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 = 144

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 = 144
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[0.0754842927382019391865894859855, 0.0754842927382019391865894859865]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 = 0.075484292

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 = 144 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": [
    [
      "0.958968989",
      "0.076395242"
    ],
    [
      "1.109832326",
      "0.082031508"
    ],
    [
      "0.809199752",
      "0.095387067"
    ],
    [
      "1.257766963",
      "0.112145572"
    ],
    [
      "0.664518239",
      "0.138500562"
    ],
    [
      "1.398828197",
      "0.165934438"
    ],
    [
      "0.528782408",
      "0.204586099"
    ],
    [
      "0.982917082",
      "0.225452287"
    ],
    [
      "1.133780418",
      "0.231088553"
    ],
    [
      "1.529254604",
      "0.241963815"
    ],
    [
      "0.79883436",
      "0.246138368"
    ],
    [
      "1.281715056",
      "0.261202617"
    ],
    [
      "0.653882077",
      "0.289094008"
    ],
    [
      "0.405611682",
      "0.291881493"
    ],
    [
      "1.645568339",
      "0.338206367"
    ],
    [
      "0.904345923",
      "0.354363446"
    ],
    [
      "1.05520926",
      "0.359999711"
    ],
    [
      "0.530487699",
      "0.378764271"
    ],
    [
      "1.504169156",
      "0.391100458"
    ],
    [
      "1.202262933",
      "0.394157155"
    ],
    [
      "1.353231519",
      "0.394157155"
    ],
    [
      "0.759510201",
      "0.396956018"
    ],
    [
      "0.298290429",
      "0.398058998"
    ],
    [
      "1.744667876",
      "0.452095768"
    ],
    [
      "0.988006369",
      "0.495185824"
    ],
    [
      "1.603268693",
      "0.504989859"
    ],
    [
      "0.387750455",
      "0.519666635"
    ],
    [
      "0.20968039",
      "0.52028737"
    ],
    [
      "0.840185048",
      "0.525851293"
    ],
    [
      "1.436966265",
      "0.526286571"
    ],
    [
      "1.135060042",
      "0.529343269"
    ],
    [
      "1.286028627",
      "0.529343269"
    ],
    [
      "0.538322591",
      "0.530600331"
    ],
    [
      "0.689291177",
      "0.530600331"
    ],
    [
      "1.823910707",
      "0.580595136"
    ],
    [
      "1.536065802",
      "0.640175972"
    ],
    [
      "0.299140417",
      "0.641895007"
    ],
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      "1.687058033",
      "0.644695434"
    ],
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      "0.14214437",
      "0.655307367"
    ],
    [
      "1.361824307",
      "0.659905624"
    ],
    [
      "0.90891898",
      "0.660265431"
    ],
    [
      "1.059887565",
      "0.660265431"
    ],
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      "1.210856151",
      "0.660265431"
    ],
    [
      "0.452738878",
      "0.661830369"
    ],
    [
      "0.607056523",
      "0.665014469"
    ],
    [
      "0.758025108",
      "0.665014469"
    ],
    [
      "1.881183807",
      "0.720278011"
    ],
    [
      "1.607545726",
      "0.773150159"
    ],
    [
      "0.231604397",
      "0.776915004"
    ],
    [
      "1.456636636",
      "0.777388117"
    ],
    [
      "1.286651602",
      "0.790860954"
    ],
    [
      "0.984403272",
      "0.791008061"
    ],
    [
      "1.135371858",
      "0.791008061"
    ],
    [
      "0.53157223",
      "0.795757099"
    ],
    [
      "0.682540816",
      "0.795757099"
    ],
    [
      "0.833509401",
      "0.795757099"
    ],
    [
      "0.380685091",
      "0.800715433"
    ],
    [
      "1.755383003",
      "0.80373861"
    ],
    [
      "0.090930487",
      "0.831708844"
    ],
    [
      "1.914959979",
      "0.867419725"
    ],
    [
      "0.285532968",
      "0.917922887"
    ],
    [
      "1.641321898",
      "0.920291873"
    ],
    [
      "1.490412808",
      "0.924529831"
    ],
    [
      "1.037704789",
      "0.932254166"
    ],
    [
      "1.188673375",
      "0.932254166"
    ],
    [
      "1.33964196",
      "0.932254166"
    ],
    [
      "0.584873747",
      "0.937003204"
    ],
    [
      "0.735842332",
      "0.937003204"
    ],
    [
      "0.886810918",
      "0.937003204"
    ],
    [
      "0.434102899",
      "0.944727538"
    ],
    [
      "1.789159175",
      "0.950880324"
    ],
    [
      "0.075661425",
      "0.981903283"
    ],
    [
      "1.924338575",
      "1.018096717"
    ],
    [
      "0.210840825",
      "1.049119676"
    ],
    [
      "1.565897101",
      "1.055272462"
    ],
    [
      "1.113189082",
      "1.062996796"
    ],
    [
      "1.264157668",
      "1.062996796"
    ],
    [
      "1.415126253",
      "1.062996796"
    ],
    [
      "0.66035804",
      "1.067745834"
    ],
    [
      "0.811326625",
      "1.067745834"
    ],
    [
      "0.962295211",
      "1.067745834"
    ],
    [
      "0.509587192",
      "1.075470169"
    ],
    [
      "0.358678102",
      "1.079708127"
    ],
    [
      "1.714467032",
      "1.082077113"
    ],
    [
      "0.085040021",
      "1.132580275"
    ],
    [
      "1.909069513",
      "1.168291156"
    ],
    [
      "0.244616997",
      "1.19626139"
    ],
    [
      "1.619314909",
      "1.199284567"
    ],
    [
      "1.166490599",
      "1.204242901"
    ],
    [
      "1.317459184",
      "1.204242901"
    ],
    [
      "1.46842777",
      "1.204242901"
    ],
    [
      "0.864628142",
      "1.208991939"
    ],
    [
      "1.015596728",
      "1.208991939"
    ],
    [
      "0.713348398",
      "1.209139046"
    ],
    [
      "0.543363364",
      "1.222611883"
    ],
    [
      "1.768395603",
      "1.223084996"
    ],
    [
      "0.392454274",
      "1.226849841"
    ],
    [
      "0.118816193",
      "1.279721989"
    ],
    [
      "1.241974892",
      "1.334985531"
    ],
    [
      "1.392943477",
      "1.334985531"
    ],
    [
      "1.547261122",
      "1.338169631"
    ],
    [
      "0.789143849",
      "1.339734569"
    ],
    [
      "0.940112435",
      "1.339734569"
    ],
    [
      "1.09108102",
      "1.339734569"
    ],
    [
      "0.638175693",
      "1.340094376"
    ],
    [
      "1.85785563",
      "1.344692633"
    ],
    [
      "0.312941967",
      "1.355304566"
    ],
    [
      "1.700859583",
      "1.358104993"
    ],
    [
      "0.463934198",
      "1.359824028"
    ],
    [
      "0.176089293",
      "1.419404864"
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    [
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    ],
    [
      "1.461677409",
      "1.469399669"
    ],
    [
      "0.713971373",
      "1.470656731"
    ],
    [
      "0.864939958",
      "1.470656731"
    ],
    [
      "0.563033735",
      "1.473713429"
    ],
    [
      "1.159814952",
      "1.474148707"
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      "1.79031961",
      "1.47971263"
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      "1.480333365"
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      "0.396731307",
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      "1.011993631",
      "1.504814176"
    ],
    [
      "0.255332124",
      "1.547904232"
    ],
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      "1.701709571",
      "1.601941002"
    ],
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      "1.240489799",
      "1.603043982"
    ],
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      "0.646768481",
      "1.605842845"
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      "0.797737067",
      "1.605842845"
    ],
    [
      "0.495830844",
      "1.608899542"
    ],
    [
      "1.469512301",
      "1.621235729"
    ],
    [
      "0.94479074",
      "1.640000289"
    ],
    [
      "1.095654077",
      "1.645636554"
    ],
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      "0.354431661",
      "1.661793633"
    ],
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      "1.594388318",
      "1.708118507"
    ],
    [
      "1.346117923",
      "1.710905992"
    ],
    [
      "0.718284944",
      "1.738797383"
    ],
    [
      "1.20116564",
      "1.753861632"
    ],
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      "0.470745396",
      "1.758036185"
    ],
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      "0.866219582",
      "1.768911447"
    ],
    [
      "1.017082918",
      "1.774547713"
    ],
    [
      "1.471217592",
      "1.795413901"
    ],
    [
      "0.601171803",
      "1.834065562"
    ],
    [
      "1.335481761",
      "1.861499438"
    ],
    [
      "0.742233037",
      "1.887854428"
    ],
    [
      "1.190800248",
      "1.904612933"
    ],
    [
      "0.890167674",
      "1.917968492"
    ],
    [
      "1.041031011",
      "1.923604758"
    ]
  ],
  "radius": "0.075484292"
}
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": 144
}

起步布局答案

{
  "centers": [
    [
      "0.958968989",
      "0.076395242"
    ],
    [
      "1.109832326",
      "0.082031508"
    ],
    [
      "0.809199752",
      "0.095387067"
    ],
    [
      "1.257766963",
      "0.112145572"
    ],
    [
      "0.664518239",
      "0.138500562"
    ],
    [
      "1.398828197",
      "0.165934438"
    ],
    [
      "0.528782408",
      "0.204586099"
    ],
    [
      "0.982917082",
      "0.225452287"
    ],
    [
      "1.133780418",
      "0.231088553"
    ],
    [
      "1.529254604",
      "0.241963815"
    ],
    [
      "0.79883436",
      "0.246138368"
    ],
    [
      "1.281715056",
      "0.261202617"
    ],
    [
      "0.653882077",
      "0.289094008"
    ],
    [
      "0.405611682",
      "0.291881493"
    ],
    [
      "1.645568339",
      "0.338206367"
    ],
    [
      "0.904345923",
      "0.354363446"
    ],
    [
      "1.05520926",
      "0.359999711"
    ],
    [
      "0.530487699",
      "0.378764271"
    ],
    [
      "1.504169156",
      "0.391100458"
    ],
    [
      "1.202262933",
      "0.394157155"
    ],
    [
      "1.353231519",
      "0.394157155"
    ],
    [
      "0.759510201",
      "0.396956018"
    ],
    [
      "0.298290429",
      "0.398058998"
    ],
    [
      "1.744667876",
      "0.452095768"
    ],
    [
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      "0.504989859"
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      "0.519666635"
    ],
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      "0.20968039",
      "0.52028737"
    ],
    [
      "0.840185048",
      "0.525851293"
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      "0.526286571"
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      "0.530600331"
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      "0.640175972"
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      "0.791008061"
    ],
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      "0.795757099"
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      "0.932254166"
    ],
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      "0.932254166"
    ],
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      "0.937003204"
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      "0.944727538"
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      "0.950880324"
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      "0.981903283"
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      "1.018096717"
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      "1.049119676"
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      "1.067745834"
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      "1.067745834"
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      "1.067745834"
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      "1.075470169"
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    [
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      "1.079708127"
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      "1.132580275"
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      "1.168291156"
    ],
    [
      "0.244616997",
      "1.19626139"
    ],
    [
      "1.619314909",
      "1.199284567"
    ],
    [
      "1.166490599",
      "1.204242901"
    ],
    [
      "1.317459184",
      "1.204242901"
    ],
    [
      "1.46842777",
      "1.204242901"
    ],
    [
      "0.864628142",
      "1.208991939"
    ],
    [
      "1.015596728",
      "1.208991939"
    ],
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      "1.209139046"
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    [
      "0.543363364",
      "1.222611883"
    ],
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  "radius": "0.075484292"
}

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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还没有帖子。第一个聊聊这道题的思路?