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 = 233

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 = 233
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[0.0599076462925, 0.0599076462935]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 = 0.059907645

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 = 233 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": [
    [
      "1.005413363",
      "0.059923232"
    ],
    [
      "1.135222152",
      "0.069683601"
    ],
    [
      "0.779084163",
      "0.086233166"
    ],
    [
      "0.896532677",
      "0.109930236"
    ],
    [
      "0.654954219",
      "0.125518987"
    ],
    [
      "1.353825366",
      "0.129034304"
    ],
    [
      "1.234280912",
      "0.137085874"
    ],
    [
      "1.062775293",
      "0.165115026"
    ],
    [
      "0.530292889",
      "0.185660474"
    ],
    [
      "1.473054917",
      "0.18760335"
    ],
    [
      "0.74737021",
      "0.201775061"
    ],
    [
      "0.953894606",
      "0.215122029"
    ],
    [
      "1.161868761",
      "0.232543638"
    ],
    [
      "0.635122517",
      "0.24368162"
    ],
    [
      "1.368825477",
      "0.247906929"
    ],
    [
      "0.422286131",
      "0.258366452"
    ],
    [
      "1.584296719",
      "0.263541598"
    ],
    [
      "0.846998687",
      "0.26924196"
    ],
    [
      "1.053955403",
      "0.284605251"
    ],
    [
      "1.260912119",
      "0.299968541"
    ],
    [
      "0.532459917",
      "0.305456168"
    ],
    [
      "1.472809262",
      "0.307430679"
    ],
    [
      "0.739085329",
      "0.321303573"
    ],
    [
      "0.946042045",
      "0.336666863"
    ],
    [
      "1.152998761",
      "0.352030154"
    ],
    [
      "1.683075892",
      "0.354104529"
    ],
    [
      "0.367734893",
      "0.36504291"
    ],
    [
      "1.359955477",
      "0.367393445"
    ],
    [
      "0.631171971",
      "0.373365185"
    ],
    [
      "1.56656205",
      "0.382037108"
    ],
    [
      "0.838128687",
      "0.388728476"
    ],
    [
      "1.045085403",
      "0.404091767"
    ],
    [
      "0.26082763",
      "0.419140429"
    ],
    [
      "1.252042119",
      "0.419455057"
    ],
    [
      "0.523258613",
      "0.425426798"
    ],
    [
      "1.458998836",
      "0.434818348"
    ],
    [
      "0.730215329",
      "0.440790089"
    ],
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      "0.937172045",
      "0.456153379"
    ],
    [
      "1.719151476",
      "0.468359754"
    ],
    [
      "1.144128761",
      "0.47151667"
    ],
    [
      "0.414367445",
      "0.475410973"
    ],
    [
      "1.351085477",
      "0.48687996"
    ],
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      "0.622301971",
      "0.492851701"
    ],
    [
      "1.559127822",
      "0.502292085"
    ],
    [
      "0.829258687",
      "0.508214992"
    ],
    [
      "1.036215403",
      "0.523578282"
    ],
    [
      "0.187675119",
      "0.526817244"
    ],
    [
      "0.307460183",
      "0.529508492"
    ],
    [
      "1.817805265",
      "0.536353387"
    ],
    [
      "1.243172119",
      "0.538941573"
    ],
    [
      "0.514388613",
      "0.544913314"
    ],
    [
      "1.450128836",
      "0.554304864"
    ],
    [
      "0.721345329",
      "0.560276604"
    ],
    [
      "1.657864208",
      "0.571313992"
    ],
    [
      "0.928302045",
      "0.575639895"
    ],
    [
      "1.135258761",
      "0.591003185"
    ],
    [
      "0.406475255",
      "0.596974926"
    ],
    [
      "1.342215478",
      "0.606366476"
    ],
    [
      "0.613431971",
      "0.612338217"
    ],
    [
      "1.549172194",
      "0.621729767"
    ],
    [
      "0.820388687",
      "0.627701507"
    ],
    [
      "1.757435897",
      "0.639848526"
    ],
    [
      "1.027345403",
      "0.643064798"
    ],
    [
      "0.17877782",
      "0.64630173"
    ],
    [
      "0.298561897",
      "0.649036539"
    ],
    [
      "1.875813869",
      "0.658351497"
    ],
    [
      "1.23430212",
      "0.658428089"
    ],
    [
      "0.505518613",
      "0.664399829"
    ],
    [
      "1.441258836",
      "0.673791379"
    ],
    [
      "0.712475329",
      "0.67976312"
    ],
    [
      "1.648994208",
      "0.690800508"
    ],
    [
      "0.919432045",
      "0.695126411"
    ],
    [
      "1.126388762",
      "0.710489701"
    ],
    [
      "0.397605255",
      "0.716461442"
    ],
    [
      "1.333345478",
      "0.725852992"
    ],
    [
      "0.604561971",
      "0.731824732"
    ],
    [
      "0.098072253",
      "0.73485858"
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    [
      "1.540302194",
      "0.741216282"
    ],
    [
      "0.811518687",
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      "1.748565897",
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      "0.768523054"
    ],
    [
      "1.866943869",
      "0.777838013"
    ],
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      "1.22543212",
      "0.777914604"
    ],
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      "0.496648613",
      "0.783886345"
    ],
    [
      "1.432388836",
      "0.793277895"
    ],
    [
      "0.703605329",
      "0.799249636"
    ],
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      "1.640124208",
      "0.810287024"
    ],
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      "0.910562046",
      "0.814612926"
    ],
    [
      "0.181778539",
      "0.820584667"
    ],
    [
      "1.117518762",
      "0.829976217"
    ],
    [
      "0.388735255",
      "0.835947958"
    ],
    [
      "1.324475478",
      "0.845339508"
    ],
    [
      "0.595691971",
      "0.851311248"
    ],
    [
      "1.531432194",
      "0.860702798"
    ],
    [
      "0.069745204",
      "0.864505196"
    ],
    [
      "0.802648687",
      "0.866674539"
    ],
    [
      "1.932185974",
      "0.87833265"
    ],
    [
      "1.751084526",
      "0.87915934"
    ],
    [
      "1.009605404",
      "0.882037829"
    ],
    [
      "0.280821897",
      "0.88800957"
    ],
    [
      "1.21656212",
      "0.89740112"
    ],
    [
      "0.487778613",
      "0.903372861"
    ],
    [
      "1.423518836",
      "0.912764411"
    ],
    [
      "0.694735329",
      "0.918736151"
    ],
    [
      "1.631254208",
      "0.92977354"
    ],
    [
      "0.901692046",
      "0.934099442"
    ],
    [
      "0.16364938",
      "0.939020464"
    ],
    [
      "1.108648762",
      "0.949462733"
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      "0.379865255",
      "0.955434473"
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    [
      "1.844013832",
      "0.959458341"
    ],
    [
      "1.315605478",
      "0.964826023"
    ],
    [
      "0.586821971",
      "0.970797764"
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      "1.522562194",
      "0.980189314"
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      "1.016887636"
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    [
      "0.478908613",
      "1.022859376"
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      "1.031759447"
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      "1.099778762",
      "1.068949248"
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      "0.370995255",
      "1.074920989"
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      "1.829170187",
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      "1.306735478",
      "1.084312539"
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    [
      "0.577951972",
      "1.09028428"
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    [
      "1.513692194",
      "1.09967583"
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      "0.784908688",
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      "0.991865404",
      "1.121010861"
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      "1.717589942",
      "1.122003251"
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      "0.261852936",
      "1.12845438"
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    [
      "0.0693129",
      "1.132646733"
    ],
    [
      "1.19882212",
      "1.136374152"
    ],
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      "0.470038614",
      "1.142345892"
    ],
    [
      "1.405778836",
      "1.151737442"
    ],
    [
      "1.927345438",
      "1.154106448"
    ],
    [
      "0.67699533",
      "1.157709183"
    ],
    [
      "0.883952046",
      "1.173072473"
    ],
    [
      "1.090908762",
      "1.188435764"
    ],
    [
      "1.622023645",
      "1.194272104"
    ],
    [
      "0.362125255",
      "1.194407505"
    ],
    [
      "1.811184364",
      "1.196808254"
    ],
    [
      "1.297865478",
      "1.203799055"
    ],
    [
      "0.569081972",
      "1.209770795"
    ],
    [
      "1.504822194",
      "1.219162345"
    ],
    [
      "0.776038688",
      "1.225134086"
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    [
      "0.136287909",
      "1.231994871"
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    [
      "0.982995404",
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    [
      "0.255015474",
      "1.248102933"
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    [
      "1.18995212",
      "1.255860667"
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    [
      "1.396908836",
      "1.271223958"
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    [
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      "1.277195699"
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      "1.313085954"
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    ],
    [
      "1.288995478",
      "1.32328557"
    ],
    [
      "0.560211972",
      "1.329257311"
    ],
    [
      "1.495952194",
      "1.338648861"
    ],
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      "0.767168688",
      "1.344620602"
    ],
    [
      "0.128105522",
      "1.351530442"
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    [
      "0.974125404",
      "1.359983892"
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      "1.368693602"
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    [
      "1.18108212",
      "1.375347183"
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      "0.45310219",
      "1.38295274"
    ],
    [
      "1.388038836",
      "1.390710474"
    ],
    [
      "0.65925533",
      "1.396682214"
    ],
    [
      "1.595074658",
      "1.405957414"
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    [
      "0.866212046",
      "1.412045505"
    ],
    [
      "1.842794244",
      "1.416499094"
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    [
      "1.721577269",
      "1.417064679"
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      "1.073168762",
      "1.427408795"
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      "1.436118505"
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      "1.450377643"
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      "1.681745118"
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      "1.697198616"
    ],
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      "1.697760779"
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      "1.709713078"
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      "0.947375101",
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      "1.15447212",
      "1.73380673"
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      "1.581768082",
      "1.738457536"
    ],
    [
      "1.363520518",
      "1.75350566"
    ],
    [
      "0.44008594",
      "1.755162154"
    ],
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      "0.649167891",
      "1.75741032"
    ],
    [
      "1.253624343",
      "1.801239665"
    ],
    [
      "0.758166114",
      "1.807160613"
    ],
    [
      "1.465163951",
      "1.816943163"
    ],
    [
      "0.543321501",
      "1.821276876"
    ],
    [
      "0.878058921",
      "1.823464142"
    ],
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      "0.998877714",
      "1.826614202"
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    [
      "1.11684772",
      "1.847561329"
    ],
    [
      "1.349911291",
      "1.872545542"
    ],
    [
      "0.660581985",
      "1.876680697"
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      "1.215906643",
      "1.914963363"
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    [
      "0.801447341",
      "1.918885453"
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    [
      "0.919934378",
      "1.936676641"
    ],
    [
      "1.039721973",
      "1.939252787"
    ]
  ],
  "radius": "0.059907645"
}
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": 233
}

起步布局答案

{
  "centers": [
    [
      "1.005413363",
      "0.059923232"
    ],
    [
      "1.135222152",
      "0.069683601"
    ],
    [
      "0.779084163",
      "0.086233166"
    ],
    [
      "0.896532677",
      "0.109930236"
    ],
    [
      "0.654954219",
      "0.125518987"
    ],
    [
      "1.353825366",
      "0.129034304"
    ],
    [
      "1.234280912",
      "0.137085874"
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  "radius": "0.059907645"
}

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?