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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 = 72

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 = 72
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[0.1055532531590669661037905813835, 0.1055532531590669661037905813845]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 = 0.105553252

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 = 72 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": [
    [
      "0.913390464",
      "0.109756342"
    ],
    [
      "1.12444907",
      "0.11425319"
    ],
    [
      "0.707156442",
      "0.154850526"
    ],
    [
      "1.32857524",
      "0.168090573"
    ],
    [
      "0.517235274",
      "0.24702377"
    ],
    [
      "1.514398122",
      "0.26826948"
    ],
    [
      "0.968829007",
      "0.313453482"
    ],
    [
      "1.179887613",
      "0.31795033"
    ],
    [
      "0.762594986",
      "0.358547666"
    ],
    [
      "0.35420652",
      "0.381141563"
    ],
    [
      "1.671566452",
      "0.409209442"
    ],
    [
      "0.572673818",
      "0.45072091"
    ],
    [
      "1.426378188",
      "0.460150818"
    ],
    [
      "1.014143429",
      "0.519639224"
    ],
    [
      "1.225202036",
      "0.524136072"
    ],
    [
      "0.227151699",
      "0.549732872"
    ],
    [
      "0.807909408",
      "0.564733408"
    ],
    [
      "1.791325166",
      "0.583059399"
    ],
    [
      "1.581224096",
      "0.603638455"
    ],
    [
      "0.430425054",
      "0.606705673"
    ],
    [
      "1.380047944",
      "0.667623708"
    ],
    [
      "0.629356759",
      "0.677360707"
    ],
    [
      "0.965802447",
      "0.725136441"
    ],
    [
      "1.176861053",
      "0.729633289"
    ],
    [
      "0.1431484",
      "0.743406316"
    ],
    [
      "1.747177459",
      "0.789498099"
    ],
    [
      "0.346421755",
      "0.800379117"
    ],
    [
      "1.536212049",
      "0.820022386"
    ],
    [
      "0.779965628",
      "0.825289493"
    ],
    [
      "1.331706961",
      "0.873120926"
    ],
    [
      "0.550129097",
      "0.874195893"
    ],
    [
      "1.115414074",
      "0.931599192"
    ],
    [
      "1.892598536",
      "0.942529689"
    ],
    [
      "0.106876029",
      "0.951373313"
    ],
    [
      "0.912446333",
      "0.989651349"
    ],
    [
      "1.684886106",
      "0.991205175"
    ],
    [
      "0.310149384",
      "1.008346115"
    ],
    [
      "0.703864438",
      "1.022202077"
    ],
    [
      "1.47675243",
      "1.026508569"
    ],
    [
      "1.270259982",
      "1.075086828"
    ],
    [
      "0.504336437",
      "1.091155205"
    ],
    [
      "1.064204847",
      "1.136400491"
    ],
    [
      "1.881206741",
      "1.153328607"
    ],
    [
      "0.120355137",
      "1.162049061"
    ],
    [
      "1.675815218",
      "1.202116711"
    ],
    [
      "0.811864364",
      "1.203591087"
    ],
    [
      "0.335528134",
      "1.217921581"
    ],
    [
      "1.424424918",
      "1.231027"
    ],
    [
      "0.612336363",
      "1.272544215"
    ],
    [
      "1.219050754",
      "1.279888127"
    ],
    [
      "0.980717972",
      "1.330297109"
    ],
    [
      "1.820695054",
      "1.35566081"
    ],
    [
      "0.182834872",
      "1.363697864"
    ],
    [
      "1.569426978",
      "1.38445568"
    ],
    [
      "0.443528061",
      "1.399310591"
    ],
    [
      "0.760548481",
      "1.423607284"
    ],
    [
      "1.364052814",
      "1.433316807"
    ],
    [
      "1.141909786",
      "1.476395704"
    ],
    [
      "0.939810399",
      "1.537402222"
    ],
    [
      "1.714480454",
      "1.53810098"
    ],
    [
      "0.290834798",
      "1.545086874"
    ],
    [
      "0.591032042",
      "1.550335531"
    ],
    [
      "1.509054873",
      "1.586745487"
    ],
    [
      "1.286911845",
      "1.629824384"
    ],
    [
      "0.769823368",
      "1.662583555"
    ],
    [
      "1.084812459",
      "1.690830901"
    ],
    [
      "0.43833878",
      "1.696111813"
    ],
    [
      "1.431913905",
      "1.783253064"
    ],
    [
      "0.617130106",
      "1.808359837"
    ],
    [
      "1.236313081",
      "1.862665121"
    ],
    [
      "0.817249203",
      "1.875578168"
    ],
    [
      "1.027548435",
      "1.894022408"
    ]
  ],
  "radius": "0.105553252"
}
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": 72
}

起步布局答案

{
  "centers": [
    [
      "0.913390464",
      "0.109756342"
    ],
    [
      "1.12444907",
      "0.11425319"
    ],
    [
      "0.707156442",
      "0.154850526"
    ],
    [
      "1.32857524",
      "0.168090573"
    ],
    [
      "0.517235274",
      "0.24702377"
    ],
    [
      "1.514398122",
      "0.26826948"
    ],
    [
      "0.968829007",
      "0.313453482"
    ],
    [
      "1.179887613",
      "0.31795033"
    ],
    [
      "0.762594986",
      "0.358547666"
    ],
    [
      "0.35420652",
      "0.381141563"
    ],
    [
      "1.671566452",
      "0.409209442"
    ],
    [
      "0.572673818",
      "0.45072091"
    ],
    [
      "1.426378188",
      "0.460150818"
    ],
    [
      "1.014143429",
      "0.519639224"
    ],
    [
      "1.225202036",
      "0.524136072"
    ],
    [
      "0.227151699",
      "0.549732872"
    ],
    [
      "0.807909408",
      "0.564733408"
    ],
    [
      "1.791325166",
      "0.583059399"
    ],
    [
      "1.581224096",
      "0.603638455"
    ],
    [
      "0.430425054",
      "0.606705673"
    ],
    [
      "1.380047944",
      "0.667623708"
    ],
    [
      "0.629356759",
      "0.677360707"
    ],
    [
      "0.965802447",
      "0.725136441"
    ],
    [
      "1.176861053",
      "0.729633289"
    ],
    [
      "0.1431484",
      "0.743406316"
    ],
    [
      "1.747177459",
      "0.789498099"
    ],
    [
      "0.346421755",
      "0.800379117"
    ],
    [
      "1.536212049",
      "0.820022386"
    ],
    [
      "0.779965628",
      "0.825289493"
    ],
    [
      "1.331706961",
      "0.873120926"
    ],
    [
      "0.550129097",
      "0.874195893"
    ],
    [
      "1.115414074",
      "0.931599192"
    ],
    [
      "1.892598536",
      "0.942529689"
    ],
    [
      "0.106876029",
      "0.951373313"
    ],
    [
      "0.912446333",
      "0.989651349"
    ],
    [
      "1.684886106",
      "0.991205175"
    ],
    [
      "0.310149384",
      "1.008346115"
    ],
    [
      "0.703864438",
      "1.022202077"
    ],
    [
      "1.47675243",
      "1.026508569"
    ],
    [
      "1.270259982",
      "1.075086828"
    ],
    [
      "0.504336437",
      "1.091155205"
    ],
    [
      "1.064204847",
      "1.136400491"
    ],
    [
      "1.881206741",
      "1.153328607"
    ],
    [
      "0.120355137",
      "1.162049061"
    ],
    [
      "1.675815218",
      "1.202116711"
    ],
    [
      "0.811864364",
      "1.203591087"
    ],
    [
      "0.335528134",
      "1.217921581"
    ],
    [
      "1.424424918",
      "1.231027"
    ],
    [
      "0.612336363",
      "1.272544215"
    ],
    [
      "1.219050754",
      "1.279888127"
    ],
    [
      "0.980717972",
      "1.330297109"
    ],
    [
      "1.820695054",
      "1.35566081"
    ],
    [
      "0.182834872",
      "1.363697864"
    ],
    [
      "1.569426978",
      "1.38445568"
    ],
    [
      "0.443528061",
      "1.399310591"
    ],
    [
      "0.760548481",
      "1.423607284"
    ],
    [
      "1.364052814",
      "1.433316807"
    ],
    [
      "1.141909786",
      "1.476395704"
    ],
    [
      "0.939810399",
      "1.537402222"
    ],
    [
      "1.714480454",
      "1.53810098"
    ],
    [
      "0.290834798",
      "1.545086874"
    ],
    [
      "0.591032042",
      "1.550335531"
    ],
    [
      "1.509054873",
      "1.586745487"
    ],
    [
      "1.286911845",
      "1.629824384"
    ],
    [
      "0.769823368",
      "1.662583555"
    ],
    [
      "1.084812459",
      "1.690830901"
    ],
    [
      "0.43833878",
      "1.696111813"
    ],
    [
      "1.431913905",
      "1.783253064"
    ],
    [
      "0.617130106",
      "1.808359837"
    ],
    [
      "1.236313081",
      "1.862665121"
    ],
    [
      "0.817249203",
      "1.875578168"
    ],
    [
      "1.027548435",
      "1.894022408"
    ]
  ],
  "radius": "0.105553252"
}

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。

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