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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 = 71

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 = 71
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[0.1062044998371123195282322402525, 0.1062044998371123195282322402535]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 = 0.106204499

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 = 71 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": [
    [
      "0.924821065",
      "0.109371838"
    ],
    [
      "1.137100764",
      "0.116782146"
    ],
    [
      "0.716787226",
      "0.15226129"
    ],
    [
      "1.365403654",
      "0.184309761"
    ],
    [
      "0.524748317",
      "0.243028247"
    ],
    [
      "1.547559321",
      "0.293565866"
    ],
    [
      "0.989076292",
      "0.311828867"
    ],
    [
      "1.20135599",
      "0.319239176"
    ],
    [
      "0.727786998",
      "0.364385282"
    ],
    [
      "0.359550067",
      "0.37654649"
    ],
    [
      "1.383511657",
      "0.428495281"
    ],
    [
      "1.698790631",
      "0.442719056"
    ],
    [
      "0.547158754",
      "0.476148517"
    ],
    [
      "0.891048493",
      "0.500264888"
    ],
    [
      "1.103328192",
      "0.507675197"
    ],
    [
      "1.534742966",
      "0.577648471"
    ],
    [
      "0.364576278",
      "0.588896014"
    ],
    [
      "0.710420249",
      "0.612028122"
    ],
    [
      "1.810556534",
      "0.623345648"
    ],
    [
      "1.299981589",
      "0.626622204"
    ],
    [
      "0.168822167",
      "0.67134546"
    ],
    [
      "1.092946582",
      "0.719830341"
    ],
    [
      "0.529302447",
      "0.722996247"
    ],
    [
      "0.881329383",
      "0.73815366"
    ],
    [
      "1.453447269",
      "0.773884535"
    ],
    [
      "1.665799611",
      "0.778790225"
    ],
    [
      "0.333548336",
      "0.805445694"
    ],
    [
      "1.87654486",
      "0.825244445"
    ],
    [
      "0.701803505",
      "0.851679174"
    ],
    [
      "1.257023097",
      "0.854724672"
    ],
    [
      "0.114742412",
      "0.876754718"
    ],
    [
      "1.058162922",
      "0.929371937"
    ],
    [
      "0.506049394",
      "0.93412862"
    ],
    [
      "1.473478339",
      "0.985346917"
    ],
    [
      "1.685830682",
      "0.990252607"
    ],
    [
      "0.856784665",
      "0.996932136"
    ],
    [
      "0.310295283",
      "1.016578066"
    ],
    [
      "1.893028804",
      "1.037012862"
    ],
    [
      "1.270246571",
      "1.06672166"
    ],
    [
      "0.661030554",
      "1.079381582"
    ],
    [
      "0.110659106",
      "1.089124466"
    ],
    [
      "1.015982642",
      "1.137550723"
    ],
    [
      "0.465276443",
      "1.161831029"
    ],
    [
      "1.439526941",
      "1.19502497"
    ],
    [
      "1.651879284",
      "1.19993066"
    ],
    [
      "0.820228531",
      "1.220000169"
    ],
    [
      "0.265640266",
      "1.234377428"
    ],
    [
      "1.859077406",
      "1.246690915"
    ],
    [
      "1.187777295",
      "1.262467418"
    ],
    [
      "0.624474421",
      "1.302449616"
    ],
    [
      "0.989508902",
      "1.348303479"
    ],
    [
      "0.424838244",
      "1.374996016"
    ],
    [
      "1.357057665",
      "1.390770728"
    ],
    [
      "1.569410008",
      "1.395676418"
    ],
    [
      "0.793754791",
      "1.430752926"
    ],
    [
      "1.77660813",
      "1.442436673"
    ],
    [
      "0.223459986",
      "1.442556215"
    ],
    [
      "1.161303554",
      "1.473220174"
    ],
    [
      "0.551194087",
      "1.545734951"
    ],
    [
      "0.96670523",
      "1.55948486"
    ],
    [
      "1.334253993",
      "1.601952109"
    ],
    [
      "1.65027857",
      "1.613195056"
    ],
    [
      "0.34981583",
      "1.61329515"
    ],
    [
      "0.742993321",
      "1.637007274"
    ],
    [
      "1.138499882",
      "1.684401555"
    ],
    [
      "1.487223408",
      "1.749322192"
    ],
    [
      "0.52390706",
      "1.756400986"
    ],
    [
      "0.91259466",
      "1.764886003"
    ],
    [
      "1.294768391",
      "1.84338822"
    ],
    [
      "0.717047763",
      "1.847825706"
    ],
    [
      "1.084389312",
      "1.889802697"
    ]
  ],
  "radius": "0.106204499"
}
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": 71
}

起步布局答案

{
  "centers": [
    [
      "0.924821065",
      "0.109371838"
    ],
    [
      "1.137100764",
      "0.116782146"
    ],
    [
      "0.716787226",
      "0.15226129"
    ],
    [
      "1.365403654",
      "0.184309761"
    ],
    [
      "0.524748317",
      "0.243028247"
    ],
    [
      "1.547559321",
      "0.293565866"
    ],
    [
      "0.989076292",
      "0.311828867"
    ],
    [
      "1.20135599",
      "0.319239176"
    ],
    [
      "0.727786998",
      "0.364385282"
    ],
    [
      "0.359550067",
      "0.37654649"
    ],
    [
      "1.383511657",
      "0.428495281"
    ],
    [
      "1.698790631",
      "0.442719056"
    ],
    [
      "0.547158754",
      "0.476148517"
    ],
    [
      "0.891048493",
      "0.500264888"
    ],
    [
      "1.103328192",
      "0.507675197"
    ],
    [
      "1.534742966",
      "0.577648471"
    ],
    [
      "0.364576278",
      "0.588896014"
    ],
    [
      "0.710420249",
      "0.612028122"
    ],
    [
      "1.810556534",
      "0.623345648"
    ],
    [
      "1.299981589",
      "0.626622204"
    ],
    [
      "0.168822167",
      "0.67134546"
    ],
    [
      "1.092946582",
      "0.719830341"
    ],
    [
      "0.529302447",
      "0.722996247"
    ],
    [
      "0.881329383",
      "0.73815366"
    ],
    [
      "1.453447269",
      "0.773884535"
    ],
    [
      "1.665799611",
      "0.778790225"
    ],
    [
      "0.333548336",
      "0.805445694"
    ],
    [
      "1.87654486",
      "0.825244445"
    ],
    [
      "0.701803505",
      "0.851679174"
    ],
    [
      "1.257023097",
      "0.854724672"
    ],
    [
      "0.114742412",
      "0.876754718"
    ],
    [
      "1.058162922",
      "0.929371937"
    ],
    [
      "0.506049394",
      "0.93412862"
    ],
    [
      "1.473478339",
      "0.985346917"
    ],
    [
      "1.685830682",
      "0.990252607"
    ],
    [
      "0.856784665",
      "0.996932136"
    ],
    [
      "0.310295283",
      "1.016578066"
    ],
    [
      "1.893028804",
      "1.037012862"
    ],
    [
      "1.270246571",
      "1.06672166"
    ],
    [
      "0.661030554",
      "1.079381582"
    ],
    [
      "0.110659106",
      "1.089124466"
    ],
    [
      "1.015982642",
      "1.137550723"
    ],
    [
      "0.465276443",
      "1.161831029"
    ],
    [
      "1.439526941",
      "1.19502497"
    ],
    [
      "1.651879284",
      "1.19993066"
    ],
    [
      "0.820228531",
      "1.220000169"
    ],
    [
      "0.265640266",
      "1.234377428"
    ],
    [
      "1.859077406",
      "1.246690915"
    ],
    [
      "1.187777295",
      "1.262467418"
    ],
    [
      "0.624474421",
      "1.302449616"
    ],
    [
      "0.989508902",
      "1.348303479"
    ],
    [
      "0.424838244",
      "1.374996016"
    ],
    [
      "1.357057665",
      "1.390770728"
    ],
    [
      "1.569410008",
      "1.395676418"
    ],
    [
      "0.793754791",
      "1.430752926"
    ],
    [
      "1.77660813",
      "1.442436673"
    ],
    [
      "0.223459986",
      "1.442556215"
    ],
    [
      "1.161303554",
      "1.473220174"
    ],
    [
      "0.551194087",
      "1.545734951"
    ],
    [
      "0.96670523",
      "1.55948486"
    ],
    [
      "1.334253993",
      "1.601952109"
    ],
    [
      "1.65027857",
      "1.613195056"
    ],
    [
      "0.34981583",
      "1.61329515"
    ],
    [
      "0.742993321",
      "1.637007274"
    ],
    [
      "1.138499882",
      "1.684401555"
    ],
    [
      "1.487223408",
      "1.749322192"
    ],
    [
      "0.52390706",
      "1.756400986"
    ],
    [
      "0.91259466",
      "1.764886003"
    ],
    [
      "1.294768391",
      "1.84338822"
    ],
    [
      "0.717047763",
      "1.847825706"
    ],
    [
      "1.084389312",
      "1.889802697"
    ]
  ],
  "radius": "0.106204499"
}

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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