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 = 120

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 = 120
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[0.0827457525728860101999239200525, 0.0827457525728860101999239200535]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 = 0.082745752

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 = 120 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": [
    [
      "1.137994218",
      "0.093185272"
    ],
    [
      "0.851932101",
      "0.094775579"
    ],
    [
      "1.298689183",
      "0.132739874"
    ],
    [
      "0.691686847",
      "0.136114386"
    ],
    [
      "0.99542591",
      "0.177219287"
    ],
    [
      "1.449661346",
      "0.200525155"
    ],
    [
      "0.541270131",
      "0.205697934"
    ],
    [
      "1.139485656",
      "0.258670057"
    ],
    [
      "0.847133394",
      "0.260197497"
    ],
    [
      "1.300180621",
      "0.298224658"
    ],
    [
      "0.405975966",
      "0.301078831"
    ],
    [
      "0.686888141",
      "0.301536303"
    ],
    [
      "1.615932239",
      "0.320398885"
    ],
    [
      "0.996917347",
      "0.342704072"
    ],
    [
      "1.451152784",
      "0.366009939"
    ],
    [
      "0.546289527",
      "0.388825869"
    ],
    [
      "0.290058581",
      "0.419191481"
    ],
    [
      "1.144931037",
      "0.42407195"
    ],
    [
      "0.853661284",
      "0.425560204"
    ],
    [
      "1.729430375",
      "0.443864511"
    ],
    [
      "1.305626002",
      "0.463626551"
    ],
    [
      "0.69341603",
      "0.46689901"
    ],
    [
      "1.568548783",
      "0.48265306"
    ],
    [
      "0.430372143",
      "0.506938519"
    ],
    [
      "1.002362728",
      "0.508105965"
    ],
    [
      "1.427583233",
      "0.575491971"
    ],
    [
      "1.817487482",
      "0.58398369"
    ],
    [
      "0.576706225",
      "0.584228715"
    ],
    [
      "0.284224198",
      "0.584580109"
    ],
    [
      "1.151327502",
      "0.589472974"
    ],
    [
      "0.859106665",
      "0.590962096"
    ],
    [
      "1.576364402",
      "0.647959909"
    ],
    [
      "0.143514208",
      "0.671690022"
    ],
    [
      "0.424537759",
      "0.672327147"
    ],
    [
      "1.294086741",
      "0.67329854"
    ],
    [
      "1.007447131",
      "0.673519347"
    ],
    [
      "0.715850602",
      "0.673818228"
    ],
    [
      "1.878934088",
      "0.737644853"
    ],
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      "1.713469687",
      "0.740639891"
    ],
    [
      "0.569792096",
      "0.75162794"
    ],
    [
      "1.443098191",
      "0.754221315"
    ],
    [
      "1.150830678",
      "0.756154671"
    ],
    [
      "0.864191068",
      "0.756375479"
    ],
    [
      "0.283827769",
      "0.75943706"
    ],
    [
      "0.098461835",
      "0.830931103"
    ],
    [
      "0.429082106",
      "0.838737854"
    ],
    [
      "1.294214224",
      "0.838789996"
    ],
    [
      "1.007574614",
      "0.839010803"
    ],
    [
      "0.720935004",
      "0.839231611"
    ],
    [
      "1.583094277",
      "0.842569723"
    ],
    [
      "1.911770008",
      "0.899846087"
    ],
    [
      "1.747142333",
      "0.91701706"
    ],
    [
      "0.574876499",
      "0.917041323"
    ],
    [
      "0.238775397",
      "0.918678141"
    ],
    [
      "1.437597771",
      "0.92142532"
    ],
    [
      "1.150958161",
      "0.921646127"
    ],
    [
      "0.864318551",
      "0.921866935"
    ],
    [
      "0.384029734",
      "0.997978935"
    ],
    [
      "0.082814284",
      "1.000748175"
    ],
    [
      "1.294341708",
      "1.004281452"
    ],
    [
      "1.007702098",
      "1.004502259"
    ],
    [
      "0.721062488",
      "1.004723067"
    ],
    [
      "1.607734717",
      "1.006216559"
    ],
    [
      "1.914926383",
      "1.06530749"
    ],
    [
      "0.575003982",
      "1.082532779"
    ],
    [
      "0.241407422",
      "1.084148715"
    ],
    [
      "1.437725254",
      "1.086916776"
    ],
    [
      "1.151085644",
      "1.087137583"
    ],
    [
      "0.864446034",
      "1.087358391"
    ],
    [
      "1.751118264",
      "1.088851883"
    ],
    [
      "0.42754156",
      "1.15764784"
    ],
    [
      "0.098042231",
      "1.166815881"
    ],
    [
      "1.581108801",
      "1.1695521"
    ],
    [
      "1.294469191",
      "1.169772908"
    ],
    [
      "1.007829581",
      "1.169993715"
    ],
    [
      "0.721189971",
      "1.170214523"
    ],
    [
      "1.888300467",
      "1.22864303"
    ],
    [
      "0.286061791",
      "1.243501862"
    ],
    [
      "0.575131466",
      "1.248024235"
    ],
    [
      "1.724492348",
      "1.252187424"
    ],
    [
      "1.437852738",
      "1.252408232"
    ],
    [
      "1.151213128",
      "1.25262904"
    ],
    [
      "0.864573518",
      "1.252849847"
    ],
    [
      "0.1426966",
      "1.326169029"
    ],
    [
      "0.429336252",
      "1.326326177"
    ],
    [
      "1.581236285",
      "1.335043556"
    ],
    [
      "1.294596674",
      "1.335264364"
    ],
    [
      "1.007957064",
      "1.335485171"
    ],
    [
      "0.72021822",
      "1.335726705"
    ],
    [
      "1.832758975",
      "1.384535881"
    ],
    [
      "0.285971062",
      "1.408993343"
    ],
    [
      "0.57411614",
      "1.413512626"
    ],
    [
      "1.437980221",
      "1.417899688"
    ],
    [
      "1.151340611",
      "1.418120496"
    ],
    [
      "0.864701001",
      "1.418341303"
    ],
    [
      "0.428767126",
      "1.492639754"
    ],
    [
      "1.581363768",
      "1.500535012"
    ],
    [
      "1.00593606",
      "1.507275937"
    ],
    [
      "1.309981786",
      "1.522798872"
    ],
    [
      "1.750109871",
      "1.527911485"
    ],
    [
      "0.688980152",
      "1.532649929"
    ],
    [
      "0.284929177",
      "1.574481568"
    ],
    [
      "0.846674731",
      "1.582848117"
    ],
    [
      "1.156042439",
      "1.583545195"
    ],
    [
      "1.453365333",
      "1.605434196"
    ],
    [
      "0.543631139",
      "1.611777057"
    ],
    [
      "1.643043512",
      "1.654102741"
    ],
    [
      "0.984552506",
      "1.674374931"
    ],
    [
      "1.302572104",
      "1.688136852"
    ],
    [
      "0.399793189",
      "1.693618871"
    ],
    [
      "0.678032825",
      "1.708335976"
    ],
    [
      "1.13304572",
      "1.747431098"
    ],
    [
      "0.837909622",
      "1.751077622"
    ],
    [
      "1.515045077",
      "1.759001926"
    ],
    [
      "0.534194876",
      "1.79017779"
    ],
    [
      "1.370281116",
      "1.839194405"
    ],
    [
      "0.710960818",
      "1.870518602"
    ],
    [
      "1.213463922",
      "1.892069789"
    ],
    [
      "0.884314611",
      "1.909929802"
    ],
    [
      "1.049698143",
      "1.9159069"
    ]
  ],
  "radius": "0.082745752"
}
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": 120
}

起步布局答案

{
  "centers": [
    [
      "1.137994218",
      "0.093185272"
    ],
    [
      "0.851932101",
      "0.094775579"
    ],
    [
      "1.298689183",
      "0.132739874"
    ],
    [
      "0.691686847",
      "0.136114386"
    ],
    [
      "0.99542591",
      "0.177219287"
    ],
    [
      "1.449661346",
      "0.200525155"
    ],
    [
      "0.541270131",
      "0.205697934"
    ],
    [
      "1.139485656",
      "0.258670057"
    ],
    [
      "0.847133394",
      "0.260197497"
    ],
    [
      "1.300180621",
      "0.298224658"
    ],
    [
      "0.405975966",
      "0.301078831"
    ],
    [
      "0.686888141",
      "0.301536303"
    ],
    [
      "1.615932239",
      "0.320398885"
    ],
    [
      "0.996917347",
      "0.342704072"
    ],
    [
      "1.451152784",
      "0.366009939"
    ],
    [
      "0.546289527",
      "0.388825869"
    ],
    [
      "0.290058581",
      "0.419191481"
    ],
    [
      "1.144931037",
      "0.42407195"
    ],
    [
      "0.853661284",
      "0.425560204"
    ],
    [
      "1.729430375",
      "0.443864511"
    ],
    [
      "1.305626002",
      "0.463626551"
    ],
    [
      "0.69341603",
      "0.46689901"
    ],
    [
      "1.568548783",
      "0.48265306"
    ],
    [
      "0.430372143",
      "0.506938519"
    ],
    [
      "1.002362728",
      "0.508105965"
    ],
    [
      "1.427583233",
      "0.575491971"
    ],
    [
      "1.817487482",
      "0.58398369"
    ],
    [
      "0.576706225",
      "0.584228715"
    ],
    [
      "0.284224198",
      "0.584580109"
    ],
    [
      "1.151327502",
      "0.589472974"
    ],
    [
      "0.859106665",
      "0.590962096"
    ],
    [
      "1.576364402",
      "0.647959909"
    ],
    [
      "0.143514208",
      "0.671690022"
    ],
    [
      "0.424537759",
      "0.672327147"
    ],
    [
      "1.294086741",
      "0.67329854"
    ],
    [
      "1.007447131",
      "0.673519347"
    ],
    [
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      "0.673818228"
    ],
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      "0.737644853"
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      "1.713469687",
      "0.740639891"
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    [
      "0.569792096",
      "0.75162794"
    ],
    [
      "1.443098191",
      "0.754221315"
    ],
    [
      "1.150830678",
      "0.756154671"
    ],
    [
      "0.864191068",
      "0.756375479"
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      "0.75943706"
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      "0.830931103"
    ],
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      "0.838737854"
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      "0.838789996"
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      "0.839010803"
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      "0.839231611"
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      "0.842569723"
    ],
    [
      "1.911770008",
      "0.899846087"
    ],
    [
      "1.747142333",
      "0.91701706"
    ],
    [
      "0.574876499",
      "0.917041323"
    ],
    [
      "0.238775397",
      "0.918678141"
    ],
    [
      "1.437597771",
      "0.92142532"
    ],
    [
      "1.150958161",
      "0.921646127"
    ],
    [
      "0.864318551",
      "0.921866935"
    ],
    [
      "0.384029734",
      "0.997978935"
    ],
    [
      "0.082814284",
      "1.000748175"
    ],
    [
      "1.294341708",
      "1.004281452"
    ],
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      "1.007702098",
      "1.004502259"
    ],
    [
      "0.721062488",
      "1.004723067"
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      "1.335726705"
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      "1.384535881"
    ],
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      "0.285971062",
      "1.408993343"
    ],
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      "0.57411614",
      "1.413512626"
    ],
    [
      "1.437980221",
      "1.417899688"
    ],
    [
      "1.151340611",
      "1.418120496"
    ],
    [
      "0.864701001",
      "1.418341303"
    ],
    [
      "0.428767126",
      "1.492639754"
    ],
    [
      "1.581363768",
      "1.500535012"
    ],
    [
      "1.00593606",
      "1.507275937"
    ],
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      "1.309981786",
      "1.522798872"
    ],
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      "1.527911485"
    ],
    [
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      "1.532649929"
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      "1.574481568"
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      "1.582848117"
    ],
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      "1.156042439",
      "1.583545195"
    ],
    [
      "1.453365333",
      "1.605434196"
    ],
    [
      "0.543631139",
      "1.611777057"
    ],
    [
      "1.643043512",
      "1.654102741"
    ],
    [
      "0.984552506",
      "1.674374931"
    ],
    [
      "1.302572104",
      "1.688136852"
    ],
    [
      "0.399793189",
      "1.693618871"
    ],
    [
      "0.678032825",
      "1.708335976"
    ],
    [
      "1.13304572",
      "1.747431098"
    ],
    [
      "0.837909622",
      "1.751077622"
    ],
    [
      "1.515045077",
      "1.759001926"
    ],
    [
      "0.534194876",
      "1.79017779"
    ],
    [
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      "1.839194405"
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      "1.892069789"
    ],
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      "1.909929802"
    ],
    [
      "1.049698143",
      "1.9159069"
    ]
  ],
  "radius": "0.082745752"
}

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

讨论区

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?