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 = 206

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 = 206
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[0.0635543769999471419973221478925, 0.0635543769999471419973221478935]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 = 0.063554376

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 = 206 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": [
    [
      "0.344056457",
      "0.670952441"
    ],
    [
      "1.553622959",
      "0.753911888"
    ],
    [
      "0.821958191",
      "1.306009438"
    ],
    [
      "1.107686876",
      "1.128652267"
    ],
    [
      "1.097151765",
      "0.421424745"
    ],
    [
      "0.569470449",
      "1.279970557"
    ],
    [
      "0.811226718",
      "0.969882553"
    ],
    [
      "1.530947424",
      "1.675362397"
    ],
    [
      "1.131641515",
      "0.575116137"
    ],
    [
      "1.073000763",
      "1.346061511"
    ],
    [
      "1.831854496",
      "1.311441822"
    ],
    [
      "0.795622095",
      "1.529646307"
    ],
    [
      "0.84591283",
      "0.752473309"
    ],
    [
      "1.39341556",
      "0.951295095"
    ],
    [
      "0.800495244",
      "0.633755668"
    ],
    [
      "1.462787786",
      "0.516476606"
    ],
    [
      "1.746146671",
      "1.565772786"
    ],
    [
      "0.32119393",
      "1.226962889"
    ],
    [
      "1.3372665",
      "0.49645057"
    ],
    [
      "0.274104592",
      "1.107952032"
    ],
    [
      "0.474037396",
      "0.225213743"
    ],
    [
      "1.006120229",
      "0.555090101"
    ],
    [
      "1.922808342",
      "0.840766938"
    ],
    [
      "1.373015156",
      "0.278640263"
    ],
    [
      "1.358729448",
      "1.16870434"
    ],
    [
      "1.599040546",
      "0.872629528"
    ],
    [
      "1.689875719",
      "1.110064809"
    ],
    [
      "0.295029808",
      "1.450747532"
    ],
    [
      "0.741854492",
      "1.404701042"
    ],
    [
      "1.062269289",
      "1.009934626"
    ],
    [
      "0.309141574",
      "0.889778407"
    ],
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      "0.753399621",
      "1.785084874"
    ],
    [
      "0.464397772",
      "1.62575946"
    ],
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      "1.287421981"
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      "1.620479583",
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      "1.167265761"
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      "0.692395199"
    ],
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      "1.347997974",
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    ],
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      "1.257162801",
      "0.595142174"
    ],
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      "1.084617234",
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      "1.732497443"
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      "0.924549956",
      "1.933401144"
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      "1.289357222",
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      "0.773937924"
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      "0.832056377",
      "1.685236238"
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      "0.425515782",
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      "1.398688197"
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      "1.500475782",
      "0.300272503"
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    [
      "0.446715216",
      "1.246988925"
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    [
      "0.992897064",
      "1.444753115"
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      "0.366851644",
      "1.345874947"
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      "0.685705432",
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    [
      "1.714738416",
      "0.523378"
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    [
      "1.233208162",
      "1.148678303"
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      "0.16927618",
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      "0.971434116",
      "0.772499345"
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      "1.815397004",
      "1.130090846"
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    [
      "0.229037875",
      "0.988470011"
    ],
    [
      "0.638255719",
      "1.731246995"
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      "0.088453928",
      "0.785490877"
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      "1.700583282",
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      "1.327363855"
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      "1.913908321"
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      "0.590893579",
      "1.613291651"
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    [
      "0.677960954",
      "0.599962427"
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      "1.302580387",
      "0.713859814"
    ],
    [
      "1.009933253",
      "0.177156554"
    ],
    [
      "1.588309072",
      "0.536502643"
    ]
  ],
  "radius": "0.063554376"
}
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": 206
}

起步布局答案

{
  "centers": [
    [
      "0.344056457",
      "0.670952441"
    ],
    [
      "1.553622959",
      "0.753911888"
    ],
    [
      "0.821958191",
      "1.306009438"
    ],
    [
      "1.107686876",
      "1.128652267"
    ],
    [
      "1.097151765",
      "0.421424745"
    ],
    [
      "0.569470449",
      "1.279970557"
    ],
    [
      "0.811226718",
      "0.969882553"
    ],
    [
      "1.530947424",
      "1.675362397"
    ],
    [
      "1.131641515",
      "0.575116137"
    ],
    [
      "1.073000763",
      "1.346061511"
    ],
    [
      "1.831854496",
      "1.311441822"
    ],
    [
      "0.795622095",
      "1.529646307"
    ],
    [
      "0.84591283",
      "0.752473309"
    ],
    [
      "1.39341556",
      "0.951295095"
    ],
    [
      "0.800495244",
      "0.633755668"
    ],
    [
      "1.462787786",
      "0.516476606"
    ],
    [
      "1.746146671",
      "1.565772786"
    ],
    [
      "0.32119393",
      "1.226962889"
    ],
    [
      "1.3372665",
      "0.49645057"
    ],
    [
      "0.274104592",
      "1.107952032"
    ],
    [
      "0.474037396",
      "0.225213743"
    ],
    [
      "1.006120229",
      "0.555090101"
    ],
    [
      "1.922808342",
      "0.840766938"
    ],
    [
      "1.373015156",
      "0.278640263"
    ],
    [
      "1.358729448",
      "1.16870434"
    ],
    [
      "1.599040546",
      "0.872629528"
    ],
    [
      "1.689875719",
      "1.110064809"
    ],
    [
      "0.295029808",
      "1.450747532"
    ],
    [
      "0.741854492",
      "1.404701042"
    ],
    [
      "1.062269289",
      "1.009934626"
    ],
    [
      "0.309141574",
      "0.889778407"
    ],
    [
      "0.753399621",
      "1.785084874"
    ],
    [
      "0.464397772",
      "1.62575946"
    ],
    [
      "1.404147034",
      "1.287421981"
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  "radius": "0.063554376"
}

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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还没有帖子。第一个聊聊这道题的思路?