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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 = 73

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 = 73
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[0.1048179996881884299913404832655, 0.1048179996881884299913404832665]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 = 0.104817999

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 = 73 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": [
    [
      "1.0104987",
      "0.104879566"
    ],
    [
      "0.80203119",
      "0.126982724"
    ],
    [
      "1.218390446",
      "0.131866124"
    ],
    [
      "0.60442058",
      "0.196963428"
    ],
    [
      "1.41430534",
      "0.206462415"
    ],
    [
      "0.950480539",
      "0.305740395"
    ],
    [
      "0.407868593",
      "0.328639626"
    ],
    [
      "1.156501561",
      "0.346300987"
    ],
    [
      "1.566602451",
      "0.350865574"
    ],
    [
      "0.602478528",
      "0.406590432"
    ],
    [
      "1.364128325",
      "0.410004841"
    ],
    [
      "0.805842236",
      "0.457486938"
    ],
    [
      "1.730124144",
      "0.482062069"
    ],
    [
      "0.259474653",
      "0.497035763"
    ],
    [
      "1.011724857",
      "0.506230759"
    ],
    [
      "1.215027828",
      "0.557369335"
    ],
    [
      "0.459470938",
      "0.559874748"
    ],
    [
      "0.662834646",
      "0.610771255"
    ],
    [
      "1.423225535",
      "0.611138558"
    ],
    [
      "0.867086553",
      "0.657977302"
    ],
    [
      "1.625397179",
      "0.66658113"
    ],
    [
      "1.834785861",
      "0.67676111"
    ],
    [
      "0.162805182",
      "0.683052606"
    ],
    [
      "1.070389525",
      "0.709115878"
    ],
    [
      "1.274125038",
      "0.758503052"
    ],
    [
      "0.512542084",
      "0.762681813"
    ],
    [
      "0.715905793",
      "0.81357832"
    ],
    [
      "1.476296681",
      "0.813945623"
    ],
    [
      "0.313575385",
      "0.828708358"
    ],
    [
      "0.920157699",
      "0.860784367"
    ],
    [
      "1.677558124",
      "0.872606375"
    ],
    [
      "1.88707162",
      "0.879772074"
    ],
    [
      "0.112048702",
      "0.886451308"
    ],
    [
      "1.123460671",
      "0.911922943"
    ],
    [
      "1.327196184",
      "0.961310117"
    ],
    [
      "0.467707628",
      "0.970801651"
    ],
    [
      "0.671838063",
      "1.018530235"
    ],
    [
      "1.528457627",
      "1.019970869"
    ],
    [
      "0.876089969",
      "1.065736282"
    ],
    [
      "1.890709073",
      "1.089376514"
    ],
    [
      "0.109988771",
      "1.096077186"
    ],
    [
      "1.079392941",
      "1.116874858"
    ],
    [
      "0.318472217",
      "1.118029517"
    ],
    [
      "1.690825992",
      "1.152574663"
    ],
    [
      "1.283128454",
      "1.166262032"
    ],
    [
      "0.523922349",
      "1.172759957"
    ],
    [
      "0.728052784",
      "1.220488541"
    ],
    [
      "1.494061943",
      "1.22676591"
    ],
    [
      "0.932304691",
      "1.267694588"
    ],
    [
      "1.845498735",
      "1.294079415"
    ],
    [
      "0.156738359",
      "1.300434051"
    ],
    [
      "1.135607662",
      "1.318833164"
    ],
    [
      "0.374686938",
      "1.319987823"
    ],
    [
      "1.339343175",
      "1.368220338"
    ],
    [
      "0.581286529",
      "1.374394783"
    ],
    [
      "0.785416964",
      "1.422123367"
    ],
    [
      "1.551426123",
      "1.428400736"
    ],
    [
      "0.98966887",
      "1.469329414"
    ],
    [
      "1.753920009",
      "1.48265457"
    ],
    [
      "0.249733651",
      "1.488314672"
    ],
    [
      "1.192971842",
      "1.52046799"
    ],
    [
      "0.452873513",
      "1.540097378"
    ],
    [
      "1.396707355",
      "1.569855164"
    ],
    [
      "0.657003949",
      "1.587825961"
    ],
    [
      "1.620995205",
      "1.644760241"
    ],
    [
      "1.04703305",
      "1.67096424"
    ],
    [
      "0.838526114",
      "1.692692331"
    ],
    [
      "1.249703992",
      "1.724552875"
    ],
    [
      "0.502025324",
      "1.743889801"
    ],
    [
      "1.454014115",
      "1.771506317"
    ],
    [
      "0.688687826",
      "1.839306585"
    ],
    [
      "1.117701403",
      "1.887410386"
    ],
    [
      "0.908087455",
      "1.890450952"
    ]
  ],
  "radius": "0.104817999"
}
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": 73
}

起步布局答案

{
  "centers": [
    [
      "1.0104987",
      "0.104879566"
    ],
    [
      "0.80203119",
      "0.126982724"
    ],
    [
      "1.218390446",
      "0.131866124"
    ],
    [
      "0.60442058",
      "0.196963428"
    ],
    [
      "1.41430534",
      "0.206462415"
    ],
    [
      "0.950480539",
      "0.305740395"
    ],
    [
      "0.407868593",
      "0.328639626"
    ],
    [
      "1.156501561",
      "0.346300987"
    ],
    [
      "1.566602451",
      "0.350865574"
    ],
    [
      "0.602478528",
      "0.406590432"
    ],
    [
      "1.364128325",
      "0.410004841"
    ],
    [
      "0.805842236",
      "0.457486938"
    ],
    [
      "1.730124144",
      "0.482062069"
    ],
    [
      "0.259474653",
      "0.497035763"
    ],
    [
      "1.011724857",
      "0.506230759"
    ],
    [
      "1.215027828",
      "0.557369335"
    ],
    [
      "0.459470938",
      "0.559874748"
    ],
    [
      "0.662834646",
      "0.610771255"
    ],
    [
      "1.423225535",
      "0.611138558"
    ],
    [
      "0.867086553",
      "0.657977302"
    ],
    [
      "1.625397179",
      "0.66658113"
    ],
    [
      "1.834785861",
      "0.67676111"
    ],
    [
      "0.162805182",
      "0.683052606"
    ],
    [
      "1.070389525",
      "0.709115878"
    ],
    [
      "1.274125038",
      "0.758503052"
    ],
    [
      "0.512542084",
      "0.762681813"
    ],
    [
      "0.715905793",
      "0.81357832"
    ],
    [
      "1.476296681",
      "0.813945623"
    ],
    [
      "0.313575385",
      "0.828708358"
    ],
    [
      "0.920157699",
      "0.860784367"
    ],
    [
      "1.677558124",
      "0.872606375"
    ],
    [
      "1.88707162",
      "0.879772074"
    ],
    [
      "0.112048702",
      "0.886451308"
    ],
    [
      "1.123460671",
      "0.911922943"
    ],
    [
      "1.327196184",
      "0.961310117"
    ],
    [
      "0.467707628",
      "0.970801651"
    ],
    [
      "0.671838063",
      "1.018530235"
    ],
    [
      "1.528457627",
      "1.019970869"
    ],
    [
      "0.876089969",
      "1.065736282"
    ],
    [
      "1.890709073",
      "1.089376514"
    ],
    [
      "0.109988771",
      "1.096077186"
    ],
    [
      "1.079392941",
      "1.116874858"
    ],
    [
      "0.318472217",
      "1.118029517"
    ],
    [
      "1.690825992",
      "1.152574663"
    ],
    [
      "1.283128454",
      "1.166262032"
    ],
    [
      "0.523922349",
      "1.172759957"
    ],
    [
      "0.728052784",
      "1.220488541"
    ],
    [
      "1.494061943",
      "1.22676591"
    ],
    [
      "0.932304691",
      "1.267694588"
    ],
    [
      "1.845498735",
      "1.294079415"
    ],
    [
      "0.156738359",
      "1.300434051"
    ],
    [
      "1.135607662",
      "1.318833164"
    ],
    [
      "0.374686938",
      "1.319987823"
    ],
    [
      "1.339343175",
      "1.368220338"
    ],
    [
      "0.581286529",
      "1.374394783"
    ],
    [
      "0.785416964",
      "1.422123367"
    ],
    [
      "1.551426123",
      "1.428400736"
    ],
    [
      "0.98966887",
      "1.469329414"
    ],
    [
      "1.753920009",
      "1.48265457"
    ],
    [
      "0.249733651",
      "1.488314672"
    ],
    [
      "1.192971842",
      "1.52046799"
    ],
    [
      "0.452873513",
      "1.540097378"
    ],
    [
      "1.396707355",
      "1.569855164"
    ],
    [
      "0.657003949",
      "1.587825961"
    ],
    [
      "1.620995205",
      "1.644760241"
    ],
    [
      "1.04703305",
      "1.67096424"
    ],
    [
      "0.838526114",
      "1.692692331"
    ],
    [
      "1.249703992",
      "1.724552875"
    ],
    [
      "0.502025324",
      "1.743889801"
    ],
    [
      "1.454014115",
      "1.771506317"
    ],
    [
      "0.688687826",
      "1.839306585"
    ],
    [
      "1.117701403",
      "1.887410386"
    ],
    [
      "0.908087455",
      "1.890450952"
    ]
  ],
  "radius": "0.104817999"
}

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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