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 = 189

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 = 189
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[0.0663609665896243687577684194395, 0.0663609665896243687577684194405]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 = 0.066360966

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 = 189 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": [
    [
      "1.743596979",
      "0.435424604"
    ],
    [
      "0.913993005",
      "1.740879736"
    ],
    [
      "0.73541819",
      "1.067338377"
    ],
    [
      "1.296898669",
      "1.631860717"
    ],
    [
      "0.650995112",
      "0.492366077"
    ],
    [
      "1.340468326",
      "1.015982712"
    ],
    [
      "0.468757369",
      "1.623580961"
    ],
    [
      "0.630985859",
      "1.460803033"
    ],
    [
      "0.489292752",
      "0.655760329"
    ],
    [
      "0.231760395",
      "0.46944392"
    ],
    [
      "0.844578364",
      "0.195836562"
    ],
    [
      "1.725131912",
      "0.911757857"
    ],
    [
      "0.871988135",
      "0.546063438"
    ],
    [
      "0.745330827",
      "0.585724978"
    ],
    [
      "0.583628467",
      "0.74911923"
    ],
    [
      "1.279815933",
      "0.794247235"
    ],
    [
      "0.664669182",
      "1.332426456"
    ],
    [
      "0.919555808",
      "1.258425114"
    ],
    [
      "1.042012042",
      "1.70586206"
    ],
    [
      "0.769101513",
      "0.938961801"
    ],
    [
      "1.716986453",
      "1.389884093"
    ],
    [
      "0.361273715",
      "0.690778005"
    ],
    [
      "1.881865222",
      "0.693413021"
    ],
    [
      "1.141910561",
      "1.316766339"
    ],
    [
      "0.411594143",
      "0.275113531"
    ],
    [
      "1.55292855",
      "1.561833646"
    ],
    [
      "1.213811018",
      "1.055644252"
    ],
    [
      "1.057461181",
      "0.735906009"
    ],
    [
      "0.446306655",
      "1.269572973"
    ],
    [
      "0.641082475",
      "0.973979477"
    ],
    [
      "0.683316705",
      "0.359345636"
    ],
    [
      "1.330581992",
      "1.503484141"
    ],
    [
      "0.98135965",
      "1.484126583"
    ],
    [
      "1.932276832",
      "0.949584202"
    ],
    [
      "0.242075193",
      "1.413249647"
    ],
    [
      "1.130061997",
      "0.075464592"
    ],
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      "1.222361926",
      "0.569386988"
    ],
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      "1.850006034",
      "1.386227377"
    ],
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      "0.785973968",
      "1.775897411"
    ],
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      "0.130816796",
      "1.340884735"
    ],
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      "0.750301224",
      "0.100965078"
    ],
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      "0.617311789",
      "0.620742654"
    ],
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      "0.81965729",
      "1.647520835"
    ],
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      "0.536510582",
      "1.366929858"
    ],
    [
      "1.430270229",
      "1.113710547"
    ],
    [
      "1.596506401",
      "0.945947361"
    ],
    [
      "0.097203296",
      "0.762048005"
    ],
    [
      "0.905671458",
      "0.417686862"
    ],
    [
      "1.407834971",
      "0.759229559"
    ],
    [
      "0.351970939",
      "1.176214072"
    ],
    [
      "0.556659397",
      "0.399007177"
    ],
    [
      "1.925317693",
      "1.095071994"
    ],
    [
      "0.802784835",
      "0.810585225"
    ],
    [
      "1.246132611",
      "0.922623811"
    ],
    [
      "1.630189724",
      "0.817570785"
    ],
    [
      "1.094342888",
      "0.604404664"
    ],
    [
      "0.863437228",
      "1.032320702"
    ],
    [
      "0.408491544",
      "1.401947534"
    ],
    [
      "1.53585414",
      "0.724211751"
    ],
    [
      "1.511067705",
      "1.781339648"
    ],
    [
      "1.780340479",
      "1.273259143"
    ],
    [
      "1.08579198",
      "1.090661928"
    ],
    [
      "0.574867279",
      "1.234698621"
    ],
    [
      "0.608550601",
      "1.106322045"
    ],
    [
      "1.108227239",
      "1.445142915"
    ],
    [
      "1.119475303",
      "0.962285351"
    ],
    [
      "0.339967922",
      "1.660119558"
    ],
    [
      "0.711647505",
      "0.714101555"
    ],
    [
      "1.558289267",
      "1.078692872"
    ],
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      "0.991456265",
      "0.997303027"
    ],
    [
      "0.89712055",
      "0.903944126"
    ],
    [
      "0.893490432",
      "0.07245617"
    ],
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      "1.236246276",
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      "1.397948636",
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      "1.013891524",
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      "1.313499256",
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    ],
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      "0.853340612",
      "1.519144258"
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    [
      "0.839666542",
      "0.679083879"
    ],
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      "1.350380963",
      "0.534369313"
    ],
    [
      "0.54674676",
      "0.880620576"
    ],
    [
      "1.390414799",
      "1.726040572"
    ],
    [
      "1.364265314",
      "1.375107565"
    ],
    [
      "0.242637354",
      "1.54597039"
    ],
    [
      "0.691638252",
      "1.68253851"
    ],
    [
      "1.683388977",
      "1.58623013"
    ],
    [
      "1.025139588",
      "0.86892645"
    ],
    [
      "1.590643049",
      "1.434582594"
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      "0.966323851",
      "0.639422339"
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      "1.067373817",
      "0.25429261"
    ],
    [
      "0.234406125",
      "0.729761672"
    ],
    [
      "1.545965468",
      "0.242633962"
    ],
    [
      "1.161709533",
      "0.347651511"
    ],
    [
      "1.569536101",
      "0.595830814"
    ],
    [
      "1.458601029",
      "1.468466466"
    ],
    [
      "0.947676327",
      "1.612503159"
    ],
    [
      "0.428640359",
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    [
      "0.930803872",
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    [
      "0.725321575",
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      "0.597419104"
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      "1.61669464",
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      "1.235046625",
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      "1.786652008",
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      "0.887023934",
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    [
      "1.195391626",
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      "0.829753906",
      "1.160697278"
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      "1.664608414",
      "0.68843949"
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    [
      "0.072557089",
      "0.892615217"
    ],
    [
      "0.597199159",
      "1.589152563"
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      "0.386195848",
      "1.047980819"
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      "0.91630825",
      "1.929880388"
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      "1.441518293",
      "0.630852983"
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      "0.134132941"
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      "0.394957037",
      "0.562401428"
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      "0.163694419",
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      "1.384063056",
      "0.405988243"
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    [
      "0.939267713",
      "0.288990711"
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    [
      "0.674765797",
      "0.8456029"
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    [
      "1.905101371",
      "1.226301799"
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    [
      "0.45560943",
      "0.784136905"
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    [
      "1.883566435",
      "0.826124051"
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    ],
    [
      "1.697273724",
      "0.559800114"
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    [
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      "0.312629342"
    ],
    [
      "0.200722803",
      "0.858138248"
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    [
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    [
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    [
      "0.419879171",
      "0.919604243"
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      "0.77765242",
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      "1.297408781"
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      "1.259843231",
      "0.103248451"
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    [
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      "1.140542743"
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    [
      "1.492284351",
      "1.340089889"
    ],
    [
      "1.61894166",
      "1.300428349"
    ],
    [
      "1.649233074",
      "0.329046869"
    ],
    [
      "0.480531564",
      "1.14133972"
    ],
    [
      "1.151796896",
      "0.82926491"
    ],
    [
      "1.829481975",
      "0.571465873"
    ],
    [
      "1.103173123",
      "1.927915086"
    ],
    [
      "0.811335742",
      "0.324327961"
    ],
    [
      "1.075695365",
      "1.577485484"
    ],
    [
      "0.374421653",
      "1.53022206"
    ],
    [
      "1.524605945",
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    ]
  ],
  "radius": "0.066360966"
}
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": 189
}

起步布局答案

{
  "centers": [
    [
      "1.743596979",
      "0.435424604"
    ],
    [
      "0.913993005",
      "1.740879736"
    ],
    [
      "0.73541819",
      "1.067338377"
    ],
    [
      "1.296898669",
      "1.631860717"
    ],
    [
      "0.650995112",
      "0.492366077"
    ],
    [
      "1.340468326",
      "1.015982712"
    ],
    [
      "0.468757369",
      "1.623580961"
    ],
    [
      "0.630985859",
      "1.460803033"
    ],
    [
      "0.489292752",
      "0.655760329"
    ],
    [
      "0.231760395",
      "0.46944392"
    ],
    [
      "0.844578364",
      "0.195836562"
    ],
    [
      "1.725131912",
      "0.911757857"
    ],
    [
      "0.871988135",
      "0.546063438"
    ],
    [
      "0.745330827",
      "0.585724978"
    ],
    [
      "0.583628467",
      "0.74911923"
    ],
    [
      "1.279815933",
      "0.794247235"
    ],
    [
      "0.664669182",
      "1.332426456"
    ],
    [
      "0.919555808",
      "1.258425114"
    ],
    [
      "1.042012042",
      "1.70586206"
    ],
    [
      "0.769101513",
      "0.938961801"
    ],
    [
      "1.716986453",
      "1.389884093"
    ],
    [
      "0.361273715",
      "0.690778005"
    ],
    [
      "1.881865222",
      "0.693413021"
    ],
    [
      "1.141910561",
      "1.316766339"
    ],
    [
      "0.411594143",
      "0.275113531"
    ],
    [
      "1.55292855",
      "1.561833646"
    ],
    [
      "1.213811018",
      "1.055644252"
    ],
    [
      "1.057461181",
      "0.735906009"
    ],
    [
      "0.446306655",
      "1.269572973"
    ],
    [
      "0.641082475",
      "0.973979477"
    ],
    [
      "0.683316705",
      "0.359345636"
    ],
    [
      "1.330581992",
      "1.503484141"
    ],
    [
      "0.98135965",
      "1.484126583"
    ],
    [
      "1.932276832",
      "0.949584202"
    ],
    [
      "0.242075193",
      "1.413249647"
    ],
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      "1.130061997",
      "0.075464592"
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      "1.222361926",
      "0.569386988"
    ],
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      "1.850006034",
      "1.386227377"
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      "1.775897411"
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      "1.340884735"
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      "0.100965078"
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      "0.620742654"
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      "1.647520835"
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      "0.536510582",
      "1.366929858"
    ],
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      "1.113710547"
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      "1.596506401",
      "0.945947361"
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      "0.097203296",
      "0.762048005"
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      "0.417686862"
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      "1.407834971",
      "0.759229559"
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    [
      "0.351970939",
      "1.176214072"
    ],
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  "radius": "0.066360966"
}

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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