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 = 234

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 = 234
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[0.0598575299212111042789635541685, 0.0598575299212111042789635541695]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 = 0.059857529

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 = 234 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": [
    [
      "1.071216135",
      "0.062558735"
    ],
    [
      "0.84519128",
      "0.072690923"
    ],
    [
      "1.201270372",
      "0.081654699"
    ],
    [
      "0.718045545",
      "0.103133483"
    ],
    [
      "0.959957811",
      "0.106754671"
    ],
    [
      "1.331910633",
      "0.120396001"
    ],
    [
      "1.121345182",
      "0.17127292"
    ],
    [
      "1.451957822",
      "0.175620239"
    ],
    [
      "0.804971339",
      "0.185447528"
    ],
    [
      "0.631162209",
      "0.18549234"
    ],
    [
      "1.238603863",
      "0.195399615"
    ],
    [
      "0.511950654",
      "0.196460517"
    ],
    [
      "0.91973787",
      "0.219511276"
    ],
    [
      "1.350212283",
      "0.238703841"
    ],
    [
      "1.034481405",
      "0.253652406"
    ],
    [
      "0.718088002",
      "0.267806385"
    ],
    [
      "0.404623813",
      "0.272413604"
    ],
    [
      "1.521040265",
      "0.27339197"
    ],
    [
      "1.149256028",
      "0.2876889"
    ],
    [
      "0.832854533",
      "0.301870133"
    ],
    [
      "1.635806796",
      "0.307455719"
    ],
    [
      "0.516441152",
      "0.316091328"
    ],
    [
      "1.264014466",
      "0.321779902"
    ],
    [
      "0.947598068",
      "0.336011263"
    ],
    [
      "0.631204665",
      "0.350165242"
    ],
    [
      "1.379060802",
      "0.354914705"
    ],
    [
      "0.30569061",
      "0.366121198"
    ],
    [
      "1.062364599",
      "0.370075011"
    ],
    [
      "0.745971196",
      "0.38422899"
    ],
    [
      "1.493547528",
      "0.389907398"
    ],
    [
      "0.42284763",
      "0.390736811"
    ],
    [
      "1.72615424",
      "0.402866947"
    ],
    [
      "1.17713113",
      "0.404138759"
    ],
    [
      "0.860714731",
      "0.41837012"
    ],
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      "1.608314059",
      "0.423971147"
    ],
    [
      "0.544314034",
      "0.432516403"
    ],
    [
      "1.291897661",
      "0.438202507"
    ],
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      "0.975481262",
      "0.452433868"
    ],
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      "0.65908786",
      "0.466587847"
    ],
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      "0.223667354",
      "0.46974017"
    ],
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      "1.406664191",
      "0.472266256"
    ],
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      "0.342951374",
      "0.47988996"
    ],
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      "1.090247793",
      "0.486497616"
    ],
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      "0.773831394",
      "0.500728977"
    ],
    [
      "1.521430722",
      "0.506330004"
    ],
    [
      "0.457437992",
      "0.514882955"
    ],
    [
      "1.754037434",
      "0.519289552"
    ],
    [
      "1.205014324",
      "0.520561365"
    ],
    [
      "0.888597925",
      "0.534792726"
    ],
    [
      "1.636197253",
      "0.540393752"
    ],
    [
      "0.572204523",
      "0.548946705"
    ],
    [
      "1.319780855",
      "0.554625113"
    ],
    [
      "1.003364456",
      "0.568856474"
    ],
    [
      "0.686948058",
      "0.583087835"
    ],
    [
      "0.206352765",
      "0.588196493"
    ],
    [
      "1.434547386",
      "0.588688861"
    ],
    [
      "1.84781533",
      "0.593703273"
    ],
    [
      "0.36988986",
      "0.596534777"
    ],
    [
      "1.118130987",
      "0.602920222"
    ],
    [
      "0.801714589",
      "0.617151583"
    ],
    [
      "1.549313917",
      "0.622752609"
    ],
    [
      "0.485321185",
      "0.631305561"
    ],
    [
      "1.232897518",
      "0.63698397"
    ],
    [
      "0.91648112",
      "0.651215331"
    ],
    [
      "1.664580109",
      "0.656697813"
    ],
    [
      "0.60006472",
      "0.665446691"
    ],
    [
      "0.119469429",
      "0.670555351"
    ],
    [
      "1.347664049",
      "0.671047718"
    ],
    [
      "0.283673892",
      "0.679592002"
    ],
    [
      "1.031247651",
      "0.685279079"
    ],
    [
      "1.780595523",
      "0.692764835"
    ],
    [
      "0.714831252",
      "0.69951044"
    ],
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      "1.46243058",
      "0.705111466"
    ],
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      "0.39843785",
      "0.713664419"
    ],
    [
      "1.146014181",
      "0.719342827"
    ],
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      "1.897302053",
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      "0.829597783",
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      "0.513181384",
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      "0.761950859"
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      "0.767637936"
    ],
    [
      "1.691963642",
      "0.773238963"
    ],
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      "0.627947915",
      "0.781869297"
    ],
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      "1.375547243",
      "0.787470323"
    ],
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      "0.311554513",
      "0.796023276"
    ],
    [
      "0.082155952",
      "0.796455979"
    ],
    [
      "1.059130845",
      "0.801701684"
    ],
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      "1.815852011",
      "0.807170585"
    ],
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      "0.742714446",
      "0.815933045"
    ],
    [
      "1.490313774",
      "0.821534072"
    ],
    [
      "0.426298048",
      "0.830164406"
    ],
    [
      "1.173897376",
      "0.835765432"
    ],
    [
      "1.927964053",
      "0.849168522"
    ],
    [
      "0.857480977",
      "0.849996793"
    ],
    [
      "1.605080305",
      "0.85559782"
    ],
    [
      "0.541064579",
      "0.864228154"
    ],
    [
      "1.288663907",
      "0.869829181"
    ],
    [
      "0.224671176",
      "0.878382133"
    ],
    [
      "0.972247508",
      "0.884060542"
    ],
    [
      "1.719846836",
      "0.889661568"
    ],
    [
      "0.65583111",
      "0.898291902"
    ],
    [
      "1.403430437",
      "0.903892929"
    ],
    [
      "0.339414711",
      "0.912523263"
    ],
    [
      "0.110034312",
      "0.912879742"
    ],
    [
      "1.087014039",
      "0.91812429"
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    [
      "1.834074562",
      "0.925490635"
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      "0.77059764",
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    ],
    [
      "1.518196968",
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      "1.20178057",
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      "0.885364171",
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    ],
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      "1.633412454",
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      "0.568947773",
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      "1.431313632",
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      "0.165066113",
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      "1.66030914",
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      "1.124902492"
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      "0.711597498",
      "1.131137113"
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      "1.459196826",
      "1.136738139"
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      "1.145368474"
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      "1.142780427",
      "1.1509695"
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      "0.072818146",
      "1.155568873"
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      "0.826364029",
      "1.165200861"
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      "0.50994763",
      "1.179432222"
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    [
      "1.257546958",
      "1.185033249"
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    [
      "0.185021674",
      "1.197306914"
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      "0.94113056",
      "1.199264609"
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      "1.917235702",
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      "0.624714161",
      "1.21349597"
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    [
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      "1.792047797",
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      "1.472250672",
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      "1.694309466"
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      "1.588402558",
      "1.733231847"
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      "0.97033932",
      "1.748189447"
    ],
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      "1.372265925",
      "1.755132903"
    ],
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      "0.657347535",
      "1.769460102"
    ],
    [
      "1.25852984",
      "1.79249329"
    ],
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      "0.772114066",
      "1.80352385"
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      "1.139334393",
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      "1.479276362",
      "1.80880284"
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      "1.834783128"
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      "1.872696192"
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      "1.932660627"
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    [
      "0.947188259",
      "1.93865797"
    ]
  ],
  "radius": "0.059857529"
}
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": 234
}

起步布局答案

{
  "centers": [
    [
      "1.071216135",
      "0.062558735"
    ],
    [
      "0.84519128",
      "0.072690923"
    ],
    [
      "1.201270372",
      "0.081654699"
    ],
    [
      "0.718045545",
      "0.103133483"
    ],
    [
      "0.959957811",
      "0.106754671"
    ],
    [
      "1.331910633",
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  "radius": "0.059857529"
}

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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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。

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