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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 = 95

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 = 95
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[0.0922491777607209227798456386505, 0.0922491777607209227798456386515]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 = 0.092249177

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 = 95 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": [
    [
      "1.064433221",
      "0.094538838"
    ],
    [
      "0.880022135",
      "0.100212877"
    ],
    [
      "1.246182591",
      "0.126269099"
    ],
    [
      "0.700567295",
      "0.143056822"
    ],
    [
      "1.417762241",
      "0.194092893"
    ],
    [
      "0.533481921",
      "0.221300805"
    ],
    [
      "1.016995223",
      "0.272834322"
    ],
    [
      "0.832584137",
      "0.278508361"
    ],
    [
      "1.572084271",
      "0.295208441"
    ],
    [
      "1.198744594",
      "0.304564583"
    ],
    [
      "0.385258034",
      "0.332213269"
    ],
    [
      "1.370324243",
      "0.372388377"
    ],
    [
      "0.568271243",
      "0.402489508"
    ],
    [
      "1.702773691",
      "0.425438694"
    ],
    [
      "0.932756065",
      "0.436978794"
    ],
    [
      "0.74834498",
      "0.442652832"
    ],
    [
      "1.524646274",
      "0.473503925"
    ],
    [
      "1.1166776",
      "0.477227302"
    ],
    [
      "0.238683636",
      "0.505622665"
    ],
    [
      "0.422885441",
      "0.516079171"
    ],
    [
      "1.29095037",
      "0.538939965"
    ],
    [
      "0.623525652",
      "0.5785196"
    ],
    [
      "1.804431763",
      "0.579403883"
    ],
    [
      "0.989167463",
      "0.612641531"
    ],
    [
      "0.804756377",
      "0.61831557"
    ],
    [
      "1.4452724",
      "0.640055513"
    ],
    [
      "1.635068861",
      "0.65300184"
    ],
    [
      "0.321728938",
      "0.67037436"
    ],
    [
      "1.163680781",
      "0.67251427"
    ],
    [
      "0.140877791",
      "0.706877865"
    ],
    [
      "0.500494762",
      "0.716007938"
    ],
    [
      "1.872859027",
      "0.750743758"
    ],
    [
      "0.680826927",
      "0.754994517"
    ],
    [
      "1.318002812",
      "0.773629818"
    ],
    [
      "0.861159093",
      "0.793981095"
    ],
    [
      "1.52934797",
      "0.804283835"
    ],
    [
      "1.044626307",
      "0.813459916"
    ],
    [
      "1.710625561",
      "0.838606979"
    ],
    [
      "0.371721394",
      "0.848293246"
    ],
    [
      "0.099354823",
      "0.886642952"
    ],
    [
      "0.556897477",
      "0.891673464"
    ],
    [
      "1.198948337",
      "0.914575465"
    ],
    [
      "0.737229643",
      "0.930660043"
    ],
    [
      "1.905228775",
      "0.932380326"
    ],
    [
      "1.381381871",
      "0.946922981"
    ],
    [
      "0.917561808",
      "0.969646621"
    ],
    [
      "1.561717718",
      "0.985920402"
    ],
    [
      "0.252647411",
      "0.989312448"
    ],
    [
      "1.742995309",
      "1.020243547"
    ],
    [
      "0.432968027",
      "1.028352411"
    ],
    [
      "1.080560666",
      "1.056081657"
    ],
    [
      "0.613300193",
      "1.06733899"
    ],
    [
      "0.096481649",
      "1.087556521"
    ],
    [
      "1.262302459",
      "1.087855292"
    ],
    [
      "0.793632358",
      "1.106325568"
    ],
    [
      "1.900203823",
      "1.116810239"
    ],
    [
      "1.442634625",
      "1.12684187"
    ],
    [
      "1.624156368",
      "1.163714905"
    ],
    [
      "0.289844603",
      "1.170953365"
    ],
    [
      "0.956631216",
      "1.192760604"
    ],
    [
      "0.485474648",
      "1.205221566"
    ],
    [
      "1.136963381",
      "1.231747183"
    ],
    [
      "0.669702908",
      "1.243004515"
    ],
    [
      "1.318705175",
      "1.263520817"
    ],
    [
      "0.13284721",
      "1.268435456"
    ],
    [
      "1.857991751",
      "1.296414761"
    ],
    [
      "1.49903734",
      "1.302507396"
    ],
    [
      "0.827909191",
      "1.337927721"
    ],
    [
      "1.679238207",
      "1.342096418"
    ],
    [
      "0.363389469",
      "1.343550349"
    ],
    [
      "1.009692284",
      "1.369464215"
    ],
    [
      "0.545697358",
      "1.379614423"
    ],
    [
      "1.200317503",
      "1.40502701"
    ],
    [
      "0.205034565",
      "1.438225412"
    ],
    [
      "1.380649669",
      "1.444013588"
    ],
    [
      "0.712161837",
      "1.481601634"
    ],
    [
      "1.566159241",
      "1.487879786"
    ],
    [
      "1.748950825",
      "1.512917359"
    ],
    [
      "0.366203026",
      "1.52802725"
    ],
    [
      "1.076217806",
      "1.541551397"
    ],
    [
      "0.887628389",
      "1.54507645"
    ],
    [
      "0.54713916",
      "1.564107144"
    ],
    [
      "1.256720219",
      "1.580692535"
    ],
    [
      "1.437052384",
      "1.619679114"
    ],
    [
      "0.701102367",
      "1.665768219"
    ],
    [
      "1.615355364",
      "1.667289074"
    ],
    [
      "0.425424988",
      "1.702762485"
    ],
    [
      "1.058410748",
      "1.725188408"
    ],
    [
      "0.873980736",
      "1.730209763"
    ],
    [
      "1.241888389",
      "1.764593759"
    ],
    [
      "1.422220555",
      "1.803580337"
    ],
    [
      "0.579388195",
      "1.80442356"
    ],
    [
      "0.750726735",
      "1.872854166"
    ],
    [
      "1.116792683",
      "1.900206101"
    ],
    [
      "0.932362671",
      "1.905227456"
    ]
  ],
  "radius": "0.092249177"
}
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": 95
}

起步布局答案

{
  "centers": [
    [
      "1.064433221",
      "0.094538838"
    ],
    [
      "0.880022135",
      "0.100212877"
    ],
    [
      "1.246182591",
      "0.126269099"
    ],
    [
      "0.700567295",
      "0.143056822"
    ],
    [
      "1.417762241",
      "0.194092893"
    ],
    [
      "0.533481921",
      "0.221300805"
    ],
    [
      "1.016995223",
      "0.272834322"
    ],
    [
      "0.832584137",
      "0.278508361"
    ],
    [
      "1.572084271",
      "0.295208441"
    ],
    [
      "1.198744594",
      "0.304564583"
    ],
    [
      "0.385258034",
      "0.332213269"
    ],
    [
      "1.370324243",
      "0.372388377"
    ],
    [
      "0.568271243",
      "0.402489508"
    ],
    [
      "1.702773691",
      "0.425438694"
    ],
    [
      "0.932756065",
      "0.436978794"
    ],
    [
      "0.74834498",
      "0.442652832"
    ],
    [
      "1.524646274",
      "0.473503925"
    ],
    [
      "1.1166776",
      "0.477227302"
    ],
    [
      "0.238683636",
      "0.505622665"
    ],
    [
      "0.422885441",
      "0.516079171"
    ],
    [
      "1.29095037",
      "0.538939965"
    ],
    [
      "0.623525652",
      "0.5785196"
    ],
    [
      "1.804431763",
      "0.579403883"
    ],
    [
      "0.989167463",
      "0.612641531"
    ],
    [
      "0.804756377",
      "0.61831557"
    ],
    [
      "1.4452724",
      "0.640055513"
    ],
    [
      "1.635068861",
      "0.65300184"
    ],
    [
      "0.321728938",
      "0.67037436"
    ],
    [
      "1.163680781",
      "0.67251427"
    ],
    [
      "0.140877791",
      "0.706877865"
    ],
    [
      "0.500494762",
      "0.716007938"
    ],
    [
      "1.872859027",
      "0.750743758"
    ],
    [
      "0.680826927",
      "0.754994517"
    ],
    [
      "1.318002812",
      "0.773629818"
    ],
    [
      "0.861159093",
      "0.793981095"
    ],
    [
      "1.52934797",
      "0.804283835"
    ],
    [
      "1.044626307",
      "0.813459916"
    ],
    [
      "1.710625561",
      "0.838606979"
    ],
    [
      "0.371721394",
      "0.848293246"
    ],
    [
      "0.099354823",
      "0.886642952"
    ],
    [
      "0.556897477",
      "0.891673464"
    ],
    [
      "1.198948337",
      "0.914575465"
    ],
    [
      "0.737229643",
      "0.930660043"
    ],
    [
      "1.905228775",
      "0.932380326"
    ],
    [
      "1.381381871",
      "0.946922981"
    ],
    [
      "0.917561808",
      "0.969646621"
    ],
    [
      "1.561717718",
      "0.985920402"
    ],
    [
      "0.252647411",
      "0.989312448"
    ],
    [
      "1.742995309",
      "1.020243547"
    ],
    [
      "0.432968027",
      "1.028352411"
    ],
    [
      "1.080560666",
      "1.056081657"
    ],
    [
      "0.613300193",
      "1.06733899"
    ],
    [
      "0.096481649",
      "1.087556521"
    ],
    [
      "1.262302459",
      "1.087855292"
    ],
    [
      "0.793632358",
      "1.106325568"
    ],
    [
      "1.900203823",
      "1.116810239"
    ],
    [
      "1.442634625",
      "1.12684187"
    ],
    [
      "1.624156368",
      "1.163714905"
    ],
    [
      "0.289844603",
      "1.170953365"
    ],
    [
      "0.956631216",
      "1.192760604"
    ],
    [
      "0.485474648",
      "1.205221566"
    ],
    [
      "1.136963381",
      "1.231747183"
    ],
    [
      "0.669702908",
      "1.243004515"
    ],
    [
      "1.318705175",
      "1.263520817"
    ],
    [
      "0.13284721",
      "1.268435456"
    ],
    [
      "1.857991751",
      "1.296414761"
    ],
    [
      "1.49903734",
      "1.302507396"
    ],
    [
      "0.827909191",
      "1.337927721"
    ],
    [
      "1.679238207",
      "1.342096418"
    ],
    [
      "0.363389469",
      "1.343550349"
    ],
    [
      "1.009692284",
      "1.369464215"
    ],
    [
      "0.545697358",
      "1.379614423"
    ],
    [
      "1.200317503",
      "1.40502701"
    ],
    [
      "0.205034565",
      "1.438225412"
    ],
    [
      "1.380649669",
      "1.444013588"
    ],
    [
      "0.712161837",
      "1.481601634"
    ],
    [
      "1.566159241",
      "1.487879786"
    ],
    [
      "1.748950825",
      "1.512917359"
    ],
    [
      "0.366203026",
      "1.52802725"
    ],
    [
      "1.076217806",
      "1.541551397"
    ],
    [
      "0.887628389",
      "1.54507645"
    ],
    [
      "0.54713916",
      "1.564107144"
    ],
    [
      "1.256720219",
      "1.580692535"
    ],
    [
      "1.437052384",
      "1.619679114"
    ],
    [
      "0.701102367",
      "1.665768219"
    ],
    [
      "1.615355364",
      "1.667289074"
    ],
    [
      "0.425424988",
      "1.702762485"
    ],
    [
      "1.058410748",
      "1.725188408"
    ],
    [
      "0.873980736",
      "1.730209763"
    ],
    [
      "1.241888389",
      "1.764593759"
    ],
    [
      "1.422220555",
      "1.803580337"
    ],
    [
      "0.579388195",
      "1.80442356"
    ],
    [
      "0.750726735",
      "1.872854166"
    ],
    [
      "1.116792683",
      "1.900206101"
    ],
    [
      "0.932362671",
      "1.905227456"
    ]
  ],
  "radius": "0.092249177"
}

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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