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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 = 52

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 = 52
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[0.1236901645924695926980281606685, 0.1236901645924695926980281606695]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 = 0.123690164

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 = 52 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": [
    [
      "0.954205259",
      "0.124887568"
    ],
    [
      "1.200599008",
      "0.14695899"
    ],
    [
      "0.711460987",
      "0.172555642"
    ],
    [
      "1.431006613",
      "0.237010991"
    ],
    [
      "0.491710977",
      "0.286164447"
    ],
    [
      "1.100750996",
      "0.373293701"
    ],
    [
      "0.853377859",
      "0.375179996"
    ],
    [
      "1.62706642",
      "0.387867145"
    ],
    [
      "0.312467557",
      "0.456660265"
    ],
    [
      "1.331158601",
      "0.463345703"
    ],
    [
      "0.633627849",
      "0.488788801"
    ],
    [
      "1.773154025",
      "0.587505416"
    ],
    [
      "1.527218408",
      "0.614201857"
    ],
    [
      "0.89600753",
      "0.619483554"
    ],
    [
      "1.143367689",
      "0.625129586"
    ],
    [
      "0.188015002",
      "0.670455935"
    ],
    [
      "0.436402433",
      "0.670756443"
    ],
    [
      "0.675947835",
      "0.732522356"
    ],
    [
      "1.357730687",
      "0.794398804"
    ],
    [
      "1.857627406",
      "0.820016222"
    ],
    [
      "1.61169179",
      "0.846712663"
    ],
    [
      "1.12346052",
      "0.873862733"
    ],
    [
      "0.87611293",
      "0.877887283"
    ],
    [
      "0.128271189",
      "0.910513645"
    ],
    [
      "0.375651514",
      "0.910561266"
    ],
    [
      "0.615196915",
      "0.972327179"
    ],
    [
      "1.38627999",
      "1.040126223"
    ],
    [
      "1.873754708",
      "1.066870304"
    ],
    [
      "1.627819092",
      "1.093566745"
    ],
    [
      "1.116203935",
      "1.121136608"
    ],
    [
      "0.868856345",
      "1.125161158"
    ],
    [
      "0.137997225",
      "1.157702705"
    ],
    [
      "0.385377549",
      "1.157750326"
    ],
    [
      "0.639754654",
      "1.218485552"
    ],
    [
      "1.309644637",
      "1.275336873"
    ],
    [
      "1.820250714",
      "1.308395354"
    ],
    [
      "1.55118374",
      "1.328777395"
    ],
    [
      "0.846608939",
      "1.371539079"
    ],
    [
      "0.216418022",
      "1.392324115"
    ],
    [
      "0.463798346",
      "1.392371736"
    ],
    [
      "1.09291194",
      "1.39460114"
    ],
    [
      "1.360043618",
      "1.517528884"
    ],
    [
      "1.701379265",
      "1.525343748"
    ],
    [
      "0.663700935",
      "1.538097446"
    ],
    [
      "0.896364388",
      "1.622149483"
    ],
    [
      "0.389004112",
      "1.628174301"
    ],
    [
      "1.143310921",
      "1.636793151"
    ],
    [
      "1.526613471",
      "1.700426427"
    ],
    [
      "0.588906701",
      "1.773900011"
    ],
    [
      "1.309880774",
      "1.819690694"
    ],
    [
      "0.821570154",
      "1.857952049"
    ],
    [
      "1.068453045",
      "1.873632136"
    ]
  ],
  "radius": "0.123690164"
}
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": 52
}

起步布局答案

{
  "centers": [
    [
      "0.954205259",
      "0.124887568"
    ],
    [
      "1.200599008",
      "0.14695899"
    ],
    [
      "0.711460987",
      "0.172555642"
    ],
    [
      "1.431006613",
      "0.237010991"
    ],
    [
      "0.491710977",
      "0.286164447"
    ],
    [
      "1.100750996",
      "0.373293701"
    ],
    [
      "0.853377859",
      "0.375179996"
    ],
    [
      "1.62706642",
      "0.387867145"
    ],
    [
      "0.312467557",
      "0.456660265"
    ],
    [
      "1.331158601",
      "0.463345703"
    ],
    [
      "0.633627849",
      "0.488788801"
    ],
    [
      "1.773154025",
      "0.587505416"
    ],
    [
      "1.527218408",
      "0.614201857"
    ],
    [
      "0.89600753",
      "0.619483554"
    ],
    [
      "1.143367689",
      "0.625129586"
    ],
    [
      "0.188015002",
      "0.670455935"
    ],
    [
      "0.436402433",
      "0.670756443"
    ],
    [
      "0.675947835",
      "0.732522356"
    ],
    [
      "1.357730687",
      "0.794398804"
    ],
    [
      "1.857627406",
      "0.820016222"
    ],
    [
      "1.61169179",
      "0.846712663"
    ],
    [
      "1.12346052",
      "0.873862733"
    ],
    [
      "0.87611293",
      "0.877887283"
    ],
    [
      "0.128271189",
      "0.910513645"
    ],
    [
      "0.375651514",
      "0.910561266"
    ],
    [
      "0.615196915",
      "0.972327179"
    ],
    [
      "1.38627999",
      "1.040126223"
    ],
    [
      "1.873754708",
      "1.066870304"
    ],
    [
      "1.627819092",
      "1.093566745"
    ],
    [
      "1.116203935",
      "1.121136608"
    ],
    [
      "0.868856345",
      "1.125161158"
    ],
    [
      "0.137997225",
      "1.157702705"
    ],
    [
      "0.385377549",
      "1.157750326"
    ],
    [
      "0.639754654",
      "1.218485552"
    ],
    [
      "1.309644637",
      "1.275336873"
    ],
    [
      "1.820250714",
      "1.308395354"
    ],
    [
      "1.55118374",
      "1.328777395"
    ],
    [
      "0.846608939",
      "1.371539079"
    ],
    [
      "0.216418022",
      "1.392324115"
    ],
    [
      "0.463798346",
      "1.392371736"
    ],
    [
      "1.09291194",
      "1.39460114"
    ],
    [
      "1.360043618",
      "1.517528884"
    ],
    [
      "1.701379265",
      "1.525343748"
    ],
    [
      "0.663700935",
      "1.538097446"
    ],
    [
      "0.896364388",
      "1.622149483"
    ],
    [
      "0.389004112",
      "1.628174301"
    ],
    [
      "1.143310921",
      "1.636793151"
    ],
    [
      "1.526613471",
      "1.700426427"
    ],
    [
      "0.588906701",
      "1.773900011"
    ],
    [
      "1.309880774",
      "1.819690694"
    ],
    [
      "0.821570154",
      "1.857952049"
    ],
    [
      "1.068453045",
      "1.873632136"
    ]
  ],
  "radius": "0.123690164"
}

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

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