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P02 · 装箱与覆盖 · 经典问题 · 基线易突破

单位圆内的等圆装箱 · n = 27

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 = 27
目标最大化 共同半径
已知最好（未证明）0.169307931值取自 Eckard Specht 的汇总表，该表未把这一项标为已证明最优

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
放大来摆，然后提交 ↗
2y0
0x2
已验证构造r = 0.169307930

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 = 2..14 与 19 已证明（Specht 汇总表 cci 的粗体标记）；15..18 与 20 以上全部开放，包括本站开到的 29。

查看来源
当前第一名

0.169307930

共同半径

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
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"。

当前第一名的答案

{
  "centers": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}
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": 27
}

当前第一名的答案

{
  "centers": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}

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