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 = 122

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 = 122
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[0.0819378309365561067175411893155, 0.0819378309365561067175411893165]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 = 0.081937830

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 = 122 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": [
    [
      "1.097552248",
      "0.08713544"
    ],
    [
      "0.816339732",
      "0.100496219"
    ],
    [
      "1.258295686",
      "0.119022427"
    ],
    [
      "0.659343523",
      "0.147479437"
    ],
    [
      "0.96092806",
      "0.177628885"
    ],
    [
      "1.410809084",
      "0.178979877"
    ],
    [
      "0.513201602",
      "0.221626397"
    ],
    [
      "1.107609776",
      "0.250702181"
    ],
    [
      "0.821835048",
      "0.264279716"
    ],
    [
      "1.550232931",
      "0.265097375"
    ],
    [
      "1.268353214",
      "0.282589167"
    ],
    [
      "0.664838838",
      "0.311262934"
    ],
    [
      "0.382570471",
      "0.320574564"
    ],
    [
      "0.970985587",
      "0.341195626"
    ],
    [
      "1.414443254",
      "0.356838294"
    ],
    [
      "1.672124784",
      "0.374630972"
    ],
    [
      "0.526446317",
      "0.399028204"
    ],
    [
      "1.118341737",
      "0.414226055"
    ],
    [
      "0.825942663",
      "0.428103891"
    ],
    [
      "0.271612408",
      "0.441171169"
    ],
    [
      "1.278594854",
      "0.448492234"
    ],
    [
      "1.536335106",
      "0.46637189"
    ],
    [
      "0.669917879",
      "0.478218754"
    ],
    [
      "0.970619251",
      "0.505070878"
    ],
    [
      "0.415488254",
      "0.519624808"
    ],
    [
      "1.683679911",
      "0.538098741"
    ],
    [
      "1.400486707",
      "0.558025831"
    ],
    [
      "1.117323686",
      "0.578098555"
    ],
    [
      "0.831625591",
      "0.591880986"
    ],
    [
      "0.560807426",
      "0.600489526"
    ],
    [
      "0.275607487",
      "0.604998126"
    ],
    [
      "1.831024715",
      "0.609825592"
    ],
    [
      "1.547890234",
      "0.629839659"
    ],
    [
      "1.264697029",
      "0.649766749"
    ],
    [
      "0.980727645",
      "0.668634483"
    ],
    [
      "0.419483333",
      "0.683451765"
    ],
    [
      "0.699145847",
      "0.688340045"
    ],
    [
      "0.135726719",
      "0.690371444"
    ],
    [
      "1.695235038",
      "0.70156651"
    ],
    [
      "1.412041834",
      "0.7214936"
    ],
    [
      "1.128878813",
      "0.741566324"
    ],
    [
      "0.845750818",
      "0.761567194"
    ],
    [
      "1.887154153",
      "0.76378896"
    ],
    [
      "0.279602565",
      "0.768825083"
    ],
    [
      "0.557821754",
      "0.771302285"
    ],
    [
      "1.559445361",
      "0.793307429"
    ],
    [
      "1.276252157",
      "0.813234518"
    ],
    [
      "0.993089171",
      "0.833307295"
    ],
    [
      "0.094447173",
      "0.848962842"
    ],
    [
      "0.709952891",
      "0.8532959"
    ],
    [
      "1.751364476",
      "0.855529879"
    ],
    [
      "0.41821858",
      "0.860043314"
    ],
    [
      "1.423596961",
      "0.884961369"
    ],
    [
      "1.140499008",
      "0.905029481"
    ],
    [
      "0.857291244",
      "0.925036001"
    ],
    [
      "1.915016323",
      "0.925278688"
    ],
    [
      "0.238323019",
      "0.927416481"
    ],
    [
      "0.567824397",
      "0.934872391"
    ],
    [
      "1.615574798",
      "0.947270797"
    ],
    [
      "1.287881869",
      "0.976812587"
    ],
    [
      "1.004709366",
      "0.996770452"
    ],
    [
      "0.082031635",
      "1.013077514"
    ],
    [
      "0.712003534",
      "1.017286798"
    ],
    [
      "1.765727925",
      "1.01877486"
    ],
    [
      "0.373836975",
      "1.019564189"
    ],
    [
      "1.435284903",
      "1.048419696"
    ],
    [
      "1.152054199",
      "1.068497246"
    ],
    [
      "0.868911439",
      "1.088499158"
    ],
    [
      "1.913723458",
      "1.08914925"
    ],
    [
      "0.226277741",
      "1.090848864"
    ],
    [
      "0.520891781",
      "1.091883735"
    ],
    [
      "1.587651227",
      "1.108749919"
    ],
    [
      "1.300341127",
      "1.141400393"
    ],
    [
      "1.016264557",
      "1.160238217"
    ],
    [
      "1.735321218",
      "1.179804872"
    ],
    [
      "0.735979548",
      "1.184334145"
    ],
    [
      "0.103969086",
      "1.19991685"
    ],
    [
      "1.451903567",
      "1.211450529"
    ],
    [
      "0.420738726",
      "1.221593403"
    ],
    [
      "1.163757137",
      "1.231954499"
    ],
    [
      "0.583147121",
      "1.243473659"
    ],
    [
      "1.883316751",
      "1.250179261"
    ],
    [
      "0.259579059",
      "1.25130526"
    ],
    [
      "0.883324156",
      "1.256061399"
    ],
    [
      "1.602575041",
      "1.275896937"
    ],
    [
      "1.316959791",
      "1.304431226"
    ],
    [
      "1.030731592",
      "1.327659446"
    ],
    [
      "1.750570574",
      "1.346271327"
    ],
    [
      "0.747534228",
      "1.347801946"
    ],
    [
      "1.464955325",
      "1.374805615"
    ],
    [
      "0.493700305",
      "1.380785347"
    ],
    [
      "0.167397562",
      "1.386796234"
    ],
    [
      "1.180375802",
      "1.394985333"
    ],
    [
      "0.330826936",
      "1.398882265"
    ],
    [
      "0.891536556",
      "1.426416895"
    ],
    [
      "1.615626799",
      "1.439252023"
    ],
    [
      "0.633805724",
      "1.465789488"
    ],
    [
      "1.047350256",
      "1.490690279"
    ],
    [
      "1.362183588",
      "1.502450446"
    ],
    [
      "1.763622332",
      "1.509626413"
    ],
    [
      "0.480774658",
      "1.546106041"
    ],
    [
      "0.263719207",
      "1.548387399"
    ],
    [
      "1.207512287",
      "1.556598586"
    ],
    [
      "1.512855062",
      "1.566896854"
    ],
    [
      "0.757505098",
      "1.573277615"
    ],
    [
      "0.920075474",
      "1.593919938"
    ],
    [
      "1.660850595",
      "1.637271243"
    ],
    [
      "0.610228511",
      "1.64748664"
    ],
    [
      "1.074486741",
      "1.652303533"
    ],
    [
      "1.381739499",
      "1.665202203"
    ],
    [
      "0.372946644",
      "1.670553678"
    ],
    [
      "1.227052374",
      "1.719305124"
    ],
    [
      "0.768001899",
      "1.736816753"
    ],
    [
      "1.536490424",
      "1.744951596"
    ],
    [
      "0.930572275",
      "1.757459076"
    ],
    [
      "0.502153793",
      "1.771354199"
    ],
    [
      "1.094026829",
      "1.815010071"
    ],
    [
      "1.395510923",
      "1.828498193"
    ],
    [
      "0.647223746",
      "1.847577171"
    ],
    [
      "1.241911725",
      "1.885616657"
    ],
    [
      "0.803534158",
      "1.896793911"
    ],
    [
      "0.966104534",
      "1.917436234"
    ]
  ],
  "radius": "0.08193783"
}
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": 122
}

起步布局答案

{
  "centers": [
    [
      "1.097552248",
      "0.08713544"
    ],
    [
      "0.816339732",
      "0.100496219"
    ],
    [
      "1.258295686",
      "0.119022427"
    ],
    [
      "0.659343523",
      "0.147479437"
    ],
    [
      "0.96092806",
      "0.177628885"
    ],
    [
      "1.410809084",
      "0.178979877"
    ],
    [
      "0.513201602",
      "0.221626397"
    ],
    [
      "1.107609776",
      "0.250702181"
    ],
    [
      "0.821835048",
      "0.264279716"
    ],
    [
      "1.550232931",
      "0.265097375"
    ],
    [
      "1.268353214",
      "0.282589167"
    ],
    [
      "0.664838838",
      "0.311262934"
    ],
    [
      "0.382570471",
      "0.320574564"
    ],
    [
      "0.970985587",
      "0.341195626"
    ],
    [
      "1.414443254",
      "0.356838294"
    ],
    [
      "1.672124784",
      "0.374630972"
    ],
    [
      "0.526446317",
      "0.399028204"
    ],
    [
      "1.118341737",
      "0.414226055"
    ],
    [
      "0.825942663",
      "0.428103891"
    ],
    [
      "0.271612408",
      "0.441171169"
    ],
    [
      "1.278594854",
      "0.448492234"
    ],
    [
      "1.536335106",
      "0.46637189"
    ],
    [
      "0.669917879",
      "0.478218754"
    ],
    [
      "0.970619251",
      "0.505070878"
    ],
    [
      "0.415488254",
      "0.519624808"
    ],
    [
      "1.683679911",
      "0.538098741"
    ],
    [
      "1.400486707",
      "0.558025831"
    ],
    [
      "1.117323686",
      "0.578098555"
    ],
    [
      "0.831625591",
      "0.591880986"
    ],
    [
      "0.560807426",
      "0.600489526"
    ],
    [
      "0.275607487",
      "0.604998126"
    ],
    [
      "1.831024715",
      "0.609825592"
    ],
    [
      "1.547890234",
      "0.629839659"
    ],
    [
      "1.264697029",
      "0.649766749"
    ],
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      "0.980727645",
      "0.668634483"
    ],
    [
      "0.419483333",
      "0.683451765"
    ],
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      "0.699145847",
      "0.688340045"
    ],
    [
      "0.135726719",
      "0.690371444"
    ],
    [
      "1.695235038",
      "0.70156651"
    ],
    [
      "1.412041834",
      "0.7214936"
    ],
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      "1.128878813",
      "0.741566324"
    ],
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      "0.845750818",
      "0.761567194"
    ],
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      "1.887154153",
      "0.76378896"
    ],
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      "0.279602565",
      "0.768825083"
    ],
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      "0.771302285"
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      "0.793307429"
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      "0.813234518"
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      "0.833307295"
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    ],
    [
      "0.709952891",
      "0.8532959"
    ],
    [
      "1.751364476",
      "0.855529879"
    ],
    [
      "0.41821858",
      "0.860043314"
    ],
    [
      "1.423596961",
      "0.884961369"
    ],
    [
      "1.140499008",
      "0.905029481"
    ],
    [
      "0.857291244",
      "0.925036001"
    ],
    [
      "1.915016323",
      "0.925278688"
    ],
    [
      "0.238323019",
      "0.927416481"
    ],
    [
      "0.567824397",
      "0.934872391"
    ],
    [
      "1.615574798",
      "0.947270797"
    ],
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      "1.287881869",
      "0.976812587"
    ],
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      "0.996770452"
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      "1.327659446"
    ],
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      "1.346271327"
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      "1.347801946"
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      "1.374805615"
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      "1.380785347"
    ],
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      "1.386796234"
    ],
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      "1.180375802",
      "1.394985333"
    ],
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      "0.330826936",
      "1.398882265"
    ],
    [
      "0.891536556",
      "1.426416895"
    ],
    [
      "1.615626799",
      "1.439252023"
    ],
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      "0.633805724",
      "1.465789488"
    ],
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      "1.047350256",
      "1.490690279"
    ],
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      "1.502450446"
    ],
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      "1.509626413"
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      "0.480774658",
      "1.546106041"
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      "1.548387399"
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    [
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      "1.556598586"
    ],
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      "1.512855062",
      "1.566896854"
    ],
    [
      "0.757505098",
      "1.573277615"
    ],
    [
      "0.920075474",
      "1.593919938"
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    [
      "1.660850595",
      "1.637271243"
    ],
    [
      "0.610228511",
      "1.64748664"
    ],
    [
      "1.074486741",
      "1.652303533"
    ],
    [
      "1.381739499",
      "1.665202203"
    ],
    [
      "0.372946644",
      "1.670553678"
    ],
    [
      "1.227052374",
      "1.719305124"
    ],
    [
      "0.768001899",
      "1.736816753"
    ],
    [
      "1.536490424",
      "1.744951596"
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    [
      "1.395510923",
      "1.828498193"
    ],
    [
      "0.647223746",
      "1.847577171"
    ],
    [
      "1.241911725",
      "1.885616657"
    ],
    [
      "0.803534158",
      "1.896793911"
    ],
    [
      "0.966104534",
      "1.917436234"
    ]
  ],
  "radius": "0.08193783"
}

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?