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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 = 65

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 = 65
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[0.1108967437229617084149485658035, 0.1108967437229617084149485658045]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 = 0.110896743

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 = 65 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": [
    [
      "1.088222507",
      "0.115284571"
    ],
    [
      "0.86650206",
      "0.120976166"
    ],
    [
      "1.304452952",
      "0.164647978"
    ],
    [
      "0.653089062",
      "0.18136858"
    ],
    [
      "1.501737578",
      "0.265994548"
    ],
    [
      "0.461264003",
      "0.29270365"
    ],
    [
      "1.153587765",
      "0.327227333"
    ],
    [
      "0.931867318",
      "0.332918928"
    ],
    [
      "0.71845432",
      "0.393311341"
    ],
    [
      "1.667799551",
      "0.413017581"
    ],
    [
      "1.353662643",
      "0.433171047"
    ],
    [
      "0.302963974",
      "0.448053101"
    ],
    [
      "0.526629262",
      "0.504646412"
    ],
    [
      "1.159796251",
      "0.548933909"
    ],
    [
      "0.933893361",
      "0.557062055"
    ],
    [
      "1.524699041",
      "0.582471395"
    ],
    [
      "1.792304992",
      "0.596567973"
    ],
    [
      "0.71896079",
      "0.615104251"
    ],
    [
      "0.188039836",
      "0.637749684"
    ],
    [
      "0.403760264",
      "0.689296244"
    ],
    [
      "1.330808223",
      "0.690168665"
    ],
    [
      "1.07537947",
      "0.75403427"
    ],
    [
      "1.649204482",
      "0.766021787"
    ],
    [
      "0.856389807",
      "0.789189334"
    ],
    [
      "0.595966826",
      "0.801914563"
    ],
    [
      "1.867506045",
      "0.805223557"
    ],
    [
      "0.123643207",
      "0.849988759"
    ],
    [
      "1.455313664",
      "0.873719057"
    ],
    [
      "0.339363636",
      "0.90153532"
    ],
    [
      "1.23602486",
      "0.906957158"
    ],
    [
      "0.744440331",
      "0.980656475"
    ],
    [
      "1.666079923",
      "0.989144819"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.526321575",
      "1.020862935"
    ],
    [
      "1.88872302",
      "1.025999894"
    ],
    [
      "0.113781431",
      "1.071562893"
    ],
    [
      "1.448723235",
      "1.095414608"
    ],
    [
      "0.329501859",
      "1.123109453"
    ],
    [
      "1.220349987",
      "1.128196055"
    ],
    [
      "0.827336042",
      "1.186376314"
    ],
    [
      "1.634740193",
      "1.216203856"
    ],
    [
      "0.600744805",
      "1.229797217"
    ],
    [
      "1.854635605",
      "1.245158282"
    ],
    [
      "1.038288825",
      "1.254869194"
    ],
    [
      "1.361452657",
      "1.299317029"
    ],
    [
      "0.166132509",
      "1.304553168"
    ],
    [
      "0.403925089",
      "1.332043734"
    ],
    [
      "0.749692267",
      "1.394135297"
    ],
    [
      "1.547469615",
      "1.420106277"
    ],
    [
      "1.179391494",
      "1.425990168"
    ],
    [
      "1.767365027",
      "1.449060704"
    ],
    [
      "0.960645049",
      "1.462628178"
    ],
    [
      "0.55287255",
      "1.496381815"
    ],
    [
      "0.271888934",
      "1.510253736"
    ],
    [
      "1.372174147",
      "1.557347111"
    ],
    [
      "0.737860814",
      "1.618740687"
    ],
    [
      "1.632342052",
      "1.625018503"
    ],
    [
      "1.168534643",
      "1.647517773"
    ],
    [
      "0.420836395",
      "1.674591817"
    ],
    [
      "0.949788198",
      "1.684155783"
    ],
    [
      "1.457969031",
      "1.762081995"
    ],
    [
      "0.605824659",
      "1.796950689"
    ],
    [
      "1.255097032",
      "1.851721847"
    ],
    [
      "0.815342079",
      "1.869716076"
    ],
    [
      "1.036350587",
      "1.888359857"
    ]
  ],
  "radius": "0.110896743"
}
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": 65
}

起步布局答案

{
  "centers": [
    [
      "1.088222507",
      "0.115284571"
    ],
    [
      "0.86650206",
      "0.120976166"
    ],
    [
      "1.304452952",
      "0.164647978"
    ],
    [
      "0.653089062",
      "0.18136858"
    ],
    [
      "1.501737578",
      "0.265994548"
    ],
    [
      "0.461264003",
      "0.29270365"
    ],
    [
      "1.153587765",
      "0.327227333"
    ],
    [
      "0.931867318",
      "0.332918928"
    ],
    [
      "0.71845432",
      "0.393311341"
    ],
    [
      "1.667799551",
      "0.413017581"
    ],
    [
      "1.353662643",
      "0.433171047"
    ],
    [
      "0.302963974",
      "0.448053101"
    ],
    [
      "0.526629262",
      "0.504646412"
    ],
    [
      "1.159796251",
      "0.548933909"
    ],
    [
      "0.933893361",
      "0.557062055"
    ],
    [
      "1.524699041",
      "0.582471395"
    ],
    [
      "1.792304992",
      "0.596567973"
    ],
    [
      "0.71896079",
      "0.615104251"
    ],
    [
      "0.188039836",
      "0.637749684"
    ],
    [
      "0.403760264",
      "0.689296244"
    ],
    [
      "1.330808223",
      "0.690168665"
    ],
    [
      "1.07537947",
      "0.75403427"
    ],
    [
      "1.649204482",
      "0.766021787"
    ],
    [
      "0.856389807",
      "0.789189334"
    ],
    [
      "0.595966826",
      "0.801914563"
    ],
    [
      "1.867506045",
      "0.805223557"
    ],
    [
      "0.123643207",
      "0.849988759"
    ],
    [
      "1.455313664",
      "0.873719057"
    ],
    [
      "0.339363636",
      "0.90153532"
    ],
    [
      "1.23602486",
      "0.906957158"
    ],
    [
      "0.744440331",
      "0.980656475"
    ],
    [
      "1.666079923",
      "0.989144819"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.526321575",
      "1.020862935"
    ],
    [
      "1.88872302",
      "1.025999894"
    ],
    [
      "0.113781431",
      "1.071562893"
    ],
    [
      "1.448723235",
      "1.095414608"
    ],
    [
      "0.329501859",
      "1.123109453"
    ],
    [
      "1.220349987",
      "1.128196055"
    ],
    [
      "0.827336042",
      "1.186376314"
    ],
    [
      "1.634740193",
      "1.216203856"
    ],
    [
      "0.600744805",
      "1.229797217"
    ],
    [
      "1.854635605",
      "1.245158282"
    ],
    [
      "1.038288825",
      "1.254869194"
    ],
    [
      "1.361452657",
      "1.299317029"
    ],
    [
      "0.166132509",
      "1.304553168"
    ],
    [
      "0.403925089",
      "1.332043734"
    ],
    [
      "0.749692267",
      "1.394135297"
    ],
    [
      "1.547469615",
      "1.420106277"
    ],
    [
      "1.179391494",
      "1.425990168"
    ],
    [
      "1.767365027",
      "1.449060704"
    ],
    [
      "0.960645049",
      "1.462628178"
    ],
    [
      "0.55287255",
      "1.496381815"
    ],
    [
      "0.271888934",
      "1.510253736"
    ],
    [
      "1.372174147",
      "1.557347111"
    ],
    [
      "0.737860814",
      "1.618740687"
    ],
    [
      "1.632342052",
      "1.625018503"
    ],
    [
      "1.168534643",
      "1.647517773"
    ],
    [
      "0.420836395",
      "1.674591817"
    ],
    [
      "0.949788198",
      "1.684155783"
    ],
    [
      "1.457969031",
      "1.762081995"
    ],
    [
      "0.605824659",
      "1.796950689"
    ],
    [
      "1.255097032",
      "1.851721847"
    ],
    [
      "0.815342079",
      "1.869716076"
    ],
    [
      "1.036350587",
      "1.888359857"
    ]
  ],
  "radius": "0.110896743"
}

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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