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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 = 33

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 = 33
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[0.1541615179470581647319376224935, 0.1541615179470581647319376224945]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 = 0.154161517

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 = 33 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": [
    [
      "1.008319109",
      "0.15420243"
    ],
    [
      "0.704622272",
      "0.207412632"
    ],
    [
      "1.310910561",
      "0.213375972"
    ],
    [
      "0.44017319",
      "0.365936375"
    ],
    [
      "1.572190362",
      "0.377070688"
    ],
    [
      "1.087538859",
      "0.452174451"
    ],
    [
      "0.783842022",
      "0.505384654"
    ],
    [
      "1.358485678",
      "0.599316953"
    ],
    [
      "0.250110031",
      "0.608710118"
    ],
    [
      "1.757441456",
      "0.623535953"
    ],
    [
      "0.544634163",
      "0.699916643"
    ],
    [
      "0.981804718",
      "0.741800738"
    ],
    [
      "1.543736771",
      "0.845782218"
    ],
    [
      "0.159687077",
      "0.90347576"
    ],
    [
      "1.2416938",
      "0.907694558"
    ],
    [
      "1.842302161",
      "0.922735469"
    ],
    [
      "0.742577726",
      "0.936309198"
    ],
    [
      "0.445549883",
      "1.018998705"
    ],
    [
      "1.51546967",
      "1.152806753"
    ],
    [
      "1.165864765",
      "1.206547451"
    ],
    [
      "0.180919106",
      "1.211066877"
    ],
    [
      "0.857941153",
      "1.222236416"
    ],
    [
      "1.81403506",
      "1.229760004"
    ],
    [
      "0.557993447",
      "1.306086739"
    ],
    [
      "1.379039207",
      "1.429302365"
    ],
    [
      "0.310984953",
      "1.490612884"
    ],
    [
      "1.677604596",
      "1.506255616"
    ],
    [
      "0.779610859",
      "1.520443491"
    ],
    [
      "1.087772154",
      "1.530429009"
    ],
    [
      "0.532602365",
      "1.704969636"
    ],
    [
      "1.418412744",
      "1.735101023"
    ],
    [
      "0.818984395",
      "1.826242149"
    ],
    [
      "1.12714569",
      "1.836227667"
    ]
  ],
  "radius": "0.154161517"
}
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": 33
}

起步布局答案

{
  "centers": [
    [
      "1.008319109",
      "0.15420243"
    ],
    [
      "0.704622272",
      "0.207412632"
    ],
    [
      "1.310910561",
      "0.213375972"
    ],
    [
      "0.44017319",
      "0.365936375"
    ],
    [
      "1.572190362",
      "0.377070688"
    ],
    [
      "1.087538859",
      "0.452174451"
    ],
    [
      "0.783842022",
      "0.505384654"
    ],
    [
      "1.358485678",
      "0.599316953"
    ],
    [
      "0.250110031",
      "0.608710118"
    ],
    [
      "1.757441456",
      "0.623535953"
    ],
    [
      "0.544634163",
      "0.699916643"
    ],
    [
      "0.981804718",
      "0.741800738"
    ],
    [
      "1.543736771",
      "0.845782218"
    ],
    [
      "0.159687077",
      "0.90347576"
    ],
    [
      "1.2416938",
      "0.907694558"
    ],
    [
      "1.842302161",
      "0.922735469"
    ],
    [
      "0.742577726",
      "0.936309198"
    ],
    [
      "0.445549883",
      "1.018998705"
    ],
    [
      "1.51546967",
      "1.152806753"
    ],
    [
      "1.165864765",
      "1.206547451"
    ],
    [
      "0.180919106",
      "1.211066877"
    ],
    [
      "0.857941153",
      "1.222236416"
    ],
    [
      "1.81403506",
      "1.229760004"
    ],
    [
      "0.557993447",
      "1.306086739"
    ],
    [
      "1.379039207",
      "1.429302365"
    ],
    [
      "0.310984953",
      "1.490612884"
    ],
    [
      "1.677604596",
      "1.506255616"
    ],
    [
      "0.779610859",
      "1.520443491"
    ],
    [
      "1.087772154",
      "1.530429009"
    ],
    [
      "0.532602365",
      "1.704969636"
    ],
    [
      "1.418412744",
      "1.735101023"
    ],
    [
      "0.818984395",
      "1.826242149"
    ],
    [
      "1.12714569",
      "1.836227667"
    ]
  ],
  "radius": "0.154161517"
}

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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