1.479946725",
      "0.703623576"
    ],
    [
      "0.971681108",
      "1.200434022"
    ],
    [
      "1.526776558",
      "1.620731655"
    ],
    [
      "0.566236366",
      "1.375283595"
    ],
    [
      "0.515761642",
      "1.25824178"
    ],
    [
      "1.578064124",
      "1.500427978"
    ],
    [
      "1.274343033",
      "0.234032341"
    ],
    [
      "1.853286225",
      "1.385360275"
    ],
    [
      "0.335597499",
      "1.474400644"
    ],
    [
      "1.427114878",
      "1.704326057"
    ],
    [
      "0.794607836",
      "1.068583692"
    ],
    [
      "1.538846096",
      "1.765666252"
    ],
    [
      "0.591885465",
      "1.156008479"
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    [
      "1.325827651",
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      "0.618375992",
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      "0.542236489",
      "0.483355197"
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    [
      "1.925514824",
      "1.141499942"
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    [
      "0.567059841",
      "0.819691547"
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      "0.922029859",
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    [
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    [
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      "1.352300225",
      "0.689267518"
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      "1.073454031",
      "0.878925605"
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      "0.849308576"
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      "0.485832947"
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      "1.225701678",
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      "0.414812194",
      "1.024158149"
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    [
      "1.833677957",
      "0.938690794"
    ],
    [
      "1.350466856",
      "0.13179904"
    ],
    [
      "0.490936017",
      "0.921924848"
    ],
    [
      "0.943819937",
      "1.934582112"
    ],
    [
      "0.819433461",
      "1.404900624"
    ],
    [
      "0.840978538",
      "0.077334244"
    ],
    [
      "0.363419666",
      "0.907315391"
    ],
    [
      "1.427600574",
      "1.142626265"
    ],
    [
      "1.190149984",
      "1.916756733"
    ],
    [
      "0.844180657",
      "1.741159111"
    ],
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      "0.411390617",
      "1.577763307"
    ],
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      "1.35147675",
      "1.244859566"
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      "0.094714417",
      "0.761131968"
    ],
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      "0.696034561",
      "0.278250275"
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      "1.793038574",
      "1.497684382"
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      "0.99596742"
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      "1.555184719",
      "0.600384002"
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    [
      "0.84508256",
      "1.185625508"
    ],
    [
      "1.222916656",
      "1.792164054"
    ],
    [
      "1.638238528",
      "1.060896514"
    ],
    [
      "1.579848221",
      "0.938159663"
    ],
    [
      "0.440461293",
      "0.804883033"
    ],
    [
      "0.972289644",
      "1.80020625"
    ],
    [
      "1.400984866",
      "0.248822179"
    ],
    [
      "1.174403479",
      "1.113009236"
    ]
  ],
  "radius": "0.063730848"
}
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": 205
}

起步布局答案

{
  "centers": [
    [
      "0.389163094",
      "1.243433265"
    ],
    [
      "1.42985832",
      "1.831758227"
    ],
    [
      "1.064705524",
      "1.894169749"
    ],
    [
      "0.744133112",
      "0.951541877"
    ],
    [
      "1.92396693",
      "0.84872199"
    ],
    [
      "1.046981456",
      "1.65379277"
    ],
    [
      "1.687263611",
      "1.426389848"
    ],
    [
      "1.275352927",
      "1.347092867"
    ],
    [
      "1.148754379",
      "1.332284352"
    ],
    [
      "1.099386689",
      "0.072313171"
    ],
    [
      "0.490112542",
      "1.477516896"
    ],
    [
      "1.048628406",
      "0.542608673"
    ],
    [
      "0.744143333",
      "0.160216253"
    ],
    [
      "0.541410741",
      "1.038966664"
    ],
    [
      "0.834955875",
      "1.868286558"
    ],
    [
      "0.480555966",
      "1.778959446"
    ],
    [
      "0.946855483",
      "0.864117091"
    ],
    [
      "1.375341753",
      "1.587852765"
    ],
    [
      "0.895557284",
      "1.302667323"
    ],
    [
      "1.098279655",
      "1.215242537"
    ],
    [
      "0.28739449",
      "1.60728355"
    ],
    [
      "0.59270894",
      "0.600416431"
    ],
    [
      "1.326651126",
      "0.908542634"
    ],
    [
      "0.190734333",
      "1.100431495"
    ],
    [
      "1.75945541",
      "1.299091726"
    ],
    [
      "0.439637818",
      "1.360475081"
    ],
    [
      "0.642360189",
      "1.273050294"
    ],
    [
      "0.070659919",
      "0.886303312"
    ],
    [
      "0.466112665",
      "0.585588498"
    ],
    [
      "0.228692944",
      "1.242693137"
    ],
    [
      "1.197290731",
      "0.335567691"
    ],
    [
      "0.516569437",
      "0.702638055"
    ],
    [
      "1.402774949",
      "0.806309333"
    ],
    [
      "0.287524666",
      "1.01749709"
    ],
    [
      "1.249703827",
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  "radius": "0.063730848"
}

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?