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P01 · 装箱与覆盖 · 经典问题

单位正方形内的等圆装箱 · n = 28

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 28
目标最大化 共同半径
已证明最优0.093672833值取自 Eckard Specht 的汇总表，粗体半径表示已证明最优

已完成已经有达到上限的答案，这道子题没有可争的了

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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
打开编辑器看看 ↗
1y0
0x1
已验证构造r = 0.093672833

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

这三十个 n 全部已证明，最优构形也已直接展示，整题作为已完成陈列，不再接受破纪录；csq 表往上收录到几百个 n，真正的前沿在那里。

查看来源
当前第一名

0.093672833

共同半径

已达最优最优值已证明，无法再提升
纪录保持者参考答案
解题方式—
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

当前第一名的答案

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}
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": 28
}

当前第一名的答案

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0