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 = 115

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 = 115
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[0.0844632117280342574621086598305, 0.0844632117280342574621086598315]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 = 0.084463211

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 = 115 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": [
    [
      "0.999938808",
      "0.084671653"
    ],
    [
      "0.830229375",
      "0.100341428"
    ],
    [
      "1.169770625",
      "0.100341428"
    ],
    [
      "0.667830423",
      "0.146846448"
    ],
    [
      "1.332169577",
      "0.146846448"
    ],
    [
      "0.915536788",
      "0.246145344"
    ],
    [
      "1.084463212",
      "0.246145344"
    ],
    [
      "0.584550243",
      "0.293817698"
    ],
    [
      "0.753476666",
      "0.293817698"
    ],
    [
      "1.246523334",
      "0.293817698"
    ],
    [
      "1.415449757",
      "0.293817698"
    ],
    [
      "0.415629616",
      "0.295217151"
    ],
    [
      "1.584370384",
      "0.295217151"
    ],
    [
      "1",
      "0.392439918"
    ],
    [
      "0.296091363",
      "0.41457687"
    ],
    [
      "1.703908637",
      "0.41457687"
    ],
    [
      "0.669013454",
      "0.440112272"
    ],
    [
      "0.837939878",
      "0.440112272"
    ],
    [
      "1.162060122",
      "0.440112272"
    ],
    [
      "1.330986546",
      "0.440112272"
    ],
    [
      "0.500092828",
      "0.441511725"
    ],
    [
      "1.499907172",
      "0.441511725"
    ],
    [
      "0.200517141",
      "0.553866872"
    ],
    [
      "1.799482859",
      "0.553866872"
    ],
    [
      "0.380554575",
      "0.560871444"
    ],
    [
      "1.619445425",
      "0.560871444"
    ],
    [
      "0.746610365",
      "0.590161783"
    ],
    [
      "0.915536788",
      "0.590161783"
    ],
    [
      "1.084463212",
      "0.590161783"
    ],
    [
      "1.253389635",
      "0.590161783"
    ],
    [
      "0.568042438",
      "0.596169369"
    ],
    [
      "1.431957562",
      "0.596169369"
    ],
    [
      "0.284980352",
      "0.700161446"
    ],
    [
      "1.715019648",
      "0.700161446"
    ],
    [
      "0.452777579",
      "0.719660862"
    ],
    [
      "1.547222421",
      "0.719660862"
    ],
    [
      "0.662147153",
      "0.736456358"
    ],
    [
      "0.831073577",
      "0.736456358"
    ],
    [
      "1",
      "0.736456358"
    ],
    [
      "1.168926423",
      "0.736456358"
    ],
    [
      "1.337852847",
      "0.736456358"
    ],
    [
      "1.878898805",
      "0.744093172"
    ],
    [
      "0.121027486",
      "0.744137184"
    ],
    [
      "0.240960291",
      "0.863251515"
    ],
    [
      "1.759039709",
      "0.863251515"
    ],
    [
      "0.408757518",
      "0.882750932"
    ],
    [
      "0.577683941",
      "0.882750932"
    ],
    [
      "0.746610365",
      "0.882750932"
    ],
    [
      "0.915536788",
      "0.882750932"
    ],
    [
      "1.084463212",
      "0.882750932"
    ],
    [
      "1.253389635",
      "0.882750932"
    ],
    [
      "1.422316059",
      "0.882750932"
    ],
    [
      "1.591242482",
      "0.882750932"
    ],
    [
      "0.086944316",
      "0.932643272"
    ],
    [
      "1.913055684",
      "0.932643272"
    ],
    [
      "0.313836491",
      "1.025209567"
    ],
    [
      "1.681663074",
      "1.025585164"
    ],
    [
      "0.49322073",
      "1.029045506"
    ],
    [
      "0.662147153",
      "1.029045506"
    ],
    [
      "0.831073577",
      "1.029045506"
    ],
    [
      "1",
      "1.029045506"
    ],
    [
      "1.168926423",
      "1.029045506"
    ],
    [
      "1.337852847",
      "1.029045506"
    ],
    [
      "1.50677927",
      "1.029045506"
    ],
    [
      "0.090111414",
      "1.101540004"
    ],
    [
      "1.909888586",
      "1.101540004"
    ],
    [
      "0.408757518",
      "1.17534008"
    ],
    [
      "0.577683941",
      "1.17534008"
    ],
    [
      "0.746610365",
      "1.17534008"
    ],
    [
      "0.915536788",
      "1.17534008"
    ],
    [
      "1.084463212",
      "1.17534008"
    ],
    [
      "1.253389635",
      "1.17534008"
    ],
    [
      "1.422316059",
      "1.17534008"
    ],
    [
      "1.591242482",
      "1.17534008"
    ],
    [
      "0.239886476",
      "1.179665335"
    ],
    [
      "1.760113524",
      "1.179665335"
    ],
    [
      "0.140939694",
      "1.316580166"
    ],
    [
      "1.859060306",
      "1.316580166"
    ],
    [
      "0.662147153",
      "1.321634654"
    ],
    [
      "0.831073577",
      "1.321634654"
    ],
    [
      "1",
      "1.321634654"
    ],
    [
      "1.168926423",
      "1.321634654"
    ],
    [
      "1.337852847",
      "1.321634654"
    ],
    [
      "0.491270692",
      "1.322743311"
    ],
    [
      "1.508729308",
      "1.322743311"
    ],
    [
      "0.308984807",
      "1.333813178"
    ],
    [
      "1.691015193",
      "1.333813178"
    ],
    [
      "0.577654384",
      "1.467912159"
    ],
    [
      "1.422345616",
      "1.467912159"
    ],
    [
      "0.746610365",
      "1.467929228"
    ],
    [
      "0.915536788",
      "1.467929228"
    ],
    [
      "1.084463212",
      "1.467929228"
    ],
    [
      "1.253389635",
      "1.467929228"
    ],
    [
      "0.408742628",
      "1.470138207"
    ],
    [
      "1.591257372",
      "1.470138207"
    ],
    [
      "1.781922348",
      "1.47619863"
    ],
    [
      "0.218779847",
      "1.476678717"
    ],
    [
      "0.662117596",
      "1.614206733"
    ],
    [
      "1.337882404",
      "1.614206733"
    ],
    [
      "1",
      "1.614223802"
    ],
    [
      "0.831044017",
      "1.614240866"
    ],
    [
      "1.168955983",
      "1.614240866"
    ],
    [
      "0.49320584",
      "1.616432781"
    ],
    [
      "1.50679416",
      "1.616432781"
    ],
    [
      "1.676172122",
      "1.617197631"
    ],
    [
      "0.323866997",
      "1.6172409"
    ],
    [
      "0.915536788",
      "1.760518376"
    ],
    [
      "1.084463212",
      "1.760518376"
    ],
    [
      "0.693490006",
      "1.780194408"
    ],
    [
      "1.306509994",
      "1.780194408"
    ],
    [
      "0.52457825",
      "1.782420456"
    ],
    [
      "1.47542175",
      "1.782420456"
    ],
    [
      "0.815717703",
      "1.896798554"
    ],
    [
      "1.184282297",
      "1.896798554"
    ],
    [
      "1.00053507",
      "1.914470742"
    ]
  ],
  "radius": "0.084463211"
}
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": 115
}

起步布局答案

{
  "centers": [
    [
      "0.999938808",
      "0.084671653"
    ],
    [
      "0.830229375",
      "0.100341428"
    ],
    [
      "1.169770625",
      "0.100341428"
    ],
    [
      "0.667830423",
      "0.146846448"
    ],
    [
      "1.332169577",
      "0.146846448"
    ],
    [
      "0.915536788",
      "0.246145344"
    ],
    [
      "1.084463212",
      "0.246145344"
    ],
    [
      "0.584550243",
      "0.293817698"
    ],
    [
      "0.753476666",
      "0.293817698"
    ],
    [
      "1.246523334",
      "0.293817698"
    ],
    [
      "1.415449757",
      "0.293817698"
    ],
    [
      "0.415629616",
      "0.295217151"
    ],
    [
      "1.584370384",
      "0.295217151"
    ],
    [
      "1",
      "0.392439918"
    ],
    [
      "0.296091363",
      "0.41457687"
    ],
    [
      "1.703908637",
      "0.41457687"
    ],
    [
      "0.669013454",
      "0.440112272"
    ],
    [
      "0.837939878",
      "0.440112272"
    ],
    [
      "1.162060122",
      "0.440112272"
    ],
    [
      "1.330986546",
      "0.440112272"
    ],
    [
      "0.500092828",
      "0.441511725"
    ],
    [
      "1.499907172",
      "0.441511725"
    ],
    [
      "0.200517141",
      "0.553866872"
    ],
    [
      "1.799482859",
      "0.553866872"
    ],
    [
      "0.380554575",
      "0.560871444"
    ],
    [
      "1.619445425",
      "0.560871444"
    ],
    [
      "0.746610365",
      "0.590161783"
    ],
    [
      "0.915536788",
      "0.590161783"
    ],
    [
      "1.084463212",
      "0.590161783"
    ],
    [
      "1.253389635",
      "0.590161783"
    ],
    [
      "0.568042438",
      "0.596169369"
    ],
    [
      "1.431957562",
      "0.596169369"
    ],
    [
      "0.284980352",
      "0.700161446"
    ],
    [
      "1.715019648",
      "0.700161446"
    ],
    [
      "0.452777579",
      "0.719660862"
    ],
    [
      "1.547222421",
      "0.719660862"
    ],
    [
      "0.662147153",
      "0.736456358"
    ],
    [
      "0.831073577",
      "0.736456358"
    ],
    [
      "1",
      "0.736456358"
    ],
    [
      "1.168926423",
      "0.736456358"
    ],
    [
      "1.337852847",
      "0.736456358"
    ],
    [
      "1.878898805",
      "0.744093172"
    ],
    [
      "0.121027486",
      "0.744137184"
    ],
    [
      "0.240960291",
      "0.863251515"
    ],
    [
      "1.759039709",
      "0.863251515"
    ],
    [
      "0.408757518",
      "0.882750932"
    ],
    [
      "0.577683941",
      "0.882750932"
    ],
    [
      "0.746610365",
      "0.882750932"
    ],
    [
      "0.915536788",
      "0.882750932"
    ],
    [
      "1.084463212",
      "0.882750932"
    ],
    [
      "1.253389635",
      "0.882750932"
    ],
    [
      "1.422316059",
      "0.882750932"
    ],
    [
      "1.591242482",
      "0.882750932"
    ],
    [
      "0.086944316",
      "0.932643272"
    ],
    [
      "1.913055684",
      "0.932643272"
    ],
    [
      "0.313836491",
      "1.025209567"
    ],
    [
      "1.681663074",
      "1.025585164"
    ],
    [
      "0.49322073",
      "1.029045506"
    ],
    [
      "0.662147153",
      "1.029045506"
    ],
    [
      "0.831073577",
      "1.029045506"
    ],
    [
      "1",
      "1.029045506"
    ],
    [
      "1.168926423",
      "1.029045506"
    ],
    [
      "1.337852847",
      "1.029045506"
    ],
    [
      "1.50677927",
      "1.029045506"
    ],
    [
      "0.090111414",
      "1.101540004"
    ],
    [
      "1.909888586",
      "1.101540004"
    ],
    [
      "0.408757518",
      "1.17534008"
    ],
    [
      "0.577683941",
      "1.17534008"
    ],
    [
      "0.746610365",
      "1.17534008"
    ],
    [
      "0.915536788",
      "1.17534008"
    ],
    [
      "1.084463212",
      "1.17534008"
    ],
    [
      "1.253389635",
      "1.17534008"
    ],
    [
      "1.422316059",
      "1.17534008"
    ],
    [
      "1.591242482",
      "1.17534008"
    ],
    [
      "0.239886476",
      "1.179665335"
    ],
    [
      "1.760113524",
      "1.179665335"
    ],
    [
      "0.140939694",
      "1.316580166"
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      "1.316580166"
    ],
    [
      "0.662147153",
      "1.321634654"
    ],
    [
      "0.831073577",
      "1.321634654"
    ],
    [
      "1",
      "1.321634654"
    ],
    [
      "1.168926423",
      "1.321634654"
    ],
    [
      "1.337852847",
      "1.321634654"
    ],
    [
      "0.491270692",
      "1.322743311"
    ],
    [
      "1.508729308",
      "1.322743311"
    ],
    [
      "0.308984807",
      "1.333813178"
    ],
    [
      "1.691015193",
      "1.333813178"
    ],
    [
      "0.577654384",
      "1.467912159"
    ],
    [
      "1.422345616",
      "1.467912159"
    ],
    [
      "0.746610365",
      "1.467929228"
    ],
    [
      "0.915536788",
      "1.467929228"
    ],
    [
      "1.084463212",
      "1.467929228"
    ],
    [
      "1.253389635",
      "1.467929228"
    ],
    [
      "0.408742628",
      "1.470138207"
    ],
    [
      "1.591257372",
      "1.470138207"
    ],
    [
      "1.781922348",
      "1.47619863"
    ],
    [
      "0.218779847",
      "1.476678717"
    ],
    [
      "0.662117596",
      "1.614206733"
    ],
    [
      "1.337882404",
      "1.614206733"
    ],
    [
      "1",
      "1.614223802"
    ],
    [
      "0.831044017",
      "1.614240866"
    ],
    [
      "1.168955983",
      "1.614240866"
    ],
    [
      "0.49320584",
      "1.616432781"
    ],
    [
      "1.50679416",
      "1.616432781"
    ],
    [
      "1.676172122",
      "1.617197631"
    ],
    [
      "0.323866997",
      "1.6172409"
    ],
    [
      "0.915536788",
      "1.760518376"
    ],
    [
      "1.084463212",
      "1.760518376"
    ],
    [
      "0.693490006",
      "1.780194408"
    ],
    [
      "1.306509994",
      "1.780194408"
    ],
    [
      "0.52457825",
      "1.782420456"
    ],
    [
      "1.47542175",
      "1.782420456"
    ],
    [
      "0.815717703",
      "1.896798554"
    ],
    [
      "1.184282297",
      "1.896798554"
    ],
    [
      "1.00053507",
      "1.914470742"
    ]
  ],
  "radius": "0.084463211"
}

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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