极值竞技场
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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 = 29

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 = 29
目标最大化 共同半径
已证明最优0.092463143值取自 Eckard Specht 的汇总表，粗体半径表示已证明最优；表值的九位截断在坐标网格上不可写，证书最好达到此值

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 = 0.092463143

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

0.092463143

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": [
    [
      "0.092463144",
      "0.092463144"
    ],
    [
      "0.420049217",
      "0.092463144"
    ],
    [
      "0.740350943",
      "0.092463144"
    ],
    [
      "0.907536856",
      "0.171498596"
    ],
    [
      "0.25625618",
      "0.178309363"
    ],
    [
      "0.58020008",
      "0.184926288"
    ],
    [
      "0.092463144",
      "0.277389432"
    ],
    [
      "0.420049217",
      "0.277389432"
    ],
    [
      "0.740350943",
      "0.277389432"
    ],
    [
      "0.907536856",
      "0.356424884"
    ],
    [
      "0.25625618",
      "0.363235651"
    ],
    [
      "0.58020008",
      "0.369852576"
    ],
    [
      "0.092463144",
      "0.46231572"
    ],
    [
      "0.420049217",
      "0.46231572"
    ],
    [
      "0.740350943",
      "0.46231572"
    ],
    [
      "0.907536856",
      "0.541351172"
    ],
    [
      "0.25625618",
      "0.548161939"
    ],
    [
      "0.58020008",
      "0.554778864"
    ],
    [
      "0.092463144",
      "0.647242008"
    ],
    [
      "0.420049217",
      "0.647242008"
    ],
    [
      "0.740350943",
      "0.647242008"
    ],
    [
      "0.907536856",
      "0.72627746"
    ],
    [
      "0.25625618",
      "0.733088227"
    ],
    [
      "0.58020008",
      "0.739705152"
    ],
    [
      "0.686719253",
      "0.890871941"
    ],
    [
      "0.108764618",
      "0.891235382"
    ],
    [
      "0.3176191",
      "0.907536856"
    ],
    [
      "0.502545388",
      "0.907536856"
    ],
    [
      "0.870893118",
      "0.907536856"
    ]
  ],
  "radius": "0.092463143"
}
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": 29
}

当前第一名的答案

{
  "centers": [
    [
      "0.092463144",
      "0.092463144"
    ],
    [
      "0.420049217",
      "0.092463144"
    ],
    [
      "0.740350943",
      "0.092463144"
    ],
    [
      "0.907536856",
      "0.171498596"
    ],
    [
      "0.25625618",
      "0.178309363"
    ],
    [
      "0.58020008",
      "0.184926288"
    ],
    [
      "0.092463144",
      "0.277389432"
    ],
    [
      "0.420049217",
      "0.277389432"
    ],
    [
      "0.740350943",
      "0.277389432"
    ],
    [
      "0.907536856",
      "0.356424884"
    ],
    [
      "0.25625618",
      "0.363235651"
    ],
    [
      "0.58020008",
      "0.369852576"
    ],
    [
      "0.092463144",
      "0.46231572"
    ],
    [
      "0.420049217",
      "0.46231572"
    ],
    [
      "0.740350943",
      "0.46231572"
    ],
    [
      "0.907536856",
      "0.541351172"
    ],
    [
      "0.25625618",
      "0.548161939"
    ],
    [
      "0.58020008",
      "0.554778864"
    ],
    [
      "0.092463144",
      "0.647242008"
    ],
    [
      "0.420049217",
      "0.647242008"
    ],
    [
      "0.740350943",
      "0.647242008"
    ],
    [
      "0.907536856",
      "0.72627746"
    ],
    [
      "0.25625618",
      "0.733088227"
    ],
    [
      "0.58020008",
      "0.739705152"
    ],
    [
      "0.686719253",
      "0.890871941"
    ],
    [
      "0.108764618",
      "0.891235382"
    ],
    [
      "0.3176191",
      "0.907536856"
    ],
    [
      "0.502545388",
      "0.907536856"
    ],
    [
      "0.870893118",
      "0.907536856"
    ]
  ],
  "radius": "0.092463143"
}

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