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 = 133

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 = 133
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[0.0785220696132153351823110424885, 0.0785220696132153351823110424895]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 = 0.078522069

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 = 133 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": [
    [
      "0.945160524",
      "0.080155335"
    ],
    [
      "1.102152524",
      "0.084201748"
    ],
    [
      "0.789761344",
      "0.102825946"
    ],
    [
      "1.256177489",
      "0.114847657"
    ],
    [
      "0.640468577",
      "0.15155511"
    ],
    [
      "1.420415927",
      "0.180017058"
    ],
    [
      "0.501618453",
      "0.224927481"
    ],
    [
      "0.977333666",
      "0.233868544"
    ],
    [
      "1.135649205",
      "0.23849074"
    ],
    [
      "0.770117052",
      "0.258636611"
    ],
    [
      "1.553548827",
      "0.263314672"
    ],
    [
      "1.290409983",
      "0.26811539"
    ],
    [
      "0.377243895",
      "0.320811948"
    ],
    [
      "0.625936628",
      "0.32088504"
    ],
    [
      "0.887654684",
      "0.362789224"
    ],
    [
      "1.67060382",
      "0.368009421"
    ],
    [
      "1.044142759",
      "0.37599316"
    ],
    [
      "1.199852681",
      "0.396420639"
    ],
    [
      "1.356246937",
      "0.410692938"
    ],
    [
      "1.513229373",
      "0.415094787"
    ],
    [
      "0.491219108",
      "0.42885135"
    ],
    [
      "0.270957381",
      "0.436423532"
    ],
    [
      "0.640622258",
      "0.477241024"
    ],
    [
      "1.768181026",
      "0.491060429"
    ],
    [
      "0.797056712",
      "0.491065758"
    ],
    [
      "0.953446435",
      "0.505387637"
    ],
    [
      "1.10929538",
      "0.524725888"
    ],
    [
      "1.6168651",
      "0.533088422"
    ],
    [
      "1.265689636",
      "0.538998187"
    ],
    [
      "0.384932594",
      "0.544462935"
    ],
    [
      "1.422446524",
      "0.553102475"
    ],
    [
      "0.185846016",
      "0.568404278"
    ],
    [
      "0.534335743",
      "0.592852609"
    ],
    [
      "0.706459063",
      "0.619818641"
    ],
    [
      "1.843446301",
      "0.628893662"
    ],
    [
      "0.862889134",
      "0.633692885"
    ],
    [
      "1.018738079",
      "0.653031137"
    ],
    [
      "1.532244622",
      "0.66538444"
    ],
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      "1.175132334",
      "0.667303436"
    ],
    [
      "1.692126688",
      "0.670958936"
    ],
    [
      "0.299821229",
      "0.676443681"
    ],
    [
      "1.331889223",
      "0.681407724"
    ],
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      "0.124381875",
      "0.712920786"
    ],
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      "0.449224379",
      "0.724833355"
    ],
    [
      "0.605521882",
      "0.740128853"
    ],
    [
      "0.77265865",
      "0.762228178"
    ],
    [
      "1.894213552",
      "0.777505733"
    ],
    [
      "0.928507595",
      "0.781566429"
    ],
    [
      "1.44168732",
      "0.793689688"
    ],
    [
      "1.084901851",
      "0.795838728"
    ],
    [
      "1.607309388",
      "0.803326973"
    ],
    [
      "1.241331922",
      "0.809712973"
    ],
    [
      "0.238357088",
      "0.820960188"
    ],
    [
      "1.756371986",
      "0.852755745"
    ],
    [
      "0.088184438",
      "0.86690621"
    ],
    [
      "0.387760237",
      "0.869349862"
    ],
    [
      "0.54405774",
      "0.884645361"
    ],
    [
      "0.700189215",
      "0.901551696"
    ],
    [
      "0.85603816",
      "0.920889947"
    ],
    [
      "1.91900824",
      "0.932580184"
    ],
    [
      "1.012432416",
      "0.935162246"
    ],
    [
      "1.325416837",
      "0.942350029"
    ],
    [
      "1.482432202",
      "0.945356152"
    ],
    [
      "1.168634091",
      "0.951407198"
    ],
    [
      "1.639034115",
      "0.957133356"
    ],
    [
      "0.271779085",
      "0.975778895"
    ],
    [
      "0.452631053",
      "1.012369578"
    ],
    [
      "1.785146398",
      "1.014701247"
    ],
    [
      "0.078826158",
      "1.02367127"
    ],
    [
      "0.608766244",
      "1.029241563"
    ],
    [
      "0.764615189",
      "1.048579814"
    ],
    [
      "0.920909343",
      "1.063909496"
    ],
    [
      "1.077216964",
      "1.079101249"
    ],
    [
      "1.393518436",
      "1.083859866"
    ],
    [
      "1.550533801",
      "1.086865989"
    ],
    [
      "1.23673569",
      "1.092917034"
    ],
    [
      "1.915840183",
      "1.101775907"
    ],
    [
      "0.204328378",
      "1.118074948"
    ],
    [
      "0.35995189",
      "1.139150627"
    ],
    [
      "0.516087081",
      "1.156022612"
    ],
    [
      "1.754563833",
      "1.168738802"
    ],
    [
      "0.671936026",
      "1.175360863"
    ],
    [
      "0.828071217",
      "1.192232848"
    ],
    [
      "0.984365371",
      "1.20756253"
    ],
    [
      "1.140672991",
      "1.222754283"
    ],
    [
      "1.453763358",
      "1.228888866"
    ],
    [
      "1.610778723",
      "1.231894989"
    ],
    [
      "1.296980612",
      "1.237946035"
    ],
    [
      "0.111649215",
      "1.244855997"
    ],
    [
      "1.885257617",
      "1.255813462"
    ],
    [
      "0.267272727",
      "1.265931675"
    ],
    [
      "0.423407918",
      "1.282803661"
    ],
    [
      "0.579256863",
      "1.302141912"
    ],
    [
      "0.735392054",
      "1.319013897"
    ],
    [
      "0.891527244",
      "1.335885882"
    ],
    [
      "1.047821398",
      "1.351215565"
    ],
    [
      "1.204129019",
      "1.366407317"
    ],
    [
      "1.517219386",
      "1.372541901"
    ],
    [
      "1.674234751",
      "1.375548024"
    ],
    [
      "1.36043664",
      "1.381599069"
    ],
    [
      "0.170051546",
      "1.399102446"
    ],
    [
      "1.828962613",
      "1.402420877"
    ],
    [
      "0.330728755",
      "1.40958471"
    ],
    [
      "0.485477459",
      "1.43714933"
    ],
    [
      "0.64271289",
      "1.445794946"
    ],
    [
      "0.798848081",
      "1.462666931"
    ],
    [
      "0.956398427",
      "1.478905431"
    ],
    [
      "1.112818765",
      "1.494177811"
    ],
    [
      "1.269126386",
      "1.509369563"
    ],
    [
      "1.593123689",
      "1.510024257"
    ],
    [
      "1.435356324",
      "1.519803343"
    ],
    [
      "1.748059404",
      "1.537847913"
    ],
    [
      "0.260816938",
      "1.55020903"
    ],
    [
      "0.557326566",
      "1.577597971"
    ],
    [
      "0.713461757",
      "1.594469956"
    ],
    [
      "0.869747927",
      "1.60988082"
    ],
    [
      "0.404476607",
      "1.613650037"
    ],
    [
      "1.026055001",
      "1.625078197"
    ],
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      "1.182362622",
      "1.640269949"
    ],
    [
      "1.343073838",
      "1.647914271"
    ],
    [
      "1.644758834",
      "1.658337013"
    ],
    [
      "1.492418352",
      "1.696484611"
    ],
    [
      "0.626301061",
      "1.725106378"
    ],
    [
      "0.782587231",
      "1.740517242"
    ],
    [
      "0.472195324",
      "1.755343498"
    ],
    [
      "1.091052368",
      "1.768040443"
    ],
    [
      "1.256310074",
      "1.778814656"
    ],
    [
      "0.934492416",
      "1.780362902"
    ],
    [
      "1.405654588",
      "1.827384996"
    ],
    [
      "0.691097947",
      "1.868159604"
    ],
    [
      "1.16499982",
      "1.906585151"
    ],
    [
      "0.843003131",
      "1.908005264"
    ],
    [
      "1.0082121",
      "1.919091094"
    ]
  ],
  "radius": "0.078522069"
}
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": 133
}

起步布局答案

{
  "centers": [
    [
      "0.945160524",
      "0.080155335"
    ],
    [
      "1.102152524",
      "0.084201748"
    ],
    [
      "0.789761344",
      "0.102825946"
    ],
    [
      "1.256177489",
      "0.114847657"
    ],
    [
      "0.640468577",
      "0.15155511"
    ],
    [
      "1.420415927",
      "0.180017058"
    ],
    [
      "0.501618453",
      "0.224927481"
    ],
    [
      "0.977333666",
      "0.233868544"
    ],
    [
      "1.135649205",
      "0.23849074"
    ],
    [
      "0.770117052",
      "0.258636611"
    ],
    [
      "1.553548827",
      "0.263314672"
    ],
    [
      "1.290409983",
      "0.26811539"
    ],
    [
      "0.377243895",
      "0.320811948"
    ],
    [
      "0.625936628",
      "0.32088504"
    ],
    [
      "0.887654684",
      "0.362789224"
    ],
    [
      "1.67060382",
      "0.368009421"
    ],
    [
      "1.044142759",
      "0.37599316"
    ],
    [
      "1.199852681",
      "0.396420639"
    ],
    [
      "1.356246937",
      "0.410692938"
    ],
    [
      "1.513229373",
      "0.415094787"
    ],
    [
      "0.491219108",
      "0.42885135"
    ],
    [
      "0.270957381",
      "0.436423532"
    ],
    [
      "0.640622258",
      "0.477241024"
    ],
    [
      "1.768181026",
      "0.491060429"
    ],
    [
      "0.797056712",
      "0.491065758"
    ],
    [
      "0.953446435",
      "0.505387637"
    ],
    [
      "1.10929538",
      "0.524725888"
    ],
    [
      "1.6168651",
      "0.533088422"
    ],
    [
      "1.265689636",
      "0.538998187"
    ],
    [
      "0.384932594",
      "0.544462935"
    ],
    [
      "1.422446524",
      "0.553102475"
    ],
    [
      "0.185846016",
      "0.568404278"
    ],
    [
      "0.534335743",
      "0.592852609"
    ],
    [
      "0.706459063",
      "0.619818641"
    ],
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      "1.843446301",
      "0.628893662"
    ],
    [
      "0.862889134",
      "0.633692885"
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      "0.653031137"
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      "0.66538444"
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      "1.692126688",
      "0.670958936"
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      "0.762228178"
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      "0.777505733"
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      "1.20756253"
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      "1.265931675"
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      "1.282803661"
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      "1.319013897"
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      "1.335885882"
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      "1.047821398",
      "1.351215565"
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      "1.366407317"
    ],
    [
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      "1.372541901"
    ],
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      "1.375548024"
    ],
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      "1.36043664",
      "1.381599069"
    ],
    [
      "0.170051546",
      "1.399102446"
    ],
    [
      "1.828962613",
      "1.402420877"
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      "0.330728755",
      "1.40958471"
    ],
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      "1.43714933"
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      "1.445794946"
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    [
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      "1.462666931"
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    [
      "1.0082121",
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  ],
  "radius": "0.078522069"
}

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?