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 = 140

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 = 140
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[0.0765709828946940516930464929835, 0.0765709828946940516930464929845]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 = 0.076570982

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 = 140 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": [
    [
      "0.22336012",
      "1.031604639"
    ],
    [
      "0.911031644",
      "1.919133168"
    ],
    [
      "1.254211499",
      "1.172760742"
    ],
    [
      "0.25039181",
      "0.46268273"
    ],
    [
      "0.448914384",
      "0.656883633"
    ],
    [
      "0.377922416",
      "1.471824961"
    ],
    [
      "0.804910317",
      "1.607860975"
    ],
    [
      "1.031882262",
      "1.383425303"
    ],
    [
      "1.218573378",
      "1.023823218"
    ],
    [
      "0.813803279",
      "1.041945011"
    ],
    [
      "1.064159696",
      "1.921197418"
    ],
    [
      "1.000494395",
      "0.682342927"
    ],
    [
      "0.627077904",
      "1.40166251"
    ],
    [
      "0.924967898",
      "0.936612731"
    ],
    [
      "0.509731643",
      "1.549791552"
    ],
    [
      "1.076020893",
      "0.428073123"
    ],
    [
      "0.591474042",
      "1.252609572"
    ],
    [
      "1.369626372",
      "0.51528361"
    ],
    [
      "1.516429112",
      "0.558888854"
    ],
    [
      "1.459073612",
      "1.801231907"
    ],
    [
      "1.740769093",
      "1.461604804"
    ],
    [
      "1.849249856",
      "0.637678437"
    ],
    [
      "0.778165158",
      "0.893007488"
    ],
    [
      "1.365376117",
      "1.067428462"
    ],
    [
      "1.320341514",
      "1.866084559"
    ],
    [
      "0.607004401",
      "1.83562887"
    ],
    [
      "1.604686308",
      "0.30209197"
    ],
    [
      "0.49919642",
      "0.381594762"
    ],
    [
      "0.5597672",
      "0.551189658"
    ],
    [
      "0.960606019",
      "1.085550255"
    ],
    [
      "0.631362418",
      "0.849402244"
    ],
    [
      "1.663231852",
      "0.602494097"
    ],
    [
      "1.175575775",
      "1.816131157"
    ],
    [
      "1.028877684",
      "1.772175133"
    ],
    [
      "1.427981762",
      "1.651279386"
    ],
    [
      "0.889329777",
      "0.787675207"
    ],
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      "1.897443667",
      "0.783073619"
    ],
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      "0.706888916",
      "0.59513244"
    ],
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      "1.405264494",
      "0.664221134"
    ],
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      "0.702638661",
      "1.147277292"
    ],
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      "0.555835921",
      "1.103672048"
    ],
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      "0.64605658",
      "0.147096016"
    ],
    [
      "0.325296669",
      "0.765091996"
    ],
    [
      "0.349122939",
      "0.34518017"
    ],
    [
      "1.258461754",
      "0.62061589"
    ],
    [
      "0.849441401",
      "1.190882535"
    ],
    [
      "0.595724297",
      "0.70046472"
    ],
    [
      "0.758934684",
      "1.891349923"
    ],
    [
      "1.512178859",
      "1.111033699"
    ],
    [
      "0.155215918",
      "1.221100833"
    ],
    [
      "0.818053534",
      "0.48980016"
    ],
    [
      "1.050432346",
      "0.125218671"
    ],
    [
      "1.595663219",
      "1.412648222"
    ],
    [
      "1.698869973",
      "0.751431621"
    ],
    [
      "1.147297135",
      "0.725948171"
    ],
    [
      "1.178685001",
      "1.427030546"
    ],
    [
      "1.921128779",
      "0.934862279"
    ],
    [
      "1.201273617",
      "0.098773014"
    ],
    [
      "1.476571069",
      "0.96208892"
    ],
    [
      "1.087204292",
      "0.275340045"
    ],
    [
      "0.920717643",
      "1.488757583"
    ],
    [
      "1.067520383",
      "1.532362826"
    ],
    [
      "0.58271126",
      "1.68442601"
    ],
    [
      "0.99624414",
      "1.234487779"
    ],
    [
      "1.552067233",
      "0.707826378"
    ],
    [
      "0.492388262",
      "0.228604205"
    ],
    [
      "1.286991853",
      "1.711064896"
    ],
    [
      "0.301979786",
      "0.613724791"
    ],
    [
      "0.62828605",
      "0.299203445"
    ],
    [
      "0.875749632",
      "1.770110884"
    ],
    [
      "1.334529627",
      "0.366217584"
    ],
    [
      "1.624727321",
      "1.007181337"
    ],
    [
      "1.851933708",
      "1.356272519"
    ],
    [
      "0.297868562",
      "1.165399085"
    ],
    [
      "1.589119504",
      "0.858236565"
    ],
    [
      "1.107408759",
      "1.129155499"
    ],
    [
      "1.550085806",
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    ],
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      "0.155805516",
      "1.374241664"
    ],
    [
      "1.628135107",
      "0.453428071"
    ],
    [
      "1.237793833",
      "0.247496702"
    ],
    [
      "0.929218153",
      "0.384467879"
    ],
    [
      "0.853691655",
      "0.638737684"
    ],
    [
      "0.964856274",
      "0.533405403"
    ],
    [
      "0.437173933",
      "1.732084673"
    ],
    [
      "0.222091257",
      "0.87846793"
    ],
    [
      "1.14304688",
      "1.278093022"
    ],
    [
      "1.440902615",
      "0.813158658"
    ],
    [
      "1.451514257",
      "1.360941951"
    ],
    [
      "1.392573433",
      "1.502287064"
    ],
    [
      "0.885079522",
      "1.339820059"
    ],
    [
      "1.35314309",
      "0.14676433"
    ],
    [
      "0.078920091",
      "0.934174853"
    ],
    [
      "0.940401552",
      "0.231734801"
    ],
    [
      "0.229265193",
      "1.508614792"
    ],
    [
      "1.222850422",
      "0.471588228"
    ],
    [
      "0.480309424",
      "1.357941852"
    ],
    [
      "0.5201978",
      "0.954734524"
    ],
    [
      "0.671250794",
      "0.446194916"
    ],
    [
      "0.667000539",
      "0.998339768"
    ],
    [
      "0.738276782",
      "1.296214815"
    ],
    [
      "0.15069537",
      "0.637504765"
    ],
    [
      "1.562678871",
      "1.255609666"
    ],
    [
      "1.294099875",
      "0.769553414"
    ],
    [
      "0.742527037",
      "0.744069964"
    ],
    [
      "1.69519168",
      "1.60780727"
    ],
    [
      "0.409033181",
      "1.060066804"
    ],
    [
      "1.706827833",
      "1.307315937"
    ],
    [
      "1.776979817",
      "0.883156008"
    ],
    [
      "0.903629607",
      "0.081613427"
    ],
    [
      "1.677629452",
      "1.154422584"
    ],
    [
      "0.323060796",
      "1.322999968"
    ],
    [
      "1.071770638",
      "0.980217975"
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      "0.371613572",
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    ],
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    ],
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    ],
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      "0.874885695"
    ],
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      "0.080707476",
      "1.087306388"
    ],
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      "0.773914903",
      "1.445152339"
    ],
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      "0.662871409",
      "1.550612304"
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      "1.292023826",
      "1.324017643"
    ],
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      "0.831280451"
    ],
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    ],
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      "1.481332367",
      "0.409822828"
    ],
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      "1.111659014",
      "0.577010647"
    ],
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      "0.793598812",
      "0.188129557"
    ],
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      "1.821778414",
      "1.206128854"
    ],
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      "1.140293762",
      "1.667108872"
    ],
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      "0.444671302",
      "1.209004328"
    ],
    [
      "1.919227111",
      "1.087992437"
    ],
    [
      "1.329737996",
      "0.918490938"
    ],
    [
      "1.585179771",
      "1.714342905"
    ],
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      "0.725803349",
      "1.738988963"
    ],
    [
      "0.102465243",
      "0.782853713"
    ],
    [
      "0.955057281",
      "1.637999839"
    ],
    [
      "0.782415413",
      "0.340862636"
    ],
    [
      "1.775078149",
      "1.036286166"
    ]
  ],
  "radius": "0.076570982"
}
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": 140
}

起步布局答案

{
  "centers": [
    [
      "0.22336012",
      "1.031604639"
    ],
    [
      "0.911031644",
      "1.919133168"
    ],
    [
      "1.254211499",
      "1.172760742"
    ],
    [
      "0.25039181",
      "0.46268273"
    ],
    [
      "0.448914384",
      "0.656883633"
    ],
    [
      "0.377922416",
      "1.471824961"
    ],
    [
      "0.804910317",
      "1.607860975"
    ],
    [
      "1.031882262",
      "1.383425303"
    ],
    [
      "1.218573378",
      "1.023823218"
    ],
    [
      "0.813803279",
      "1.041945011"
    ],
    [
      "1.064159696",
      "1.921197418"
    ],
    [
      "1.000494395",
      "0.682342927"
    ],
    [
      "0.627077904",
      "1.40166251"
    ],
    [
      "0.924967898",
      "0.936612731"
    ],
    [
      "0.509731643",
      "1.549791552"
    ],
    [
      "1.076020893",
      "0.428073123"
    ],
    [
      "0.591474042",
      "1.252609572"
    ],
    [
      "1.369626372",
      "0.51528361"
    ],
    [
      "1.516429112",
      "0.558888854"
    ],
    [
      "1.459073612",
      "1.801231907"
    ],
    [
      "1.740769093",
      "1.461604804"
    ],
    [
      "1.849249856",
      "0.637678437"
    ],
    [
      "0.778165158",
      "0.893007488"
    ],
    [
      "1.365376117",
      "1.067428462"
    ],
    [
      "1.320341514",
      "1.866084559"
    ],
    [
      "0.607004401",
      "1.83562887"
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      "1.604686308",
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    ],
    [
      "0.49919642",
      "0.381594762"
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      "0.5597672",
      "0.551189658"
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      "1.085550255"
    ],
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      "0.849402244"
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      "0.602494097"
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      "1.816131157"
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      "1.651279386"
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      "0.366217584"
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      "0.858236565"
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      "0.453428071"
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      "1.360941951"
    ],
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      "0.14676433"
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      "0.934174853"
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      "1.508614792"
    ],
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      "1.222850422",
      "0.471588228"
    ],
    [
      "0.480309424",
      "1.357941852"
    ],
    [
      "0.5201978",
      "0.954734524"
    ],
    [
      "0.671250794",
      "0.446194916"
    ],
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      "0.998339768"
    ],
    [
      "0.738276782",
      "1.296214815"
    ],
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  "radius": "0.076570982"
}

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?