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 = 182

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 = 182
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[0.0674812525203308248039823836625, 0.0674812525203308248039823836635]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 = 0.067481252

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 = 182 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": [
    [
      "1.008708281",
      "0.067521914"
    ],
    [
      "0.87401428",
      "0.076030946"
    ],
    [
      "1.216424921",
      "0.092943515"
    ],
    [
      "0.733234132",
      "0.106501973"
    ],
    [
      "0.605119405",
      "0.15521569"
    ],
    [
      "1.102211039",
      "0.164846691"
    ],
    [
      "1.44858829",
      "0.182467273"
    ],
    [
      "1.31368641",
      "0.186512102"
    ],
    [
      "0.972547521",
      "0.202293318"
    ],
    [
      "0.837635877",
      "0.20599821"
    ],
    [
      "0.487310845",
      "0.221064159"
    ],
    [
      "0.707791575",
      "0.242813083"
    ],
    [
      "1.199472528",
      "0.258415278"
    ],
    [
      "1.569808358",
      "0.261879964"
    ],
    [
      "1.06980901",
      "0.295861905"
    ],
    [
      "0.594890838",
      "0.316761048"
    ],
    [
      "1.438591886",
      "0.317059062"
    ],
    [
      "0.940145492",
      "0.333308531"
    ],
    [
      "0.805233848",
      "0.337013424"
    ],
    [
      "0.330301013",
      "0.351082069"
    ],
    [
      "1.307972912",
      "0.354034259"
    ],
    [
      "0.46522732",
      "0.354207674"
    ],
    [
      "1.678680113",
      "0.360481025"
    ],
    [
      "1.178266562",
      "0.391701379"
    ],
    [
      "1.548361468",
      "0.395579866"
    ],
    [
      "0.692014519",
      "0.410472672"
    ],
    [
      "1.048603044",
      "0.429148006"
    ],
    [
      "1.405191569",
      "0.44782334"
    ],
    [
      "0.562351",
      "0.447919299"
    ],
    [
      "1.763886412",
      "0.465146035"
    ],
    [
      "0.918939526",
      "0.466594633"
    ],
    [
      "0.297765371",
      "0.482064166"
    ],
    [
      "1.275528051",
      "0.485269967"
    ],
    [
      "0.432687482",
      "0.485365925"
    ],
    [
      "1.63150544",
      "0.501890531"
    ],
    [
      "0.789276007",
      "0.504041259"
    ],
    [
      "1.145864533",
      "0.522716593"
    ],
    [
      "1.502453058",
      "0.541391927"
    ],
    [
      "0.659612489",
      "0.541487886"
    ],
    [
      "0.182235817",
      "0.551833786"
    ],
    [
      "1.016201015",
      "0.56016322"
    ],
    [
      "1.37278954",
      "0.578838554"
    ],
    [
      "0.529948971",
      "0.578934513"
    ],
    [
      "1.728766929",
      "0.595459119"
    ],
    [
      "0.886537496",
      "0.597609847"
    ],
    [
      "1.243126022",
      "0.616285181"
    ],
    [
      "0.400285453",
      "0.616381139"
    ],
    [
      "1.857936788",
      "0.634574657"
    ],
    [
      "1.599714547",
      "0.634960515"
    ],
    [
      "0.756873978",
      "0.635056473"
    ],
    [
      "1.113462504",
      "0.653731807"
    ],
    [
      "0.270621935",
      "0.653827766"
    ],
    [
      "1.470051029",
      "0.672407141"
    ],
    [
      "0.62721046",
      "0.6725031"
    ],
    [
      "0.126107865",
      "0.674571436"
    ],
    [
      "0.983798985",
      "0.691178434"
    ],
    [
      "1.340387511",
      "0.709853768"
    ],
    [
      "0.497546942",
      "0.709949726"
    ],
    [
      "1.696976036",
      "0.728529102"
    ],
    [
      "0.854135467",
      "0.728625061"
    ],
    [
      "1.210723993",
      "0.747300395"
    ],
    [
      "0.367883424",
      "0.747396353"
    ],
    [
      "1.826671272",
      "0.765865727"
    ],
    [
      "1.567312518",
      "0.765975729"
    ],
    [
      "0.724471949",
      "0.766071687"
    ],
    [
      "0.214493983",
      "0.776565416"
    ],
    [
      "1.081060474",
      "0.784747021"
    ],
    [
      "1.437649",
      "0.803422355"
    ],
    [
      "0.594808431",
      "0.803518314"
    ],
    [
      "0.084747601",
      "0.821620838"
    ],
    [
      "0.951396956",
      "0.822193648"
    ],
    [
      "1.307985481",
      "0.840868982"
    ],
    [
      "0.465144913",
      "0.84096494"
    ],
    [
      "1.729508878",
      "0.859586546"
    ],
    [
      "0.821733438",
      "0.859640274"
    ],
    [
      "1.922140481",
      "0.861262309"
    ],
    [
      "1.178321963",
      "0.878315609"
    ],
    [
      "0.335481395",
      "0.878411567"
    ],
    [
      "1.599659145",
      "0.897004632"
    ],
    [
      "0.69206992",
      "0.897086901"
    ],
    [
      "0.20455266",
      "0.911161284"
    ],
    [
      "1.048658445",
      "0.915762235"
    ],
    [
      "1.469995627",
      "0.934451258"
    ],
    [
      "0.562406402",
      "0.934533528"
    ],
    [
      "0.918994927",
      "0.953208862"
    ],
    [
      "0.06858022",
      "0.955629448"
    ],
    [
      "1.82784937",
      "0.957823502"
    ],
    [
      "1.340332109",
      "0.971897885"
    ],
    [
      "0.432742884",
      "0.971980154"
    ],
    [
      "1.696920634",
      "0.990573219"
    ],
    [
      "0.789331409",
      "0.990655488"
    ],
    [
      "1.210668591",
      "1.009344512"
    ],
    [
      "0.303079366",
      "1.009426781"
    ],
    [
      "1.567257116",
      "1.028019846"
    ],
    [
      "0.659667891",
      "1.028102115"
    ],
    [
      "0.17215063",
      "1.042176498"
    ],
    [
      "1.931421412",
      "1.044384931"
    ],
    [
      "1.081005073",
      "1.046791138"
    ],
    [
      "1.437593598",
      "1.065466472"
    ],
    [
      "0.530004373",
      "1.065548742"
    ],
    [
      "0.951341555",
      "1.084237765"
    ],
    [
      "1.79544734",
      "1.088838716"
    ],
    [
      "1.30793008",
      "1.102913099"
    ],
    [
      "0.400340855",
      "1.102995368"
    ],
    [
      "1.664518605",
      "1.121588433"
    ],
    [
      "0.821678037",
      "1.121684391"
    ],
    [
      "0.077859519",
      "1.138737691"
    ],
    [
      "1.178266562",
      "1.140359726"
    ],
    [
      "0.27049162",
      "1.140413403"
    ],
    [
      "1.534855087",
      "1.15903506"
    ],
    [
      "0.692014519",
      "1.159131018"
    ],
    [
      "1.048603044",
      "1.177806352"
    ],
    [
      "1.915248777",
      "1.178397343"
    ],
    [
      "1.405191569",
      "1.196481686"
    ],
    [
      "0.562351",
      "1.196577645"
    ],
    [
      "0.918939526",
      "1.215252979"
    ],
    [
      "1.785506017",
      "1.223434584"
    ],
    [
      "1.275528051",
      "1.233928313"
    ],
    [
      "0.432687482",
      "1.234024271"
    ],
    [
      "0.173328728",
      "1.234134273"
    ],
    [
      "1.632116576",
      "1.252603647"
    ],
    [
      "0.789276007",
      "1.252699605"
    ],
    [
      "1.145864533",
      "1.271374939"
    ],
    [
      "0.303023964",
      "1.271470898"
    ],
    [
      "1.502453058",
      "1.290050274"
    ],
    [
      "0.659612489",
      "1.290146232"
    ],
    [
      "1.016201015",
      "1.308821566"
    ],
    [
      "1.873892135",
      "1.325428564"
    ],
    [
      "1.37278954",
      "1.3274969"
    ],
    [
      "0.529948971",
      "1.327592859"
    ],
    [
      "1.729378065",
      "1.346172234"
    ],
    [
      "0.886537496",
      "1.346268193"
    ],
    [
      "1.243126022",
      "1.364943527"
    ],
    [
      "0.400285453",
      "1.365039485"
    ],
    [
      "0.142063212",
      "1.365425343"
    ],
    [
      "1.599714547",
      "1.383618861"
    ],
    [
      "0.756873978",
      "1.383714819"
    ],
    [
      "1.113462504",
      "1.402390153"
    ],
    [
      "0.271233071",
      "1.404540881"
    ],
    [
      "1.470051029",
      "1.421065487"
    ],
    [
      "0.62721046",
      "1.421161446"
    ],
    [
      "0.983798985",
      "1.43983678"
    ],
    [
      "1.817764183",
      "1.448166214"
    ],
    [
      "1.340387511",
      "1.458512114"
    ],
    [
      "0.497546942",
      "1.458608073"
    ],
    [
      "0.854135467",
      "1.477283407"
    ],
    [
      "1.210723993",
      "1.495958741"
    ],
    [
      "0.36849456",
      "1.498109469"
    ],
    [
      "1.567312518",
      "1.514634075"
    ],
    [
      "0.724471949",
      "1.514730033"
    ],
    [
      "1.702234629",
      "1.517935834"
    ],
    [
      "1.081060474",
      "1.533405367"
    ],
    [
      "0.236113588",
      "1.534853965"
    ],
    [
      "1.437649",
      "1.552080701"
    ],
    [
      "0.594808431",
      "1.55217666"
    ],
    [
      "0.951396956",
      "1.570851994"
    ],
    [
      "1.307985481",
      "1.589527328"
    ],
    [
      "0.451638532",
      "1.604420134"
    ],
    [
      "0.821733438",
      "1.608298621"
    ],
    [
      "0.321319887",
      "1.639518975"
    ],
    [
      "1.53477268",
      "1.645792326"
    ],
    [
      "0.692035507",
      "1.645986524"
    ],
    [
      "1.669698987",
      "1.648917931"
    ],
    [
      "1.194766152",
      "1.662986576"
    ],
    [
      "1.059854508",
      "1.666691469"
    ],
    [
      "0.561408114",
      "1.682940938"
    ],
    [
      "1.405109162",
      "1.683238952"
    ],
    [
      "0.93019099",
      "1.704138095"
    ],
    [
      "0.430184584",
      "1.738114587"
    ],
    [
      "0.800527472",
      "1.741584722"
    ],
    [
      "1.292208425",
      "1.757186917"
    ],
    [
      "1.512689155",
      "1.778935841"
    ],
    [
      "1.162364123",
      "1.79400179"
    ],
    [
      "1.027452479",
      "1.797706682"
    ],
    [
      "0.68631359",
      "1.813487898"
    ],
    [
      "0.55141171",
      "1.817532727"
    ],
    [
      "0.897788961",
      "1.835153309"
    ],
    [
      "1.394880595",
      "1.84478431"
    ],
    [
      "1.261359962",
      "1.895094253"
    ],
    [
      "0.783575079",
      "1.907056485"
    ],
    [
      "1.12598572",
      "1.923969054"
    ],
    [
      "0.991291719",
      "1.932478086"
    ]
  ],
  "radius": "0.067481252"
}
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": 182
}

起步布局答案

{
  "centers": [
    [
      "1.008708281",
      "0.067521914"
    ],
    [
      "0.87401428",
      "0.076030946"
    ],
    [
      "1.216424921",
      "0.092943515"
    ],
    [
      "0.733234132",
      "0.106501973"
    ],
    [
      "0.605119405",
      "0.15521569"
    ],
    [
      "1.102211039",
      "0.164846691"
    ],
    [
      "1.44858829",
      "0.182467273"
    ],
    [
      "1.31368641",
      "0.186512102"
    ],
    [
      "0.972547521",
      "0.202293318"
    ],
    [
      "0.837635877",
      "0.20599821"
    ],
    [
      "0.487310845",
      "0.221064159"
    ],
    [
      "0.707791575",
      "0.242813083"
    ],
    [
      "1.199472528",
      "0.258415278"
    ],
    [
      "1.569808358",
      "0.261879964"
    ],
    [
      "1.06980901",
      "0.295861905"
    ],
    [
      "0.594890838",
      "0.316761048"
    ],
    [
      "1.438591886",
      "0.317059062"
    ],
    [
      "0.940145492",
      "0.333308531"
    ],
    [
      "0.805233848",
      "0.337013424"
    ],
    [
      "0.330301013",
      "0.351082069"
    ],
    [
      "1.307972912",
      "0.354034259"
    ],
    [
      "0.46522732",
      "0.354207674"
    ],
    [
      "1.678680113",
      "0.360481025"
    ],
    [
      "1.178266562",
      "0.391701379"
    ],
    [
      "1.548361468",
      "0.395579866"
    ],
    [
      "0.692014519",
      "0.410472672"
    ],
    [
      "1.048603044",
      "0.429148006"
    ],
    [
      "1.405191569",
      "0.44782334"
    ],
    [
      "0.562351",
      "0.447919299"
    ],
    [
      "1.763886412",
      "0.465146035"
    ],
    [
      "0.918939526",
      "0.466594633"
    ],
    [
      "0.297765371",
      "0.482064166"
    ],
    [
      "1.275528051",
      "0.485269967"
    ],
    [
      "0.432687482",
      "0.485365925"
    ],
    [
      "1.63150544",
      "0.501890531"
    ],
    [
      "0.789276007",
      "0.504041259"
    ],
    [
      "1.145864533",
      "0.522716593"
    ],
    [
      "1.502453058",
      "0.541391927"
    ],
    [
      "0.659612489",
      "0.541487886"
    ],
    [
      "0.182235817",
      "0.551833786"
    ],
    [
      "1.016201015",
      "0.56016322"
    ],
    [
      "1.37278954",
      "0.578838554"
    ],
    [
      "0.529948971",
      "0.578934513"
    ],
    [
      "1.728766929",
      "0.595459119"
    ],
    [
      "0.886537496",
      "0.597609847"
    ],
    [
      "1.243126022",
      "0.616285181"
    ],
    [
      "0.400285453",
      "0.616381139"
    ],
    [
      "1.857936788",
      "0.634574657"
    ],
    [
      "1.599714547",
      "0.634960515"
    ],
    [
      "0.756873978",
      "0.635056473"
    ],
    [
      "1.113462504",
      "0.653731807"
    ],
    [
      "0.270621935",
      "0.653827766"
    ],
    [
      "1.470051029",
      "0.672407141"
    ],
    [
      "0.62721046",
      "0.6725031"
    ],
    [
      "0.126107865",
      "0.674571436"
    ],
    [
      "0.983798985",
      "0.691178434"
    ],
    [
      "1.340387511",
      "0.709853768"
    ],
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  "radius": "0.067481252"
}

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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