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 = 298

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 = 298
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[0.0533067659534028021856338596605, 0.0533067659534028021856338596615]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 = 0.053306765

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 = 298 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": [
    [
      "1.215985405",
      "0.770958181"
    ],
    [
      "0.903926958",
      "0.700833062"
    ],
    [
      "0.845919281",
      "1.445235714"
    ],
    [
      "1.039712803",
      "0.825979136"
    ],
    [
      "1.288238524",
      "0.692562187"
    ],
    [
      "0.323471053",
      "1.326126834"
    ],
    [
      "1.255207271",
      "1.819718327"
    ],
    [
      "1.890278147",
      "1.321920643"
    ],
    [
      "0.542581227",
      "1.093172532"
    ],
    [
      "0.750620192",
      "1.139922612"
    ],
    [
      "1.623342941",
      "1.146396402"
    ],
    [
      "0.502094471",
      "1.273339561"
    ],
    [
      "1.604717572",
      "1.728384874"
    ],
    [
      "0.904194908",
      "1.941130077"
    ],
    [
      "0.986103462",
      "0.340127403"
    ],
    [
      "0.733179405",
      "1.703798736"
    ],
    [
      "0.557518293",
      "0.163078249"
    ],
    [
      "1.116459416",
      "1.898356499"
    ],
    [
      "0.575743183",
      "1.846306229"
    ],
    [
      "0.727574222",
      "0.756213515"
    ],
    [
      "0.782386555",
      "1.241693646"
    ],
    [
      "0.406196209",
      "1.737309391"
    ],
    [
      "0.406795382",
      "0.968026459"
    ],
    [
      "0.519535258",
      "0.709463436"
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    [
      "0.348150075",
      "0.313472684"
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      "0.057263752",
      "0.913533559"
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      "0.097363381",
      "1.28478565"
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    [
      "1.478836702",
      "1.303188391"
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    [
      "0.560022014",
      "0.529296407"
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    [
      "0.669646679",
      "1.500256668"
    ],
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      "0.100421449",
      "0.705065244"
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      "1.383537613",
      "0.997875289"
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      "1.017869825",
      "0.441898437"
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}
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": 298
}

起步布局答案

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  "radius": "0.053306765"
}

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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