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 = 124

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 = 124
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[0.0811575777005538408773539366175, 0.0811575777005538408773539366185]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 = 0.081157577

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 = 124 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": [
    [
      "1.020633354",
      "0.081389276"
    ],
    [
      "0.858671416",
      "0.092091619"
    ],
    [
      "1.181951411",
      "0.099352965"
    ],
    [
      "1.337591519",
      "0.145422114"
    ],
    [
      "0.589362837",
      "0.178021583"
    ],
    [
      "0.748275282",
      "0.211082809"
    ],
    [
      "1.482696796",
      "0.218159095"
    ],
    [
      "1.06821676",
      "0.236573132"
    ],
    [
      "0.906311796",
      "0.248105367"
    ],
    [
      "0.451133534",
      "0.263103128"
    ],
    [
      "1.227996278",
      "0.265151362"
    ],
    [
      "0.636318619",
      "0.333396498"
    ],
    [
      "1.373101555",
      "0.337888343"
    ],
    [
      "0.795231064",
      "0.366457724"
    ],
    [
      "1.546044999",
      "0.367602111"
    ],
    [
      "0.330032077",
      "0.37118017"
    ],
    [
      "1.115172542",
      "0.391948047"
    ],
    [
      "0.953267578",
      "0.403480282"
    ],
    [
      "1.703126069",
      "0.408489114"
    ],
    [
      "0.498089316",
      "0.418478043"
    ],
    [
      "1.263580214",
      "0.457685194"
    ],
    [
      "0.679387086",
      "0.489893517"
    ],
    [
      "1.425192195",
      "0.491617894"
    ],
    [
      "0.229837541",
      "0.49888007"
    ],
    [
      "0.838299531",
      "0.522954743"
    ],
    [
      "1.583946542",
      "0.525430127"
    ],
    [
      "0.387534787",
      "0.537322083"
    ],
    [
      "0.996336045",
      "0.5599773"
    ],
    [
      "1.748188486",
      "0.564423678"
    ],
    [
      "0.54420852",
      "0.579743197"
    ],
    [
      "1.157359024",
      "0.580417702"
    ],
    [
      "1.316532769",
      "0.612197075"
    ],
    [
      "0.153676582",
      "0.642217846"
    ],
    [
      "0.730211447",
      "0.644046344"
    ],
    [
      "1.475287116",
      "0.646009308"
    ],
    [
      "1.634041462",
      "0.679821541"
    ],
    [
      "1.861542475",
      "0.680600625"
    ],
    [
      "0.311373828",
      "0.680659859"
    ],
    [
      "0.888247962",
      "0.681068901"
    ],
    [
      "1.045810249",
      "0.72006038"
    ],
    [
      "0.468047562",
      "0.723080973"
    ],
    [
      "1.207453945",
      "0.734809116"
    ],
    [
      "1.36662769",
      "0.766588488"
    ],
    [
      "0.624295982",
      "0.767042782"
    ],
    [
      "1.747265072",
      "0.796125555"
    ],
    [
      "0.198089413",
      "0.798338665"
    ],
    [
      "1.525382036",
      "0.800400721"
    ],
    [
      "0.782332497",
      "0.80406534"
    ],
    [
      "1.904301841",
      "0.83718238"
    ],
    [
      "0.355067018",
      "0.839621122"
    ],
    [
      "0.939894784",
      "0.843056819"
    ],
    [
      "1.097311785",
      "0.882630765"
    ],
    [
      "0.511315439",
      "0.883582932"
    ],
    [
      "1.259519073",
      "0.888547309"
    ],
    [
      "0.085108869",
      "0.914878815"
    ],
    [
      "1.638605646",
      "0.916704736"
    ],
    [
      "0.66873244",
      "0.923156878"
    ],
    [
      "1.418159021",
      "0.923175847"
    ],
    [
      "0.242086474",
      "0.956161272"
    ],
    [
      "1.795642415",
      "0.957761561"
    ],
    [
      "0.826768955",
      "0.960179436"
    ],
    [
      "0.984331242",
      "0.999170915"
    ],
    [
      "0.398334895",
      "1.000123081"
    ],
    [
      "1.141725233",
      "1.038836281"
    ],
    [
      "1.531997945",
      "1.039101663"
    ],
    [
      "0.555751897",
      "1.039697027"
    ],
    [
      "1.303935626",
      "1.044667069"
    ],
    [
      "1.916457549",
      "1.066158577"
    ],
    [
      "0.084405809",
      "1.077192447"
    ],
    [
      "0.715688222",
      "1.078531793"
    ],
    [
      "1.704877839",
      "1.092327535"
    ],
    [
      "0.873250509",
      "1.117523272"
    ],
    [
      "0.241383414",
      "1.118474904"
    ],
    [
      "1.0306445",
      "1.157188638"
    ],
    [
      "1.420783977",
      "1.15732883"
    ],
    [
      "0.399738973",
      "1.162432164"
    ],
    [
      "1.578778691",
      "1.194529368"
    ],
    [
      "0.602707678",
      "1.195071943"
    ],
    [
      "1.200894623",
      "1.210860264"
    ],
    [
      "1.890516776",
      "1.226387432"
    ],
    [
      "0.112274646",
      "1.237097224"
    ],
    [
      "0.759267011",
      "1.237913338"
    ],
    [
      "1.730337649",
      "1.252633517"
    ],
    [
      "0.91759629",
      "1.27366315"
    ],
    [
      "0.283120797",
      "1.275332166"
    ],
    [
      "0.481278123",
      "1.302780219"
    ],
    [
      "1.367053868",
      "1.310493072"
    ],
    [
      "1.074990281",
      "1.313328516"
    ],
    [
      "1.525048581",
      "1.34769361"
    ],
    [
      "0.64067403",
      "1.352884381"
    ],
    [
      "1.836786666",
      "1.379551674"
    ],
    [
      "1.22170982",
      "1.382751757"
    ],
    [
      "0.167845712",
      "1.389603179"
    ],
    [
      "0.797233362",
      "1.395725777"
    ],
    [
      "1.676607539",
      "1.405797759"
    ],
    [
      "0.364659947",
      "1.415680222"
    ],
    [
      "0.955562641",
      "1.431475588"
    ],
    [
      "0.52029521",
      "1.461765736"
    ],
    [
      "1.413834614",
      "1.465920777"
    ],
    [
      "1.104787275",
      "1.495336517"
    ],
    [
      "0.681851959",
      "1.509889438"
    ],
    [
      "1.569666193",
      "1.527903279"
    ],
    [
      "0.249384862",
      "1.529951235"
    ],
    [
      "1.268490566",
      "1.538179462"
    ],
    [
      "1.728883349",
      "1.559464441"
    ],
    [
      "0.405020125",
      "1.576036749"
    ],
    [
      "0.819419563",
      "1.596036777"
    ],
    [
      "0.981655883",
      "1.601095062"
    ],
    [
      "0.561577339",
      "1.618885884"
    ],
    [
      "1.440385502",
      "1.626049663"
    ],
    [
      "1.138882338",
      "1.654030362"
    ],
    [
      "1.611436367",
      "1.685850334"
    ],
    [
      "1.295041454",
      "1.698308348"
    ],
    [
      "0.699144943",
      "1.705033223"
    ],
    [
      "0.446047528",
      "1.733081207"
    ],
    [
      "0.8534995",
      "1.755241592"
    ],
    [
      "1.015750946",
      "1.759788907"
    ],
    [
      "1.480119199",
      "1.783426417"
    ],
    [
      "1.178616036",
      "1.811407117"
    ],
    [
      "0.583684779",
      "1.819117228"
    ],
    [
      "1.334775152",
      "1.855685102"
    ],
    [
      "0.734313501",
      "1.879591997"
    ],
    [
      "0.893233198",
      "1.912618347"
    ],
    [
      "1.055484643",
      "1.917165662"
    ]
  ],
  "radius": "0.081157577"
}
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": 124
}

起步布局答案

{
  "centers": [
    [
      "1.020633354",
      "0.081389276"
    ],
    [
      "0.858671416",
      "0.092091619"
    ],
    [
      "1.181951411",
      "0.099352965"
    ],
    [
      "1.337591519",
      "0.145422114"
    ],
    [
      "0.589362837",
      "0.178021583"
    ],
    [
      "0.748275282",
      "0.211082809"
    ],
    [
      "1.482696796",
      "0.218159095"
    ],
    [
      "1.06821676",
      "0.236573132"
    ],
    [
      "0.906311796",
      "0.248105367"
    ],
    [
      "0.451133534",
      "0.263103128"
    ],
    [
      "1.227996278",
      "0.265151362"
    ],
    [
      "0.636318619",
      "0.333396498"
    ],
    [
      "1.373101555",
      "0.337888343"
    ],
    [
      "0.795231064",
      "0.366457724"
    ],
    [
      "1.546044999",
      "0.367602111"
    ],
    [
      "0.330032077",
      "0.37118017"
    ],
    [
      "1.115172542",
      "0.391948047"
    ],
    [
      "0.953267578",
      "0.403480282"
    ],
    [
      "1.703126069",
      "0.408489114"
    ],
    [
      "0.498089316",
      "0.418478043"
    ],
    [
      "1.263580214",
      "0.457685194"
    ],
    [
      "0.679387086",
      "0.489893517"
    ],
    [
      "1.425192195",
      "0.491617894"
    ],
    [
      "0.229837541",
      "0.49888007"
    ],
    [
      "0.838299531",
      "0.522954743"
    ],
    [
      "1.583946542",
      "0.525430127"
    ],
    [
      "0.387534787",
      "0.537322083"
    ],
    [
      "0.996336045",
      "0.5599773"
    ],
    [
      "1.748188486",
      "0.564423678"
    ],
    [
      "0.54420852",
      "0.579743197"
    ],
    [
      "1.157359024",
      "0.580417702"
    ],
    [
      "1.316532769",
      "0.612197075"
    ],
    [
      "0.153676582",
      "0.642217846"
    ],
    [
      "0.730211447",
      "0.644046344"
    ],
    [
      "1.475287116",
      "0.646009308"
    ],
    [
      "1.634041462",
      "0.679821541"
    ],
    [
      "1.861542475",
      "0.680600625"
    ],
    [
      "0.311373828",
      "0.680659859"
    ],
    [
      "0.888247962",
      "0.681068901"
    ],
    [
      "1.045810249",
      "0.72006038"
    ],
    [
      "0.468047562",
      "0.723080973"
    ],
    [
      "1.207453945",
      "0.734809116"
    ],
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      "0.766588488"
    ],
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      "0.624295982",
      "0.767042782"
    ],
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      "1.747265072",
      "0.796125555"
    ],
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      "0.198089413",
      "0.798338665"
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      "0.800400721"
    ],
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      "0.782332497",
      "0.80406534"
    ],
    [
      "1.904301841",
      "0.83718238"
    ],
    [
      "0.355067018",
      "0.839621122"
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    [
      "0.939894784",
      "0.843056819"
    ],
    [
      "1.097311785",
      "0.882630765"
    ],
    [
      "0.511315439",
      "0.883582932"
    ],
    [
      "1.259519073",
      "0.888547309"
    ],
    [
      "0.085108869",
      "0.914878815"
    ],
    [
      "1.638605646",
      "0.916704736"
    ],
    [
      "0.66873244",
      "0.923156878"
    ],
    [
      "1.418159021",
      "0.923175847"
    ],
    [
      "0.242086474",
      "0.956161272"
    ],
    [
      "1.795642415",
      "0.957761561"
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      "0.960179436"
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      "1.077192447"
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      "1.078531793"
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      "1.117523272"
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      "1.118474904"
    ],
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      "1.157188638"
    ],
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      "1.194529368"
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      "1.730337649",
      "1.252633517"
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      "1.27366315"
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      "1.275332166"
    ],
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      "0.481278123",
      "1.302780219"
    ],
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      "1.367053868",
      "1.310493072"
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      "1.074990281",
      "1.313328516"
    ],
    [
      "1.525048581",
      "1.34769361"
    ],
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      "0.64067403",
      "1.352884381"
    ],
    [
      "1.836786666",
      "1.379551674"
    ],
    [
      "1.22170982",
      "1.382751757"
    ],
    [
      "0.167845712",
      "1.389603179"
    ],
    [
      "0.797233362",
      "1.395725777"
    ],
    [
      "1.676607539",
      "1.405797759"
    ],
    [
      "0.364659947",
      "1.415680222"
    ],
    [
      "0.955562641",
      "1.431475588"
    ],
    [
      "0.52029521",
      "1.461765736"
    ],
    [
      "1.413834614",
      "1.465920777"
    ],
    [
      "1.104787275",
      "1.495336517"
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    [
      "0.681851959",
      "1.509889438"
    ],
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      "1.527903279"
    ],
    [
      "0.249384862",
      "1.529951235"
    ],
    [
      "1.268490566",
      "1.538179462"
    ],
    [
      "1.728883349",
      "1.559464441"
    ],
    [
      "0.405020125",
      "1.576036749"
    ],
    [
      "0.819419563",
      "1.596036777"
    ],
    [
      "0.981655883",
      "1.601095062"
    ],
    [
      "0.561577339",
      "1.618885884"
    ],
    [
      "1.440385502",
      "1.626049663"
    ],
    [
      "1.138882338",
      "1.654030362"
    ],
    [
      "1.611436367",
      "1.685850334"
    ],
    [
      "1.295041454",
      "1.698308348"
    ],
    [
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      "1.705033223"
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    [
      "0.446047528",
      "1.733081207"
    ],
    [
      "0.8534995",
      "1.755241592"
    ],
    [
      "1.015750946",
      "1.759788907"
    ],
    [
      "1.480119199",
      "1.783426417"
    ],
    [
      "1.178616036",
      "1.811407117"
    ],
    [
      "0.583684779",
      "1.819117228"
    ],
    [
      "1.334775152",
      "1.855685102"
    ],
    [
      "0.734313501",
      "1.879591997"
    ],
    [
      "0.893233198",
      "1.912618347"
    ],
    [
      "1.055484643",
      "1.917165662"
    ]
  ],
  "radius": "0.081157577"
}

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?