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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 = 35

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 = 35
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[0.1493167766351160226017211775365, 0.1493167766351160226017211775375]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 = 0.149316776

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 = 35 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": [
    [
      "1.096080624",
      "0.154760117"
    ],
    [
      "0.798044279",
      "0.173636985"
    ],
    [
      "1.384053188",
      "0.240944603"
    ],
    [
      "0.524896371",
      "0.294352433"
    ],
    [
      "1.622718914",
      "0.420445774"
    ],
    [
      "1.166078273",
      "0.445074306"
    ],
    [
      "0.868041927",
      "0.463951174"
    ],
    [
      "0.30641492",
      "0.507453877"
    ],
    [
      "0.59489402",
      "0.584666622"
    ],
    [
      "1.404743999",
      "0.624575477"
    ],
    [
      "1.784642567",
      "0.671369527"
    ],
    [
      "1.041129427",
      "0.716311813"
    ],
    [
      "0.17892805",
      "0.777507756"
    ],
    [
      "0.765024104",
      "0.83010043"
    ],
    [
      "0.46740715",
      "0.854720501"
    ],
    [
      "1.563709628",
      "0.877383562"
    ],
    [
      "1.267044757",
      "0.911617375"
    ],
    [
      "1.849869147",
      "0.962792745"
    ],
    [
      "0.990939434",
      "1.025405992"
    ],
    [
      "0.152627705",
      "1.074980933"
    ],
    [
      "0.442157441",
      "1.152284698"
    ],
    [
      "1.524200799",
      "1.173392095"
    ],
    [
      "0.73892402",
      "1.185625312"
    ],
    [
      "1.227535928",
      "1.207625908"
    ],
    [
      "1.810360318",
      "1.258801278"
    ],
    [
      "0.230755064",
      "1.363213677"
    ],
    [
      "0.975520515",
      "1.367845228"
    ],
    [
      "0.67845124",
      "1.478074755"
    ],
    [
      "1.238936005",
      "1.508918306"
    ],
    [
      "1.670985034",
      "1.522915893"
    ],
    [
      "0.403681938",
      "1.606685021"
    ],
    [
      "0.926544784",
      "1.662435413"
    ],
    [
      "1.448919501",
      "1.722587869"
    ],
    [
      "0.65009732",
      "1.775390263"
    ],
    [
      "1.171530431",
      "1.833210212"
    ]
  ],
  "radius": "0.149316776"
}
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": 35
}

起步布局答案

{
  "centers": [
    [
      "1.096080624",
      "0.154760117"
    ],
    [
      "0.798044279",
      "0.173636985"
    ],
    [
      "1.384053188",
      "0.240944603"
    ],
    [
      "0.524896371",
      "0.294352433"
    ],
    [
      "1.622718914",
      "0.420445774"
    ],
    [
      "1.166078273",
      "0.445074306"
    ],
    [
      "0.868041927",
      "0.463951174"
    ],
    [
      "0.30641492",
      "0.507453877"
    ],
    [
      "0.59489402",
      "0.584666622"
    ],
    [
      "1.404743999",
      "0.624575477"
    ],
    [
      "1.784642567",
      "0.671369527"
    ],
    [
      "1.041129427",
      "0.716311813"
    ],
    [
      "0.17892805",
      "0.777507756"
    ],
    [
      "0.765024104",
      "0.83010043"
    ],
    [
      "0.46740715",
      "0.854720501"
    ],
    [
      "1.563709628",
      "0.877383562"
    ],
    [
      "1.267044757",
      "0.911617375"
    ],
    [
      "1.849869147",
      "0.962792745"
    ],
    [
      "0.990939434",
      "1.025405992"
    ],
    [
      "0.152627705",
      "1.074980933"
    ],
    [
      "0.442157441",
      "1.152284698"
    ],
    [
      "1.524200799",
      "1.173392095"
    ],
    [
      "0.73892402",
      "1.185625312"
    ],
    [
      "1.227535928",
      "1.207625908"
    ],
    [
      "1.810360318",
      "1.258801278"
    ],
    [
      "0.230755064",
      "1.363213677"
    ],
    [
      "0.975520515",
      "1.367845228"
    ],
    [
      "0.67845124",
      "1.478074755"
    ],
    [
      "1.238936005",
      "1.508918306"
    ],
    [
      "1.670985034",
      "1.522915893"
    ],
    [
      "0.403681938",
      "1.606685021"
    ],
    [
      "0.926544784",
      "1.662435413"
    ],
    [
      "1.448919501",
      "1.722587869"
    ],
    [
      "0.65009732",
      "1.775390263"
    ],
    [
      "1.171530431",
      "1.833210212"
    ]
  ],
  "radius": "0.149316776"
}

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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