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 = 111

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 = 111
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[0.0857426210697467305755234393175, 0.0857426210697467305755234393185]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 = 0.085742620

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 = 111 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": [
    [
      "0.915850346",
      "0.08962349"
    ],
    [
      "1.160510696",
      "0.099942851"
    ],
    [
      "0.747325892",
      "0.121351976"
    ],
    [
      "1.328085934",
      "0.146730194"
    ],
    [
      "1.033120829",
      "0.214742721"
    ],
    [
      "0.52599009",
      "0.218219238"
    ],
    [
      "1.482766809",
      "0.223596263"
    ],
    [
      "1.196235366",
      "0.267665647"
    ],
    [
      "0.685456633",
      "0.281287535"
    ],
    [
      "0.888841927",
      "0.307546961"
    ],
    [
      "0.388337718",
      "0.320488552"
    ],
    [
      "1.353430711",
      "0.336198936"
    ],
    [
      "1.060199513",
      "0.384076513"
    ],
    [
      "0.552762545",
      "0.389912888"
    ],
    [
      "1.51575401",
      "0.391878861"
    ],
    [
      "1.687164315",
      "0.396947963"
    ],
    [
      "0.767086066",
      "0.432098091"
    ],
    [
      "1.220618872",
      "0.444680288"
    ],
    [
      "0.241609131",
      "0.489555976"
    ],
    [
      "0.413590432",
      "0.490104261"
    ],
    [
      "0.927505425",
      "0.492701866"
    ],
    [
      "1.381352517",
      "0.505395748"
    ],
    [
      "0.634391978",
      "0.540723444"
    ],
    [
      "1.087924784",
      "0.553305641"
    ],
    [
      "1.543538903",
      "0.56109821"
    ],
    [
      "1.714949208",
      "0.566167312"
    ],
    [
      "0.794811337",
      "0.601327219"
    ],
    [
      "1.248540678",
      "0.613877099"
    ],
    [
      "0.330462559",
      "0.640094078"
    ],
    [
      "0.158982696",
      "0.641452305"
    ],
    [
      "0.50169789",
      "0.649348797"
    ],
    [
      "0.955230696",
      "0.661930993"
    ],
    [
      "1.852217047",
      "0.668952178"
    ],
    [
      "1.409077788",
      "0.674624875"
    ],
    [
      "0.662117249",
      "0.709952571"
    ],
    [
      "1.11584659",
      "0.722502452"
    ],
    [
      "1.571264174",
      "0.730327338"
    ],
    [
      "0.822536608",
      "0.770556346"
    ],
    [
      "1.276265949",
      "0.783106227"
    ],
    [
      "1.731683533",
      "0.790931112"
    ],
    [
      "0.278301166",
      "0.803453734"
    ],
    [
      "0.106821303",
      "0.804811962"
    ],
    [
      "0.529423161",
      "0.818577924"
    ],
    [
      "0.983152502",
      "0.831127805"
    ],
    [
      "1.436803059",
      "0.843854003"
    ],
    [
      "1.901388851",
      "0.847658473"
    ],
    [
      "0.68984252",
      "0.879181699"
    ],
    [
      "1.143571861",
      "0.89173158"
    ],
    [
      "1.598989445",
      "0.899556465"
    ],
    [
      "0.396855327",
      "0.927357323"
    ],
    [
      "0.850458413",
      "0.939753158"
    ],
    [
      "1.30399122",
      "0.952335354"
    ],
    [
      "1.759408804",
      "0.96016024"
    ],
    [
      "0.086083436",
      "0.975038661"
    ],
    [
      "0.557274686",
      "0.987961098"
    ],
    [
      "1.010877773",
      "1.000356933"
    ],
    [
      "1.46452833",
      "1.01308313"
    ],
    [
      "0.251859646",
      "1.018918455"
    ],
    [
      "1.913577382",
      "1.035255099"
    ],
    [
      "0.717764325",
      "1.048378511"
    ],
    [
      "1.171297132",
      "1.060960707"
    ],
    [
      "1.626714716",
      "1.068785593"
    ],
    [
      "0.409639534",
      "1.099282096"
    ],
    [
      "0.878183685",
      "1.108982285"
    ],
    [
      "1.331716491",
      "1.121564482"
    ],
    [
      "0.097498687",
      "1.146143542"
    ],
    [
      "0.584999957",
      "1.157190225"
    ],
    [
      "1.038603044",
      "1.16958606"
    ],
    [
      "1.822999706",
      "1.180867162"
    ],
    [
      "1.497381776",
      "1.181391892"
    ],
    [
      "0.263274897",
      "1.190023336"
    ],
    [
      "0.745489596",
      "1.217607638"
    ],
    [
      "1.199022403",
      "1.230189835"
    ],
    [
      "1.660902429",
      "1.23682841"
    ],
    [
      "0.452305869",
      "1.265815578"
    ],
    [
      "0.905908956",
      "1.278211413"
    ],
    [
      "1.359441762",
      "1.29079361"
    ],
    [
      "0.140665449",
      "1.312106848"
    ],
    [
      "0.612725228",
      "1.326419353"
    ],
    [
      "1.066328315",
      "1.338815188"
    ],
    [
      "1.844178774",
      "1.351039531"
    ],
    [
      "1.519861121",
      "1.351397384"
    ],
    [
      "0.30644166",
      "1.355986642"
    ],
    [
      "0.773214868",
      "1.386836766"
    ],
    [
      "1.226747674",
      "1.399418962"
    ],
    [
      "1.682081498",
      "1.40700078"
    ],
    [
      "0.933634227",
      "1.447440541"
    ],
    [
      "0.470906244",
      "1.450113023"
    ],
    [
      "1.387167033",
      "1.460022737"
    ],
    [
      "0.640647033",
      "1.495616164"
    ],
    [
      "1.094053586",
      "1.508044315"
    ],
    [
      "0.245452176",
      "1.516259756"
    ],
    [
      "1.547586392",
      "1.520626512"
    ],
    [
      "0.798132436",
      "1.563808432"
    ],
    [
      "1.254472945",
      "1.56864809"
    ],
    [
      "1.709806769",
      "1.576229907"
    ],
    [
      "0.961359498",
      "1.616669668"
    ],
    [
      "0.524602644",
      "1.621873394"
    ],
    [
      "1.416186709",
      "1.630814132"
    ],
    [
      "0.355132223",
      "1.648083409"
    ],
    [
      "1.121778857",
      "1.677273443"
    ],
    [
      "0.681763834",
      "1.69048497"
    ],
    [
      "1.587595079",
      "1.700315513"
    ],
    [
      "1.282225381",
      "1.737872765"
    ],
    [
      "0.845097783",
      "1.742726806"
    ],
    [
      "0.543763871",
      "1.792284765"
    ],
    [
      "1.44192154",
      "1.800357362"
    ],
    [
      "1.044564845",
      "1.830391648"
    ],
    [
      "0.722559589",
      "1.871054885"
    ],
    [
      "1.204989964",
      "1.890980174"
    ],
    [
      "0.89154574",
      "1.907801866"
    ]
  ],
  "radius": "0.08574262"
}
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": 111
}

起步布局答案

{
  "centers": [
    [
      "0.915850346",
      "0.08962349"
    ],
    [
      "1.160510696",
      "0.099942851"
    ],
    [
      "0.747325892",
      "0.121351976"
    ],
    [
      "1.328085934",
      "0.146730194"
    ],
    [
      "1.033120829",
      "0.214742721"
    ],
    [
      "0.52599009",
      "0.218219238"
    ],
    [
      "1.482766809",
      "0.223596263"
    ],
    [
      "1.196235366",
      "0.267665647"
    ],
    [
      "0.685456633",
      "0.281287535"
    ],
    [
      "0.888841927",
      "0.307546961"
    ],
    [
      "0.388337718",
      "0.320488552"
    ],
    [
      "1.353430711",
      "0.336198936"
    ],
    [
      "1.060199513",
      "0.384076513"
    ],
    [
      "0.552762545",
      "0.389912888"
    ],
    [
      "1.51575401",
      "0.391878861"
    ],
    [
      "1.687164315",
      "0.396947963"
    ],
    [
      "0.767086066",
      "0.432098091"
    ],
    [
      "1.220618872",
      "0.444680288"
    ],
    [
      "0.241609131",
      "0.489555976"
    ],
    [
      "0.413590432",
      "0.490104261"
    ],
    [
      "0.927505425",
      "0.492701866"
    ],
    [
      "1.381352517",
      "0.505395748"
    ],
    [
      "0.634391978",
      "0.540723444"
    ],
    [
      "1.087924784",
      "0.553305641"
    ],
    [
      "1.543538903",
      "0.56109821"
    ],
    [
      "1.714949208",
      "0.566167312"
    ],
    [
      "0.794811337",
      "0.601327219"
    ],
    [
      "1.248540678",
      "0.613877099"
    ],
    [
      "0.330462559",
      "0.640094078"
    ],
    [
      "0.158982696",
      "0.641452305"
    ],
    [
      "0.50169789",
      "0.649348797"
    ],
    [
      "0.955230696",
      "0.661930993"
    ],
    [
      "1.852217047",
      "0.668952178"
    ],
    [
      "1.409077788",
      "0.674624875"
    ],
    [
      "0.662117249",
      "0.709952571"
    ],
    [
      "1.11584659",
      "0.722502452"
    ],
    [
      "1.571264174",
      "0.730327338"
    ],
    [
      "0.822536608",
      "0.770556346"
    ],
    [
      "1.276265949",
      "0.783106227"
    ],
    [
      "1.731683533",
      "0.790931112"
    ],
    [
      "0.278301166",
      "0.803453734"
    ],
    [
      "0.106821303",
      "0.804811962"
    ],
    [
      "0.529423161",
      "0.818577924"
    ],
    [
      "0.983152502",
      "0.831127805"
    ],
    [
      "1.436803059",
      "0.843854003"
    ],
    [
      "1.901388851",
      "0.847658473"
    ],
    [
      "0.68984252",
      "0.879181699"
    ],
    [
      "1.143571861",
      "0.89173158"
    ],
    [
      "1.598989445",
      "0.899556465"
    ],
    [
      "0.396855327",
      "0.927357323"
    ],
    [
      "0.850458413",
      "0.939753158"
    ],
    [
      "1.30399122",
      "0.952335354"
    ],
    [
      "1.759408804",
      "0.96016024"
    ],
    [
      "0.086083436",
      "0.975038661"
    ],
    [
      "0.557274686",
      "0.987961098"
    ],
    [
      "1.010877773",
      "1.000356933"
    ],
    [
      "1.46452833",
      "1.01308313"
    ],
    [
      "0.251859646",
      "1.018918455"
    ],
    [
      "1.913577382",
      "1.035255099"
    ],
    [
      "0.717764325",
      "1.048378511"
    ],
    [
      "1.171297132",
      "1.060960707"
    ],
    [
      "1.626714716",
      "1.068785593"
    ],
    [
      "0.409639534",
      "1.099282096"
    ],
    [
      "0.878183685",
      "1.108982285"
    ],
    [
      "1.331716491",
      "1.121564482"
    ],
    [
      "0.097498687",
      "1.146143542"
    ],
    [
      "0.584999957",
      "1.157190225"
    ],
    [
      "1.038603044",
      "1.16958606"
    ],
    [
      "1.822999706",
      "1.180867162"
    ],
    [
      "1.497381776",
      "1.181391892"
    ],
    [
      "0.263274897",
      "1.190023336"
    ],
    [
      "0.745489596",
      "1.217607638"
    ],
    [
      "1.199022403",
      "1.230189835"
    ],
    [
      "1.660902429",
      "1.23682841"
    ],
    [
      "0.452305869",
      "1.265815578"
    ],
    [
      "0.905908956",
      "1.278211413"
    ],
    [
      "1.359441762",
      "1.29079361"
    ],
    [
      "0.140665449",
      "1.312106848"
    ],
    [
      "0.612725228",
      "1.326419353"
    ],
    [
      "1.066328315",
      "1.338815188"
    ],
    [
      "1.844178774",
      "1.351039531"
    ],
    [
      "1.519861121",
      "1.351397384"
    ],
    [
      "0.30644166",
      "1.355986642"
    ],
    [
      "0.773214868",
      "1.386836766"
    ],
    [
      "1.226747674",
      "1.399418962"
    ],
    [
      "1.682081498",
      "1.40700078"
    ],
    [
      "0.933634227",
      "1.447440541"
    ],
    [
      "0.470906244",
      "1.450113023"
    ],
    [
      "1.387167033",
      "1.460022737"
    ],
    [
      "0.640647033",
      "1.495616164"
    ],
    [
      "1.094053586",
      "1.508044315"
    ],
    [
      "0.245452176",
      "1.516259756"
    ],
    [
      "1.547586392",
      "1.520626512"
    ],
    [
      "0.798132436",
      "1.563808432"
    ],
    [
      "1.254472945",
      "1.56864809"
    ],
    [
      "1.709806769",
      "1.576229907"
    ],
    [
      "0.961359498",
      "1.616669668"
    ],
    [
      "0.524602644",
      "1.621873394"
    ],
    [
      "1.416186709",
      "1.630814132"
    ],
    [
      "0.355132223",
      "1.648083409"
    ],
    [
      "1.121778857",
      "1.677273443"
    ],
    [
      "0.681763834",
      "1.69048497"
    ],
    [
      "1.587595079",
      "1.700315513"
    ],
    [
      "1.282225381",
      "1.737872765"
    ],
    [
      "0.845097783",
      "1.742726806"
    ],
    [
      "0.543763871",
      "1.792284765"
    ],
    [
      "1.44192154",
      "1.800357362"
    ],
    [
      "1.044564845",
      "1.830391648"
    ],
    [
      "0.722559589",
      "1.871054885"
    ],
    [
      "1.204989964",
      "1.890980174"
    ],
    [
      "0.89154574",
      "1.907801866"
    ]
  ],
  "radius": "0.08574262"
}

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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