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 = 157

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 = 157
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[0.0726058656851315766086957064955, 0.0726058656851315766086957064965]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 = 0.072605865

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 = 157 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": [
    [
      "0.995031701",
      "0.072619174"
    ],
    [
      "1.139856549",
      "0.083212115"
    ],
    [
      "0.850328664",
      "0.084763216"
    ],
    [
      "1.281252476",
      "0.116282327"
    ],
    [
      "0.70929518",
      "0.119346499"
    ],
    [
      "1.415752818",
      "0.171019014"
    ],
    [
      "0.575389028",
      "0.175521132"
    ],
    [
      "0.954464962",
      "0.212049395"
    ],
    [
      "0.809761924",
      "0.224193437"
    ],
    [
      "1.101407707",
      "0.225969657"
    ],
    [
      "0.451893241",
      "0.251909858"
    ],
    [
      "1.552659245",
      "0.256711083"
    ],
    [
      "1.242866632",
      "0.259856521"
    ],
    [
      "1.419394672",
      "0.31618507"
    ],
    [
      "0.579237805",
      "0.321693873"
    ],
    [
      "0.723190137",
      "0.340777195"
    ],
    [
      "1.012788724",
      "0.345033551"
    ],
    [
      "0.341835615",
      "0.346639821"
    ],
    [
      "1.665381666",
      "0.353991394"
    ],
    [
      "0.868085686",
      "0.357177593"
    ],
    [
      "1.153878741",
      "0.379385466"
    ],
    [
      "1.29812913",
      "0.396066954"
    ],
    [
      "1.53101862",
      "0.409064245"
    ],
    [
      "0.46918018",
      "0.416423837"
    ],
    [
      "0.63661835",
      "0.457360952"
    ],
    [
      "0.247914481",
      "0.457388489"
    ],
    [
      "1.758066747",
      "0.465776557"
    ],
    [
      "0.781513899",
      "0.473761351"
    ],
    [
      "0.933911247",
      "0.486612682"
    ],
    [
      "1.078161637",
      "0.50329417"
    ],
    [
      "1.222412027",
      "0.519975659"
    ],
    [
      "1.367683548",
      "0.523763743"
    ],
    [
      "0.376461019",
      "0.531599798"
    ],
    [
      "0.525143138",
      "0.550418591"
    ],
    [
      "1.510137347",
      "0.552766783"
    ],
    [
      "1.654584514",
      "0.56764841"
    ],
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      "0.17243254",
      "0.58144059"
    ],
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      "0.694942111",
      "0.590345108"
    ],
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      "1.83216599",
      "0.590659499"
    ],
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      "0.84733946",
      "0.60319644"
    ],
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      "0.99158985",
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      "0.653240904"
    ],
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      "0.657545297"
    ],
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      "0.44088723",
      "0.668686883"
    ],
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      "1.424341018",
      "0.669922393"
    ],
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      "0.58513762",
      "0.685368371"
    ],
    [
      "1.568698917",
      "0.685646388"
    ],
    [
      "1.728683757",
      "0.692531352"
    ],
    [
      "0.117240417",
      "0.715754686"
    ],
    [
      "0.760767673",
      "0.719780198"
    ],
    [
      "1.885862672",
      "0.725578413"
    ],
    [
      "0.905018062",
      "0.736461686"
    ],
    [
      "1.049268452",
      "0.753143174"
    ],
    [
      "1.193518842",
      "0.769824662"
    ],
    [
      "1.337769231",
      "0.78650615"
    ],
    [
      "0.240911434",
      "0.791859394"
    ],
    [
      "0.385695107",
      "0.803000979"
    ],
    [
      "1.482019621",
      "0.803187638"
    ],
    [
      "1.633099565",
      "0.815796314"
    ],
    [
      "0.529945496",
      "0.819682468"
    ],
    [
      "1.777689123",
      "0.829224149"
    ],
    [
      "0.674195886",
      "0.836363956"
    ],
    [
      "0.818446275",
      "0.853045444"
    ],
    [
      "0.113167315",
      "0.860909282"
    ],
    [
      "1.917840289",
      "0.867225435"
    ],
    [
      "0.962696665",
      "0.869726932"
    ],
    [
      "1.106947054",
      "0.88640842"
    ],
    [
      "1.251197444",
      "0.903089908"
    ],
    [
      "1.395447833",
      "0.919771396"
    ],
    [
      "0.298590036",
      "0.925124639"
    ],
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      "1.54725872",
      "0.932919311"
    ],
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      "0.443373709",
      "0.936266225"
    ],
    [
      "1.693765498",
      "0.947728462"
    ],
    [
      "0.587624098",
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    ],
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      "0.731874488",
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      "0.876124878",
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    ],
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      "0.072605928",
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    ],
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    ],
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      "1.036355154"
    ],
    [
      "0.211840043",
      "1.041575855"
    ],
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      "1.453126436",
      "1.053036642"
    ],
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      "0.356268638",
      "1.058389885"
    ],
    [
      "1.597437394",
      "1.069185769"
    ],
    [
      "0.501052311",
      "1.069531471"
    ],
    [
      "1.74181137",
      "1.084761455"
    ],
    [
      "0.645302701",
      "1.086212959"
    ],
    [
      "0.78955309",
      "1.102894447"
    ],
    [
      "0.93380348",
      "1.119575936"
    ],
    [
      "1.917706254",
      "1.133697835"
    ],
    [
      "1.078053869",
      "1.136257424"
    ],
    [
      "0.084027822",
      "1.14510289"
    ],
    [
      "1.222304259",
      "1.152938912"
    ],
    [
      "1.366554648",
      "1.1696204"
    ],
    [
      "1.510805038",
      "1.186301888"
    ],
    [
      "0.223261936",
      "1.186337685"
    ],
    [
      "1.655115996",
      "1.202451015"
    ],
    [
      "0.367495154",
      "1.203166998"
    ],
    [
      "0.512278827",
      "1.214308584"
    ],
    [
      "1.799489972",
      "1.218026701"
    ],
    [
      "0.702981303",
      "1.219478205"
    ],
    [
      "0.847231693",
      "1.236159693"
    ],
    [
      "0.991482082",
      "1.252841181"
    ],
    [
      "1.135732472",
      "1.26952267"
    ],
    [
      "1.279982861",
      "1.286204158"
    ],
    [
      "0.125994217",
      "1.298823021"
    ],
    [
      "1.424642503",
      "1.303188433"
    ],
    [
      "1.568974963",
      "1.319353407"
    ],
    [
      "0.270415721",
      "1.323680204"
    ],
    [
      "0.604663085",
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    ],
    [
      "1.713285921",
      "1.335502534"
    ],
    [
      "0.414648939",
      "1.340509517"
    ],
    [
      "1.858529678",
      "1.350694557"
    ],
    [
      "0.760659905",
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      "0.904910295",
      "1.369424939"
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      "1.437258371"
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      "1.459607734"
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      "1.772850243",
      "1.467935681"
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      "0.817701583",
      "1.486447687"
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      "0.962588897",
      "1.502690185"
    ],
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      "1.106839287",
      "1.519371673"
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      "1.251051409",
      "1.538121412"
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      "1.395867528",
      "1.552505117"
    ],
    [
      "0.430796377",
      "1.559525987"
    ],
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      "1.539679301",
      "1.572620307"
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      "0.575051856",
      "1.576163405"
    ],
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      "1.593199114"
    ],
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      "0.290590827",
      "1.59732613"
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    [
      "1.708675047",
      "1.598196923"
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    [
      "0.857077886",
      "1.638947439"
    ],
    [
      "1.002249018",
      "1.642380993"
    ],
    [
      "1.146499407",
      "1.659062481"
    ],
    [
      "1.376625441",
      "1.696436312"
    ],
    [
      "0.392540985",
      "1.700731218"
    ],
    [
      "1.606609198",
      "1.70148782"
    ],
    [
      "0.632402991",
      "1.709569904"
    ],
    [
      "0.770220035",
      "1.755318229"
    ],
    [
      "1.247681636",
      "1.763219111"
    ],
    [
      "0.916052181",
      "1.776165888"
    ],
    [
      "1.061263223",
      "1.776950546"
    ],
    [
      "0.509366528",
      "1.786976447"
    ],
    [
      "1.489670858",
      "1.787580048"
    ],
    [
      "1.360727053",
      "1.854362846"
    ],
    [
      "0.639816776",
      "1.854592257"
    ],
    [
      "0.77763382",
      "1.900340581"
    ],
    [
      "1.209698142",
      "1.903375099"
    ],
    [
      "0.920902709",
      "1.92401488"
    ],
    [
      "1.066110859",
      "1.925034721"
    ]
  ],
  "radius": "0.072605865"
}
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": 157
}

起步布局答案

{
  "centers": [
    [
      "0.995031701",
      "0.072619174"
    ],
    [
      "1.139856549",
      "0.083212115"
    ],
    [
      "0.850328664",
      "0.084763216"
    ],
    [
      "1.281252476",
      "0.116282327"
    ],
    [
      "0.70929518",
      "0.119346499"
    ],
    [
      "1.415752818",
      "0.171019014"
    ],
    [
      "0.575389028",
      "0.175521132"
    ],
    [
      "0.954464962",
      "0.212049395"
    ],
    [
      "0.809761924",
      "0.224193437"
    ],
    [
      "1.101407707",
      "0.225969657"
    ],
    [
      "0.451893241",
      "0.251909858"
    ],
    [
      "1.552659245",
      "0.256711083"
    ],
    [
      "1.242866632",
      "0.259856521"
    ],
    [
      "1.419394672",
      "0.31618507"
    ],
    [
      "0.579237805",
      "0.321693873"
    ],
    [
      "0.723190137",
      "0.340777195"
    ],
    [
      "1.012788724",
      "0.345033551"
    ],
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      "0.346639821"
    ],
    [
      "1.665381666",
      "0.353991394"
    ],
    [
      "0.868085686",
      "0.357177593"
    ],
    [
      "1.153878741",
      "0.379385466"
    ],
    [
      "1.29812913",
      "0.396066954"
    ],
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      "1.53101862",
      "0.409064245"
    ],
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      "0.416423837"
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      "0.753143174"
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      "1.193518842",
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      "0.78650615"
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      "0.240911434",
      "0.791859394"
    ],
    [
      "0.385695107",
      "0.803000979"
    ],
    [
      "1.482019621",
      "0.803187638"
    ],
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      "1.633099565",
      "0.815796314"
    ],
    [
      "0.529945496",
      "0.819682468"
    ],
    [
      "1.777689123",
      "0.829224149"
    ],
    [
      "0.674195886",
      "0.836363956"
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      "0.860909282"
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      "0.962696665",
      "0.869726932"
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      "0.88640842"
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      "0.903089908"
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      "0.919771396"
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      "0.925124639"
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      "0.936266225"
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      "0.947728462"
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      "0.952947713"
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      "0.969629202"
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      "0.98631069"
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      "1.00034106"
    ],
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      "1.860027652",
      "1.000432589"
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    [
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    [
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      "1.041575855"
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  "radius": "0.072605865"
}

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

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还没有帖子。第一个聊聊这道题的思路?