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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 = 29

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 = 29
目标最大化 共同半径
已知最好（未证明）0.162903649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 = 0.162903648

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

0.162903648

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": [
    [
      "1.054795022",
      "0.164698973"
    ],
    [
      "0.727950535",
      "0.208446976"
    ],
    [
      "1.369537674",
      "0.248886022"
    ],
    [
      "0.390213198",
      "0.426510369"
    ],
    [
      "1.641914347",
      "0.462713976"
    ],
    [
      "1.020558335",
      "0.488702437"
    ],
    [
      "0.697876125",
      "0.533720067"
    ],
    [
      "1.335300987",
      "0.572889486"
    ],
    [
      "0.217458885",
      "0.702746397"
    ],
    [
      "1.798413778",
      "0.748473182"
    ],
    [
      "0.525121812",
      "0.809956095"
    ],
    [
      "0.850726386",
      "0.821447704"
    ],
    [
      "1.174659788",
      "0.856341072"
    ],
    [
      "1.50300251",
      "0.886421894"
    ],
    [
      "0.163248111",
      "1.024012"
    ],
    [
      "1.833965194",
      "1.072335024"
    ],
    [
      "0.470911038",
      "1.131221698"
    ],
    [
      "0.796515612",
      "1.142713307"
    ],
    [
      "1.120449013",
      "1.177606675"
    ],
    [
      "1.455030305",
      "1.209664415"
    ],
    [
      "1.743183076",
      "1.385239167"
    ],
    [
      "0.623761299",
      "1.418949335"
    ],
    [
      "0.291779744",
      "1.422703042"
    ],
    [
      "1.260740658",
      "1.471662197"
    ],
    [
      "0.937021897",
      "1.5084937"
    ],
    [
      "1.539819587",
      "1.639785209"
    ],
    [
      "0.617701964",
      "1.744700283"
    ],
    [
      "1.254681324",
      "1.797413145"
    ],
    [
      "0.930962563",
      "1.834244648"
    ]
  ],
  "radius": "0.162903648"
}
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": [
    [
      "1.054795022",
      "0.164698973"
    ],
    [
      "0.727950535",
      "0.208446976"
    ],
    [
      "1.369537674",
      "0.248886022"
    ],
    [
      "0.390213198",
      "0.426510369"
    ],
    [
      "1.641914347",
      "0.462713976"
    ],
    [
      "1.020558335",
      "0.488702437"
    ],
    [
      "0.697876125",
      "0.533720067"
    ],
    [
      "1.335300987",
      "0.572889486"
    ],
    [
      "0.217458885",
      "0.702746397"
    ],
    [
      "1.798413778",
      "0.748473182"
    ],
    [
      "0.525121812",
      "0.809956095"
    ],
    [
      "0.850726386",
      "0.821447704"
    ],
    [
      "1.174659788",
      "0.856341072"
    ],
    [
      "1.50300251",
      "0.886421894"
    ],
    [
      "0.163248111",
      "1.024012"
    ],
    [
      "1.833965194",
      "1.072335024"
    ],
    [
      "0.470911038",
      "1.131221698"
    ],
    [
      "0.796515612",
      "1.142713307"
    ],
    [
      "1.120449013",
      "1.177606675"
    ],
    [
      "1.455030305",
      "1.209664415"
    ],
    [
      "1.743183076",
      "1.385239167"
    ],
    [
      "0.623761299",
      "1.418949335"
    ],
    [
      "0.291779744",
      "1.422703042"
    ],
    [
      "1.260740658",
      "1.471662197"
    ],
    [
      "0.937021897",
      "1.5084937"
    ],
    [
      "1.539819587",
      "1.639785209"
    ],
    [
      "0.617701964",
      "1.744700283"
    ],
    [
      "1.254681324",
      "1.797413145"
    ],
    [
      "0.930962563",
      "1.834244648"
    ]
  ],
  "radius": "0.162903648"
}

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