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 = 228

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 = 228
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[0.0605143792272492351463711024695, 0.0605143792272492351463711024705]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 = 0.060514378

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 = 228 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": [
    [
      "1.184486865",
      "1.535991"
    ],
    [
      "1.589065348",
      "0.99219747"
    ],
    [
      "1.189365245",
      "0.660946523"
    ],
    [
      "0.398085029",
      "1.445083359"
    ],
    [
      "0.419593851",
      "0.682368445"
    ],
    [
      "0.68585612",
      "0.851270319"
    ],
    [
      "1.717363028",
      "0.599869273"
    ],
    [
      "1.086805594",
      "0.188384239"
    ],
    [
      "0.986376548",
      "0.275596067"
    ],
    [
      "0.332676026",
      "0.766589891"
    ],
    [
      "1.813678472",
      "0.821391467"
    ],
    [
      "0.69663292",
      "0.411920507"
    ],
    [
      "0.095675796",
      "0.745380744"
    ],
    [
      "0.244693693",
      "0.851481045"
    ],
    [
      "1.74252164",
      "1.15505265"
    ],
    [
      "0.975599747",
      "0.714945879"
    ],
    [
      "0.775560179",
      "1.203100995"
    ],
    [
      "0.921554623",
      "1.353227878"
    ],
    [
      "0.97029015",
      "0.155641125"
    ],
    [
      "0.978689715",
      "1.151310333"
    ],
    [
      "0.370613812",
      "0.445578434"
    ],
    [
      "1.486217722",
      "0.528010459"
    ],
    [
      "1.366038042",
      "0.490990042"
    ],
    [
      "1.151917898",
      "0.982242778"
    ],
    [
      "1.683767862",
      "0.789646118"
    ],
    [
      "1.301002309",
      "1.568734115"
    ],
    [
      "1.471562878",
      "0.96319523"
    ],
    [
      "0.344100254",
      "1.672628244"
    ],
    [
      "1.531365584",
      "1.197749015"
    ],
    [
      "0.422380085",
      "1.118420568"
    ],
    [
      "0.653488662",
      "0.126751396"
    ],
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      "1.417517754",
      "1.60147723"
    ],
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      "0.22830139",
      "0.553055051"
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      "1.392609176"
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      "0.135912846",
      "0.631236361"
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    ],
    [
      "0.66643064",
      "1.620099643"
    ],
    [
      "0.856425579",
      "1.590557189"
    ],
    [
      "0.30228526",
      "0.648837864"
    ],
    [
      "1.238531989",
      "0.897709001"
    ],
    [
      "1.518703459",
      "0.216686555"
    ],
    [
      "1.0055011",
      "0.832222771"
    ],
    [
      "0.887603493",
      "0.067261957"
    ],
    [
      "1.384948787",
      "1.047729008"
    ],
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      "1.223917261",
      "0.309258224"
    ],
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      "1.501464231",
      "1.080472123"
    ],
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      "0.071526773",
      "1.143424889"
    ],
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      "0.626396631",
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      "1.095205159",
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      "1.42544756",
      "0.293831209"
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      "0.487966317",
      "0.475181641"
    ],
    [
      "1.214204444",
      "1.653314595"
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      "0.389667168",
      "0.565098014"
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      "0.204038026",
      "1.499076915"
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      "0.772470212",
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      "0.275963287",
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      "1.628460922",
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      "0.598579615",
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    ],
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      "1.395725587",
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      "1.132262785"
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      "1.137965092",
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      "1.162694698",
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      "1.30588069",
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      "1.034199122"
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      "1.181819251",
      "1.09951967"
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      "1.335495241"
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      "1.556304624"
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      "1.132793345",
      "0.425616074"
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      "0.101969599",
      "1.275838952"
    ],
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      "0.572020563",
      "0.216255079"
    ],
    [
      "0.46129577",
      "0.357128084"
    ],
    [
      "1.249308789",
      "0.458359189"
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    [
      "0.745658826",
      "1.085824103"
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    [
      "1.861572073",
      "0.625397818"
    ],
    [
      "1.325146081",
      "0.813175223"
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    [
      "0.070446684",
      "0.863750724"
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    [
      "0.656551768",
      "1.181079221"
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    [
      "1.072849801",
      "0.628203408"
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    [
      "1.459547174",
      "0.409956902"
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  ],
  "radius": "0.060514378"
}
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": 228
}

起步布局答案

{
  "centers": [
    [
      "1.184486865",
      "1.535991"
    ],
    [
      "1.589065348",
      "0.99219747"
    ],
    [
      "1.189365245",
      "0.660946523"
    ],
    [
      "0.398085029",
      "1.445083359"
    ],
    [
      "0.419593851",
      "0.682368445"
    ],
    [
      "0.68585612",
      "0.851270319"
    ],
    [
      "1.717363028",
      "0.599869273"
    ],
    [
      "1.086805594",
      "0.188384239"
    ],
    [
      "0.986376548",
      "0.275596067"
    ],
    [
      "0.332676026",
      "0.766589891"
    ],
    [
      "1.813678472",
      "0.821391467"
    ],
    [
      "0.69663292",
      "0.411920507"
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  "radius": "0.060514378"
}

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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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。

还没有帖子。第一个聊聊这道题的思路?