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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 = 24

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 = 24
目标最大化 共同半径
已知最好（未证明）0.176939130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 = 0.176939130

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

0.176939130

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": [
    [
      "1.11770021",
      "0.185398346"
    ],
    [
      "0.764768916",
      "0.211269671"
    ],
    [
      "1.448873398",
      "0.310114598"
    ],
    [
      "0.455322532",
      "0.382945991"
    ],
    [
      "1.023343536",
      "0.526465262"
    ],
    [
      "1.697067675",
      "0.562363334"
    ],
    [
      "0.690697098",
      "0.647197078"
    ],
    [
      "1.354516723",
      "0.651181515"
    ],
    [
      "0.246565435",
      "0.668691154"
    ],
    [
      "1.816401751",
      "0.895513753"
    ],
    [
      "1.155987809",
      "0.944125356"
    ],
    [
      "0.526680514",
      "0.960770648"
    ],
    [
      "0.177088545",
      "1.015682199"
    ],
    [
      "1.495364291",
      "1.044392136"
    ],
    [
      "0.845610213",
      "1.11411233"
    ],
    [
      "1.784815483",
      "1.247979541"
    ],
    [
      "1.307188142",
      "1.357180569"
    ],
    [
      "0.259735385",
      "1.359774227"
    ],
    [
      "0.612989037",
      "1.380790437"
    ],
    [
      "0.965647553",
      "1.463375673"
    ],
    [
      "1.60814792",
      "1.554603734"
    ],
    [
      "0.628278926",
      "1.734338231"
    ],
    [
      "1.319057897",
      "1.758703666"
    ],
    [
      "0.970986707",
      "1.822549344"
    ]
  ],
  "radius": "0.17693913"
}
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": 24
}

当前第一名的答案

{
  "centers": [
    [
      "1.11770021",
      "0.185398346"
    ],
    [
      "0.764768916",
      "0.211269671"
    ],
    [
      "1.448873398",
      "0.310114598"
    ],
    [
      "0.455322532",
      "0.382945991"
    ],
    [
      "1.023343536",
      "0.526465262"
    ],
    [
      "1.697067675",
      "0.562363334"
    ],
    [
      "0.690697098",
      "0.647197078"
    ],
    [
      "1.354516723",
      "0.651181515"
    ],
    [
      "0.246565435",
      "0.668691154"
    ],
    [
      "1.816401751",
      "0.895513753"
    ],
    [
      "1.155987809",
      "0.944125356"
    ],
    [
      "0.526680514",
      "0.960770648"
    ],
    [
      "0.177088545",
      "1.015682199"
    ],
    [
      "1.495364291",
      "1.044392136"
    ],
    [
      "0.845610213",
      "1.11411233"
    ],
    [
      "1.784815483",
      "1.247979541"
    ],
    [
      "1.307188142",
      "1.357180569"
    ],
    [
      "0.259735385",
      "1.359774227"
    ],
    [
      "0.612989037",
      "1.380790437"
    ],
    [
      "0.965647553",
      "1.463375673"
    ],
    [
      "1.60814792",
      "1.554603734"
    ],
    [
      "0.628278926",
      "1.734338231"
    ],
    [
      "1.319057897",
      "1.758703666"
    ],
    [
      "0.970986707",
      "1.822549344"
    ]
  ],
  "radius": "0.17693913"
}

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