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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 = 25

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 = 25
目标最大化 共同半径
已知最好（未证明）0.173827661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 = 0.173827661

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

0.173827661

共同半径

已追平已知最好
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": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.46579153",
      "0.369776196"
    ],
    [
      "1.57268885",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468233",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.85492095",
      "1.4579161"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}
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": 25
}

当前第一名的答案

{
  "centers": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.46579153",
      "0.369776196"
    ],
    [
      "1.57268885",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468233",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.85492095",
      "1.4579161"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}

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