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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 = 74

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 = 74
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[0.1042836298352085606249560372925, 0.1042836298352085606249560372935]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 = 0.104283629

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 = 74 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": [
    [
      "1.005234809",
      "0.106913555"
    ],
    [
      "0.796616987",
      "0.127679436"
    ],
    [
      "1.230383476",
      "0.134418537"
    ],
    [
      "0.456788337",
      "0.287799954"
    ],
    [
      "0.665346438",
      "0.289754557"
    ],
    [
      "1.357126558",
      "0.300058185"
    ],
    [
      "0.906561343",
      "0.304915269"
    ],
    [
      "1.565623701",
      "0.30546588"
    ],
    [
      "1.115102255",
      "0.308230337"
    ],
    [
      "0.306806744",
      "0.432733814"
    ],
    [
      "1.710912184",
      "0.455103971"
    ],
    [
      "0.775290794",
      "0.46699039"
    ],
    [
      "0.559374652",
      "0.469393869"
    ],
    [
      "1.241845337",
      "0.473869985"
    ],
    [
      "1.456691928",
      "0.483325855"
    ],
    [
      "1.007960866",
      "0.487174531"
    ],
    [
      "0.194409374",
      "0.608424261"
    ],
    [
      "0.409393059",
      "0.614327728"
    ],
    [
      "1.601980411",
      "0.632963946"
    ],
    [
      "1.817655742",
      "0.634285763"
    ],
    [
      "0.669319008",
      "0.646629702"
    ],
    [
      "0.877855932",
      "0.650186825"
    ],
    [
      "1.132864053",
      "0.654205952"
    ],
    [
      "1.341410708",
      "0.657137655"
    ],
    [
      "0.125690297",
      "0.80534554"
    ],
    [
      "0.334227612",
      "0.808879669"
    ],
    [
      "0.542764927",
      "0.812413797"
    ],
    [
      "1.002759119",
      "0.817218245"
    ],
    [
      "1.462986597",
      "0.82660625"
    ],
    [
      "0.770668304",
      "0.829103325"
    ],
    [
      "1.671532477",
      "0.829592561"
    ],
    [
      "1.88006685",
      "0.833296204"
    ],
    [
      "1.234216474",
      "0.836491411"
    ],
    [
      "0.226898309",
      "0.987711217"
    ],
    [
      "0.435435624",
      "0.991245345"
    ],
    [
      "0.643972939",
      "0.994779474"
    ],
    [
      "0.895571491",
      "0.996134746"
    ],
    [
      "1.10411154",
      "0.999503705"
    ],
    [
      "1.356127437",
      "1.005719124"
    ],
    [
      "1.564673316",
      "1.008705435"
    ],
    [
      "1.773207689",
      "1.012409078"
    ],
    [
      "0.768876127",
      "1.161810895"
    ],
    [
      "0.11994533",
      "1.166768085"
    ],
    [
      "1.226022503",
      "1.168731418"
    ],
    [
      "0.328482645",
      "1.170302214"
    ],
    [
      "0.53701996",
      "1.173836343"
    ],
    [
      "0.996923912",
      "1.178420205"
    ],
    [
      "1.457183942",
      "1.188168801"
    ],
    [
      "1.665729821",
      "1.191155112"
    ],
    [
      "1.874264194",
      "1.194858755"
    ],
    [
      "0.661683586",
      "1.341046636"
    ],
    [
      "0.870228547",
      "1.344096354"
    ],
    [
      "1.118834875",
      "1.347647919"
    ],
    [
      "1.327368176",
      "1.351411469"
    ],
    [
      "0.182370976",
      "1.365773966"
    ],
    [
      "0.397887518",
      "1.366982832"
    ],
    [
      "1.592487775",
      "1.386439275"
    ],
    [
      "1.803338548",
      "1.395074668"
    ],
    [
      "0.99213951",
      "1.513324068"
    ],
    [
      "0.545543236",
      "1.51428555"
    ],
    [
      "0.759093704",
      "1.525468898"
    ],
    [
      "1.43896199",
      "1.527613406"
    ],
    [
      "1.219842195",
      "1.53012483"
    ],
    [
      "0.289127623",
      "1.54494796"
    ],
    [
      "1.690498845",
      "1.570542866"
    ],
    [
      "0.884613529",
      "1.692037428"
    ],
    [
      "0.434427036",
      "1.694575438"
    ],
    [
      "1.09314683",
      "1.695800979"
    ],
    [
      "0.642953354",
      "1.698707812"
    ],
    [
      "1.331436009",
      "1.706326767"
    ],
    [
      "1.539832511",
      "1.71476477"
    ],
    [
      "0.76847318",
      "1.865276342"
    ],
    [
      "1.204740644",
      "1.872002915"
    ],
    [
      "0.985439964",
      "1.878176331"
    ]
  ],
  "radius": "0.104283629"
}
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": 74
}

起步布局答案

{
  "centers": [
    [
      "1.005234809",
      "0.106913555"
    ],
    [
      "0.796616987",
      "0.127679436"
    ],
    [
      "1.230383476",
      "0.134418537"
    ],
    [
      "0.456788337",
      "0.287799954"
    ],
    [
      "0.665346438",
      "0.289754557"
    ],
    [
      "1.357126558",
      "0.300058185"
    ],
    [
      "0.906561343",
      "0.304915269"
    ],
    [
      "1.565623701",
      "0.30546588"
    ],
    [
      "1.115102255",
      "0.308230337"
    ],
    [
      "0.306806744",
      "0.432733814"
    ],
    [
      "1.710912184",
      "0.455103971"
    ],
    [
      "0.775290794",
      "0.46699039"
    ],
    [
      "0.559374652",
      "0.469393869"
    ],
    [
      "1.241845337",
      "0.473869985"
    ],
    [
      "1.456691928",
      "0.483325855"
    ],
    [
      "1.007960866",
      "0.487174531"
    ],
    [
      "0.194409374",
      "0.608424261"
    ],
    [
      "0.409393059",
      "0.614327728"
    ],
    [
      "1.601980411",
      "0.632963946"
    ],
    [
      "1.817655742",
      "0.634285763"
    ],
    [
      "0.669319008",
      "0.646629702"
    ],
    [
      "0.877855932",
      "0.650186825"
    ],
    [
      "1.132864053",
      "0.654205952"
    ],
    [
      "1.341410708",
      "0.657137655"
    ],
    [
      "0.125690297",
      "0.80534554"
    ],
    [
      "0.334227612",
      "0.808879669"
    ],
    [
      "0.542764927",
      "0.812413797"
    ],
    [
      "1.002759119",
      "0.817218245"
    ],
    [
      "1.462986597",
      "0.82660625"
    ],
    [
      "0.770668304",
      "0.829103325"
    ],
    [
      "1.671532477",
      "0.829592561"
    ],
    [
      "1.88006685",
      "0.833296204"
    ],
    [
      "1.234216474",
      "0.836491411"
    ],
    [
      "0.226898309",
      "0.987711217"
    ],
    [
      "0.435435624",
      "0.991245345"
    ],
    [
      "0.643972939",
      "0.994779474"
    ],
    [
      "0.895571491",
      "0.996134746"
    ],
    [
      "1.10411154",
      "0.999503705"
    ],
    [
      "1.356127437",
      "1.005719124"
    ],
    [
      "1.564673316",
      "1.008705435"
    ],
    [
      "1.773207689",
      "1.012409078"
    ],
    [
      "0.768876127",
      "1.161810895"
    ],
    [
      "0.11994533",
      "1.166768085"
    ],
    [
      "1.226022503",
      "1.168731418"
    ],
    [
      "0.328482645",
      "1.170302214"
    ],
    [
      "0.53701996",
      "1.173836343"
    ],
    [
      "0.996923912",
      "1.178420205"
    ],
    [
      "1.457183942",
      "1.188168801"
    ],
    [
      "1.665729821",
      "1.191155112"
    ],
    [
      "1.874264194",
      "1.194858755"
    ],
    [
      "0.661683586",
      "1.341046636"
    ],
    [
      "0.870228547",
      "1.344096354"
    ],
    [
      "1.118834875",
      "1.347647919"
    ],
    [
      "1.327368176",
      "1.351411469"
    ],
    [
      "0.182370976",
      "1.365773966"
    ],
    [
      "0.397887518",
      "1.366982832"
    ],
    [
      "1.592487775",
      "1.386439275"
    ],
    [
      "1.803338548",
      "1.395074668"
    ],
    [
      "0.99213951",
      "1.513324068"
    ],
    [
      "0.545543236",
      "1.51428555"
    ],
    [
      "0.759093704",
      "1.525468898"
    ],
    [
      "1.43896199",
      "1.527613406"
    ],
    [
      "1.219842195",
      "1.53012483"
    ],
    [
      "0.289127623",
      "1.54494796"
    ],
    [
      "1.690498845",
      "1.570542866"
    ],
    [
      "0.884613529",
      "1.692037428"
    ],
    [
      "0.434427036",
      "1.694575438"
    ],
    [
      "1.09314683",
      "1.695800979"
    ],
    [
      "0.642953354",
      "1.698707812"
    ],
    [
      "1.331436009",
      "1.706326767"
    ],
    [
      "1.539832511",
      "1.71476477"
    ],
    [
      "0.76847318",
      "1.865276342"
    ],
    [
      "1.204740644",
      "1.872002915"
    ],
    [
      "0.985439964",
      "1.878176331"
    ]
  ],
  "radius": "0.104283629"
}

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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