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 = 238

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 = 238
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[0.0594286211638122363160894327635, 0.0594286211638122363160894327645]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 = 0.059428620

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 = 238 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": [
    [
      "0.569123024",
      "0.725862953"
    ],
    [
      "1.360665666",
      "1.054597704"
    ],
    [
      "0.377376674",
      "0.800792714"
    ],
    [
      "0.989451585",
      "1.518684126"
    ],
    [
      "1.348387047",
      "0.740370805"
    ],
    [
      "1.732758759",
      "0.583235874"
    ],
    [
      "0.964894345",
      "0.890230328"
    ],
    [
      "1.661416258",
      "0.463471213"
    ],
    [
      "0.878372909",
      "0.668820714"
    ],
    [
      "0.398844525",
      "0.588784818"
    ],
    [
      "1.787418939",
      "1.015443953"
    ],
    [
      "1.552412017",
      "0.979667942"
    ],
    [
      "1.391647765",
      "0.851075612"
    ],
    [
      "0.507158828",
      "1.132907136"
    ],
    [
      "0.089739826",
      "0.763143643"
    ],
    [
      "0.482601588",
      "0.504453338"
    ],
    [
      "0.491082441",
      "1.652697853"
    ],
    [
      "1.583809826",
      "1.737455494"
    ],
    [
      "0.346394575",
      "1.004314806"
    ],
    [
      "1.595672735",
      "1.090372749"
    ],
    [
      "1.125658598",
      "1.018822658"
    ],
    [
      "1.020433683",
      "1.315162034"
    ],
    [
      "1.713176122",
      "1.108261237"
    ],
    [
      "1.850618822",
      "0.598599778"
    ],
    [
      "1.150215837",
      "1.647276456"
    ],
    [
      "1.29870147",
      "1.461641887"
    ],
    [
      "1.533660378",
      "1.497730487"
    ],
    [
      "1.317404948",
      "0.943892897"
    ],
    [
      "0.66792308",
      "1.261499466"
    ],
    [
      "0.334348602",
      "0.688621284"
    ],
    [
      "1.063694401",
      "1.425866841"
    ],
    [
      "1.583394115",
      "0.776145851"
    ],
    [
      "1.051415781",
      "1.111639942"
    ],
    [
      "0.525862306",
      "0.615158145"
    ],
    [
      "0.791851473",
      "0.447411099"
    ],
    [
      "0.600105123",
      "0.522340861"
    ],
    [
      "1.212930536",
      "0.08386335"
    ],
    [
      "0.742337823",
      "0.095409086"
    ],
    [
      "1.895046813",
      "0.965012964"
    ],
    [
      "0.760869375",
      "0.650933191"
    ],
    [
      "1.928697538",
      "0.851018785"
    ],
    [
      "0.550419546",
      "1.243611943"
    ],
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      "1.379369145",
      "0.536848713"
    ],
    [
      "0.698905178",
      "1.057977374"
    ],
    [
      "1.101101358",
      "0.390368861"
    ],
    [
      "1.638933306",
      "1.201078522"
    ],
    [
      "1.458311617",
      "1.590678306"
    ],
    [
      "0.686626559",
      "0.743750475"
    ],
    [
      "1.265415538",
      "1.676535179"
    ],
    [
      "0.390975422",
      "1.716773034"
    ],
    [
      "1.454648723",
      "0.323871057"
    ],
    [
      "0.401933913",
      "1.429246512"
    ],
    [
      "1.212180034",
      "1.240232272"
    ],
    [
      "0.19329932",
      "1.186066543"
    ],
    [
      "0.804130093",
      "0.761637998"
    ],
    [
      "1.69106418",
      "1.638047661"
    ],
    [
      "1.133389835",
      "0.172313133"
    ],
    [
      "1.540133397",
      "0.665441043"
    ],
    [
      "1.422629863",
      "0.647553521"
    ],
    [
      "1.509151299",
      "0.868963135"
    ],
    [
      "1.341345749",
      "1.876446004"
    ],
    [
      "0.303133857",
      "0.893609998"
    ],
    [
      "0.995876444",
      "0.686708237"
    ],
    [
      "0.603586764",
      "1.806705757"
    ],
    [
      "1.075981096",
      "1.740100198"
    ],
    [
      "1.199901414",
      "0.926005374"
    ],
    [
      "1.218604893",
      "0.408256384"
    ],
    [
      "0.940337106",
      "0.261776531"
    ],
    [
      "0.785426614",
      "1.279386988"
    ],
    [
      "0.260105786",
      "0.781438568"
    ],
    [
      "0.946190867",
      "1.407979318"
    ],
    [
      "0.612383742",
      "0.83656776"
    ],
    [
      "0.958477561",
      "1.722212676"
    ],
    [
      "0.921633627",
      "0.779525521"
    ],
    [
      "0.556844405",
      "0.411636054"
    ],
    [
      "1.490470046",
      "0.197433085"
    ],
    [
      "0.65564446",
      "0.947272567"
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    [
      "0.828687332",
      "1.390091796"
    ],
    [
      "0.305442543",
      "1.634243217"
    ],
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      "0.524191542",
      "0.188654925"
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      "0.538140926",
      "0.929385044"
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    [
      "0.952615726",
      "0.576003429"
    ],
    [
      "0.827024274",
      "0.225898317"
    ],
    [
      "1.039137162",
      "0.797413044"
    ],
    [
      "1.113379978",
      "0.704595759"
    ],
    [
      "1.690798109",
      "1.319734114"
    ],
    [
      "1.564938102",
      "1.294093326"
    ],
    [
      "1.830438235",
      "1.127668475"
    ],
    [
      "1.626654833",
      "0.886850658"
    ],
    [
      "0.203288664",
      "1.499912828"
    ],
    [
      "0.773147995",
      "0.96516009"
    ],
    [
      "0.630312984",
      "0.135126582"
    ],
    [
      "0.146606388",
      "1.395398174"
    ],
    [
      "0.067136594",
      "0.879831851"
    ],
    [
      "0.909355008",
      "0.465298622"
    ],
    [
      "0.643365841",
      "0.633045668"
    ],
    [
      "0.729887277",
      "0.854455283"
    ],
    [
      "0.312872692",
      "1.208800726"
    ],
    [
      "1.07011926",
      "0.593890952"
    ],
    [
      "0.494880208",
      "0.818680237"
    ],
    [
      "0.86609429",
      "0.354593815"
    ],
    [
      "0.932203253",
      "1.937894496"
    ],
    [
      "1.573738956",
      "1.619025677"
    ],
    [
      "1.27414423",
      "0.833188089"
    ],
    [
      "1.64887509",
      "1.526930044"
    ],
    [
      "1.746821024",
      "0.903726433"
    ],
    [
      "0.18586297",
      "0.874255853"
    ],
    [
      "0.074377734",
      "1.167026762"
    ],
    [
      "0.890651529",
      "0.983047613"
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    [
      "1.496872679",
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      "1.766829514",
      "1.544640312"
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      "0.519437447",
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      "0.389655293",
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      "1.181197935",
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    [
      "1.607903194",
      "1.404913202"
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      "0.420637392",
      "0.911497521"
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      "0.486233592",
      "1.787856964"
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      "0.87194805",
      "1.500796603"
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      "1.106955119",
      "1.536571648"
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      "0.983597824",
      "0.372481338"
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      "0.183719192",
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      "1.156640696",
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      "0.690209949",
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      "1.478169201",
      "1.072485227"
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      "0.999979043",
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      "1.224458653",
      "1.554459171"
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      "1.521429919",
      "1.183190034"
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      "0.636940982",
      "1.465021557"
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      "1.403926384",
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      "0.708170952",
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      "1.261865611",
      "0.518961191"
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      "0.804684558",
      "1.920068691"
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      "0.358673195",
      "1.318541704"
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      "1.292847709",
      "0.315439099"
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      "0.432916011",
      "1.22572442"
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    [
      "1.008155063",
      "1.000935135"
    ],
    [
      "0.280135648",
      "0.582847943"
    ],
    [
      "0.748590755",
      "0.336706292"
    ],
    [
      "0.277887016",
      "1.405723338"
    ],
    [
      "0.742165896",
      "1.168682181"
    ],
    [
      "1.228087724",
      "1.912496854"
    ],
    [
      "1.014579922",
      "0.168959246"
    ],
    [
      "1.351907812",
      "1.758058982"
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    [
      "0.816408713",
      "1.075864897"
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    [
      "0.711183798",
      "1.372204273"
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    [
      "1.329683568",
      "1.258119795"
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    [
      "0.581401644",
      "1.040089851"
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      "0.300669872",
      "0.37102314"
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      "0.12032483"
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      "1.186569704"
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      "0.228891041",
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      "1.255440752",
      "1.35093708"
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      "0.967082846",
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      "1.829120681"
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      "1.47952941"
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      "1.036710181"
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      "1.094676499",
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      "1.209076418"
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      "0.573589002"
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      "1.543304686"
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      "1.147414988"
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    [
      "1.372944286",
      "1.368824602"
    ]
  ],
  "radius": "0.05942862"
}
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": 238
}

起步布局答案

{
  "centers": [
    [
      "0.569123024",
      "0.725862953"
    ],
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  "radius": "0.05942862"
}

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?