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 = 146

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 = 146
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[0.0750118119146680603319182609585, 0.0750118119146680603319182609595]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 = 0.075011811

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 = 146 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": [
    [
      "1.074072081",
      "0.077982389"
    ],
    [
      "0.917581281",
      "0.078758396"
    ],
    [
      "1.222147122",
      "0.102083632"
    ],
    [
      "0.769001215",
      "0.104319974"
    ],
    [
      "1.438665764",
      "0.185644122"
    ],
    [
      "0.552968161",
      "0.190206395"
    ],
    [
      "0.998231815",
      "0.207424815"
    ],
    [
      "0.85000799",
      "0.230593447"
    ],
    [
      "1.146147454",
      "0.23248577"
    ],
    [
      "1.296171052",
      "0.23257327"
    ],
    [
      "0.696023541",
      "0.23539761"
    ],
    [
      "1.513601786",
      "0.315612119"
    ],
    [
      "0.477956349",
      "0.320130664"
    ],
    [
      "1.659332461",
      "0.35124438"
    ],
    [
      "0.333070676",
      "0.359056613"
    ],
    [
      "1.071059877",
      "0.362366267"
    ],
    [
      "1.221083476",
      "0.362453767"
    ],
    [
      "1.371107074",
      "0.362541267"
    ],
    [
      "0.921036713",
      "0.36273734"
    ],
    [
      "0.621011729",
      "0.365321879"
    ],
    [
      "0.771035353",
      "0.365321879"
    ],
    [
      "1.755535386",
      "0.466362081"
    ],
    [
      "0.238230562",
      "0.475299636"
    ],
    [
      "1.595973893",
      "0.48723254"
    ],
    [
      "1.145995899",
      "0.492334264"
    ],
    [
      "1.296019498",
      "0.492421764"
    ],
    [
      "1.446043096",
      "0.492509264"
    ],
    [
      "0.995972735",
      "0.492705337"
    ],
    [
      "0.545924153",
      "0.495202377"
    ],
    [
      "0.395900555",
      "0.495289847"
    ],
    [
      "0.695947751",
      "0.495289877"
    ],
    [
      "0.845971375",
      "0.495289877"
    ],
    [
      "1.692176818",
      "0.60235024"
    ],
    [
      "0.30106044",
      "0.611532871"
    ],
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      "1.525622006",
      "0.619777809"
    ],
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      "1.84156758",
      "0.620616408"
    ],
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      "1.220931814",
      "0.622302601"
    ],
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      "1.37095552",
      "0.622389761"
    ],
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      "1.07090854",
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    ],
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      "0.620859933",
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    ],
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      "0.470836309",
      "0.625257999"
    ],
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      "0.770883557",
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    ],
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    ],
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      "0.152190764",
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    ],
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      "1.745693492",
      "0.742503918"
    ],
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      "0.245858311",
      "0.751031303"
    ],
    [
      "1.295943708",
      "0.752314031"
    ],
    [
      "1.445967332",
      "0.752314031"
    ],
    [
      "1.595990956",
      "0.752314031"
    ],
    [
      "1.145920352",
      "0.752597728"
    ],
    [
      "0.395824497",
      "0.755182268"
    ],
    [
      "0.545848121",
      "0.755182268"
    ],
    [
      "0.695871745",
      "0.755182268"
    ],
    [
      "0.845895369",
      "0.755182268"
    ],
    [
      "0.995918993",
      "0.755182268"
    ],
    [
      "1.894241525",
      "0.76349367"
    ],
    [
      "0.100037524",
      "0.786292981"
    ],
    [
      "1.37095552",
      "0.8822383"
    ],
    [
      "1.520979144",
      "0.8822383"
    ],
    [
      "1.671002767",
      "0.8822383"
    ],
    [
      "1.220932164",
      "0.882521998"
    ],
    [
      "0.319312449",
      "0.884240376"
    ],
    [
      "0.470836309",
      "0.885106537"
    ],
    [
      "0.620859933",
      "0.885106537"
    ],
    [
      "0.770883557",
      "0.885106537"
    ],
    [
      "0.920907181",
      "0.885106537"
    ],
    [
      "1.070930805",
      "0.885106537"
    ],
    [
      "1.820551011",
      "0.894171886"
    ],
    [
      "0.118820862",
      "0.935136099"
    ],
    [
      "1.924986284",
      "1.00187686"
    ],
    [
      "1.445967332",
      "1.01216257"
    ],
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      "1.595990956",
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    ],
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      "1.295943976",
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    ],
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    ],
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    ],
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      "0.695871745",
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    ],
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      "1.015030807"
    ],
    [
      "0.995918993",
      "1.015030807"
    ],
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      "1.145942617",
      "1.015030807"
    ],
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      "1.779493404",
      "1.038467972"
    ],
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      "0.078445289",
      "1.07962451"
    ],
    [
      "1.520979144",
      "1.142086839"
    ],
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      "1.671002767",
      "1.142086839"
    ],
    [
      "1.370955788",
      "1.142370537"
    ],
    [
      "0.320812685",
      "1.144955076"
    ],
    [
      "0.470836309",
      "1.144955076"
    ],
    [
      "0.620859933",
      "1.144955076"
    ],
    [
      "0.770883557",
      "1.144955076"
    ],
    [
      "0.920907181",
      "1.144955076"
    ],
    [
      "1.070930805",
      "1.144955076"
    ],
    [
      "1.220954428",
      "1.144955076"
    ],
    [
      "1.912516755",
      "1.15138137"
    ],
    [
      "0.103438024",
      "1.227551691"
    ],
    [
      "1.788341019",
      "1.235568508"
    ],
    [
      "1.595990956",
      "1.272011108"
    ],
    [
      "1.4459676",
      "1.272294806"
    ],
    [
      "0.245800874",
      "1.274879346"
    ],
    [
      "0.395824497",
      "1.274879346"
    ],
    [
      "0.545848121",
      "1.274879346"
    ],
    [
      "0.695871745",
      "1.274879346"
    ],
    [
      "0.845895369",
      "1.274879346"
    ],
    [
      "0.995918993",
      "1.274879346"
    ],
    [
      "1.145942617",
      "1.274879346"
    ],
    [
      "1.29596624",
      "1.274879346"
    ],
    [
      "1.845970407",
      "1.374081833"
    ],
    [
      "1.696249922",
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    [
      "1.520978895",
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      "1.667846355"
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      "1.745657597"
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      "0.8380472",
      "1.789819616"
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      "1.791811095"
    ],
    [
      "1.142294468",
      "1.796648704"
    ],
    [
      "0.580383581",
      "1.824333191"
    ],
    [
      "1.401934237",
      "1.833097844"
    ],
    [
      "0.719160758",
      "1.88132427"
    ],
    [
      "1.261972775",
      "1.88711522"
    ],
    [
      "0.911327732",
      "1.920728178"
    ],
    [
      "1.061334866",
      "1.922952427"
    ]
  ],
  "radius": "0.075011811"
}
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": 146
}

起步布局答案

{
  "centers": [
    [
      "1.074072081",
      "0.077982389"
    ],
    [
      "0.917581281",
      "0.078758396"
    ],
    [
      "1.222147122",
      "0.102083632"
    ],
    [
      "0.769001215",
      "0.104319974"
    ],
    [
      "1.438665764",
      "0.185644122"
    ],
    [
      "0.552968161",
      "0.190206395"
    ],
    [
      "0.998231815",
      "0.207424815"
    ],
    [
      "0.85000799",
      "0.230593447"
    ],
    [
      "1.146147454",
      "0.23248577"
    ],
    [
      "1.296171052",
      "0.23257327"
    ],
    [
      "0.696023541",
      "0.23539761"
    ],
    [
      "1.513601786",
      "0.315612119"
    ],
    [
      "0.477956349",
      "0.320130664"
    ],
    [
      "1.659332461",
      "0.35124438"
    ],
    [
      "0.333070676",
      "0.359056613"
    ],
    [
      "1.071059877",
      "0.362366267"
    ],
    [
      "1.221083476",
      "0.362453767"
    ],
    [
      "1.371107074",
      "0.362541267"
    ],
    [
      "0.921036713",
      "0.36273734"
    ],
    [
      "0.621011729",
      "0.365321879"
    ],
    [
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      "0.365321879"
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    ],
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      "1.144955076"
    ],
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      "1.144955076"
    ],
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  "radius": "0.075011811"
}

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?