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 = 130

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 = 130
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[0.0793629380137397231460953935005, 0.0793629380137397231460953935015]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 = 0.079362937

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 = 130 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": [
    [
      "0.39088471",
      "0.518632748"
    ],
    [
      "0.874703664",
      "1.384072299"
    ],
    [
      "0.926913448",
      "0.392326234"
    ],
    [
      "0.654464486",
      "0.146666649"
    ],
    [
      "1.569882958",
      "0.733145315"
    ],
    [
      "0.112711394",
      "0.754456664"
    ],
    [
      "0.838724575",
      "1.229477963"
    ],
    [
      "0.338584951",
      "0.361431228"
    ],
    [
      "1.680510053",
      "0.619322866"
    ],
    [
      "1.517946834",
      "1.761120015"
    ],
    [
      "0.766587223",
      "0.920098147"
    ],
    [
      "0.235259143",
      "0.487414377"
    ],
    [
      "0.231235906",
      "1.044924511"
    ],
    [
      "1.47025131",
      "0.400461092"
    ],
    [
      "1.640984167",
      "1.66084181"
    ],
    [
      "1.142468909",
      "1.321754688"
    ],
    [
      "1.547446729",
      "0.25981443"
    ],
    [
      "1.06255492",
      "1.584804998"
    ],
    [
      "1.540309316",
      "0.544905802"
    ],
    [
      "0.138273458",
      "1.323452483"
    ],
    [
      "1.045907644",
      "0.226698244"
    ],
    [
      "0.521071723",
      "1.585413487"
    ],
    [
      "0.722831497",
      "1.337933936"
    ],
    [
      "0.882480301",
      "0.811642173"
    ],
    [
      "0.578735966",
      "0.719365447"
    ],
    [
      "1.150245546",
      "0.749324562"
    ],
    [
      "1.529272499",
      "1.154329237"
    ],
    [
      "0.473612563",
      "0.244692866"
    ],
    [
      "1.026575831",
      "1.430210662"
    ],
    [
      "0.195077643",
      "0.890139032"
    ],
    [
      "0.543048173",
      "1.742610622"
    ],
    [
      "1.641851816",
      "1.042298822"
    ],
    [
      "1.106310647",
      "1.166969209"
    ],
    [
      "1.122197783",
      "0.087508739"
    ],
    [
      "1.077030787",
      "1.742869394"
    ],
    [
      "1.114266457",
      "0.594730226"
    ],
    [
      "0.53498024",
      "1.137201237"
    ],
    [
      "1.078287367",
      "0.44013589"
    ],
    [
      "0.674286256",
      "1.626876854"
    ],
    [
      "0.310970721",
      "0.781683058"
    ],
    [
      "1.826808597",
      "1.404919922"
    ],
    [
      "1.759262867",
      "0.935488014"
    ],
    [
      "1.761730632",
      "0.482951604"
    ],
    [
      "1.716861147",
      "1.290440833"
    ],
    [
      "1.916559775",
      "1.086549281"
    ],
    [
      "0.462842888",
      "0.827821421"
    ],
    [
      "0.92515862",
      "1.696731031"
    ],
    [
      "0.694629044",
      "0.610909474"
    ],
    [
      "1.302117713",
      "0.795462925"
    ],
    [
      "1.178447998",
      "1.476349025"
    ],
    [
      "0.614715055",
      "0.873959784"
    ],
    [
      "1.608056468",
      "0.887212465"
    ],
    [
      "1.120305736",
      "1.912742642"
    ],
    [
      "0.392648091",
      "1.691878789"
    ],
    [
      "1.55964879",
      "1.312308126"
    ],
    [
      "0.954438479",
      "1.120830846"
    ],
    [
      "0.910682753",
      "1.538666636"
    ],
    [
      "0.570959329",
      "1.291795573"
    ],
    [
      "1.338096803",
      "0.950057262"
    ],
    [
      "1.451412608",
      "1.428406501"
    ],
    [
      "1.891766344",
      "0.771252567"
    ],
    [
      "0.998373379",
      "0.7031862"
    ],
    [
      "1.410054981",
      "1.259245935"
    ],
    [
      "0.774543033",
      "0.347859164"
    ],
    [
      "0.686852407",
      "1.1833396"
    ],
    [
      "0.990596742",
      "1.275616326"
    ],
    [
      "0.91845939",
      "0.966236509"
    ],
    [
      "1.382031702",
      "0.532412615"
    ],
    [
      "1.45398988",
      "0.841601288"
    ],
    [
      "0.97043289",
      "1.860473781"
    ],
    [
      "1.502090617",
      "1.584014175"
    ],
    [
      "1.375746733",
      "1.680092929"
    ],
    [
      "1.379513648",
      "1.8387741"
    ],
    [
      "0.887473784",
      "0.236321954"
    ],
    [
      "1.60607484",
      "1.464092597"
    ],
    [
      "0.426863799",
      "0.673227084"
    ],
    [
      "0.34694981",
      "0.936277394"
    ],
    [
      "1.74496839",
      "1.540920232"
    ],
    [
      "0.300843907",
      "1.351546003"
    ],
    [
      "0.092838091",
      "1.156939067"
    ],
    [
      "1.186224635",
      "0.903918899"
    ],
    [
      "1.070331557",
      "1.012374872"
    ],
    [
      "0.079363738",
      "0.998786148"
    ],
    [
      "1.76705441",
      "1.139860157"
    ],
    [
      "1.258182814",
      "1.213107572"
    ],
    [
      "0.383108073",
      "1.091062874"
    ],
    [
      "1.917803815",
      "0.92782828"
    ],
    [
      "0.962394289",
      "0.548591863"
    ],
    [
      "1.374075892",
      "1.104651598"
    ],
    [
      "0.810522122",
      "0.5024535"
    ],
    [
      "1.295762797",
      "1.369430837"
    ],
    [
      "0.758810586",
      "1.492528273"
    ],
    [
      "0.228604472",
      "0.646000691"
    ],
    [
      "1.226903633",
      "1.795138254"
    ],
    [
      "0.656122096",
      "1.854002686"
    ],
    [
      "1.266138624",
      "0.640868589"
    ],
    [
      "0.606443022",
      "1.44800076"
    ],
    [
      "0.542756877",
      "0.564771111"
    ],
    [
      "0.802566312",
      "1.074692483"
    ],
    [
      "0.244710258",
      "1.203077429"
    ],
    [
      "1.41217095",
      "0.176782102"
    ],
    [
      "1.335880812",
      "0.315971608"
    ],
    [
      "0.650694145",
      "1.02855412"
    ],
    [
      "1.83922132",
      "0.62147632"
    ],
    [
      "1.418010791",
      "0.687006952"
    ],
    [
      "1.034352468",
      "0.857780536"
    ],
    [
      "1.235668208",
      "0.464775996"
    ],
    [
      "1.177154936",
      "0.31596219"
    ],
    [
      "1.628915534",
      "0.396037546"
    ],
    [
      "1.885233009",
      "1.24860923"
    ],
    [
      "0.963645796",
      "0.080080997"
    ],
    [
      "0.658649955",
      "0.456315137"
    ],
    [
      "1.48996897",
      "0.996195625"
    ],
    [
      "0.370671641",
      "1.534681653"
    ],
    [
      "1.253445075",
      "0.176772685"
    ],
    [
      "0.80617443",
      "0.09999764"
    ],
    [
      "0.455066251",
      "1.400251547"
    ],
    [
      "1.212427766",
      "1.637073859"
    ],
    [
      "1.222203725",
      "1.058513235"
    ],
    [
      "0.219089836",
      "1.487598109"
    ],
    [
      "0.498032361",
      "0.401529014"
    ],
    [
      "0.846501211",
      "0.657047837"
    ],
    [
      "1.732309141",
      "0.77666115"
    ],
    [
      "0.62164655",
      "0.301962775"
    ],
    [
      "0.817325709",
      "1.902331814"
    ],
    [
      "1.328320844",
      "1.528617885"
    ],
    [
      "0.419087162",
      "1.245657211"
    ],
    [
      "0.730608133",
      "0.76550381"
    ],
    [
      "0.498821977",
      "0.982415757"
    ],
    [
      "0.77537339",
      "1.749250442"
    ]
  ],
  "radius": "0.079362937"
}
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": 130
}

起步布局答案

{
  "centers": [
    [
      "0.39088471",
      "0.518632748"
    ],
    [
      "0.874703664",
      "1.384072299"
    ],
    [
      "0.926913448",
      "0.392326234"
    ],
    [
      "0.654464486",
      "0.146666649"
    ],
    [
      "1.569882958",
      "0.733145315"
    ],
    [
      "0.112711394",
      "0.754456664"
    ],
    [
      "0.838724575",
      "1.229477963"
    ],
    [
      "0.338584951",
      "0.361431228"
    ],
    [
      "1.680510053",
      "0.619322866"
    ],
    [
      "1.517946834",
      "1.761120015"
    ],
    [
      "0.766587223",
      "0.920098147"
    ],
    [
      "0.235259143",
      "0.487414377"
    ],
    [
      "0.231235906",
      "1.044924511"
    ],
    [
      "1.47025131",
      "0.400461092"
    ],
    [
      "1.640984167",
      "1.66084181"
    ],
    [
      "1.142468909",
      "1.321754688"
    ],
    [
      "1.547446729",
      "0.25981443"
    ],
    [
      "1.06255492",
      "1.584804998"
    ],
    [
      "1.540309316",
      "0.544905802"
    ],
    [
      "0.138273458",
      "1.323452483"
    ],
    [
      "1.045907644",
      "0.226698244"
    ],
    [
      "0.521071723",
      "1.585413487"
    ],
    [
      "0.722831497",
      "1.337933936"
    ],
    [
      "0.882480301",
      "0.811642173"
    ],
    [
      "0.578735966",
      "0.719365447"
    ],
    [
      "1.150245546",
      "0.749324562"
    ],
    [
      "1.529272499",
      "1.154329237"
    ],
    [
      "0.473612563",
      "0.244692866"
    ],
    [
      "1.026575831",
      "1.430210662"
    ],
    [
      "0.195077643",
      "0.890139032"
    ],
    [
      "0.543048173",
      "1.742610622"
    ],
    [
      "1.641851816",
      "1.042298822"
    ],
    [
      "1.106310647",
      "1.166969209"
    ],
    [
      "1.122197783",
      "0.087508739"
    ],
    [
      "1.077030787",
      "1.742869394"
    ],
    [
      "1.114266457",
      "0.594730226"
    ],
    [
      "0.53498024",
      "1.137201237"
    ],
    [
      "1.078287367",
      "0.44013589"
    ],
    [
      "0.674286256",
      "1.626876854"
    ],
    [
      "0.310970721",
      "0.781683058"
    ],
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      "1.826808597",
      "1.404919922"
    ],
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      "1.759262867",
      "0.935488014"
    ],
    [
      "1.761730632",
      "0.482951604"
    ],
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      "1.290440833"
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      "1.086549281"
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      "0.827821421"
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      "0.92515862",
      "1.696731031"
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      "0.694629044",
      "0.610909474"
    ],
    [
      "1.302117713",
      "0.795462925"
    ],
    [
      "1.178447998",
      "1.476349025"
    ],
    [
      "0.614715055",
      "0.873959784"
    ],
    [
      "1.608056468",
      "0.887212465"
    ],
    [
      "1.120305736",
      "1.912742642"
    ],
    [
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      "1.691878789"
    ],
    [
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      "1.312308126"
    ],
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    ],
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      "1.538666636"
    ],
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      "0.570959329",
      "1.291795573"
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      "0.950057262"
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      "1.259245935"
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      "0.966236509"
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      "0.532412615"
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      "1.860473781"
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      "1.584014175"
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      "0.936277394"
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      "1.156939067"
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      "0.903918899"
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      "1.012374872"
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    [
      "0.079363738",
      "0.998786148"
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    [
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      "1.139860157"
    ],
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      "1.213107572"
    ],
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      "1.091062874"
    ],
    [
      "1.917803815",
      "0.92782828"
    ],
    [
      "0.962394289",
      "0.548591863"
    ],
    [
      "1.374075892",
      "1.104651598"
    ],
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      "0.5024535"
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    [
      "1.295762797",
      "1.369430837"
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    [
      "0.758810586",
      "1.492528273"
    ],
    [
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      "0.646000691"
    ],
    [
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      "1.795138254"
    ],
    [
      "0.656122096",
      "1.854002686"
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    [
      "1.266138624",
      "0.640868589"
    ],
    [
      "0.606443022",
      "1.44800076"
    ],
    [
      "0.542756877",
      "0.564771111"
    ],
    [
      "0.802566312",
      "1.074692483"
    ],
    [
      "0.244710258",
      "1.203077429"
    ],
    [
      "1.41217095",
      "0.176782102"
    ],
    [
      "1.335880812",
      "0.315971608"
    ],
    [
      "0.650694145",
      "1.02855412"
    ],
    [
      "1.83922132",
      "0.62147632"
    ],
    [
      "1.418010791",
      "0.687006952"
    ],
    [
      "1.034352468",
      "0.857780536"
    ],
    [
      "1.235668208",
      "0.464775996"
    ],
    [
      "1.177154936",
      "0.31596219"
    ],
    [
      "1.628915534",
      "0.396037546"
    ],
    [
      "1.885233009",
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      "0.455066251",
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      "1.212427766",
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    [
      "1.222203725",
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      "0.498032361",
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    [
      "1.732309141",
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      "0.62164655",
      "0.301962775"
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    [
      "0.817325709",
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    [
      "1.328320844",
      "1.528617885"
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    [
      "0.419087162",
      "1.245657211"
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    [
      "0.730608133",
      "0.76550381"
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    [
      "0.498821977",
      "0.982415757"
    ],
    [
      "0.77537339",
      "1.749250442"
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  ],
  "radius": "0.079362937"
}

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?