极值竞技场
中/EN
P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 50

文献中也称circles in a circlepacking equal circles in a circlecci

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 = 50
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.1258254895304041886805677050265, 0.1258254895304041886805677050275]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.125825489

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 = 1..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 50 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
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"。

起步布局答案

{
  "centers": [
    [
      "0.90133979",
      "0.131410777"
    ],
    [
      "1.152867225",
      "0.139295239"
    ],
    [
      "0.657513593",
      "0.195708924"
    ],
    [
      "1.391726447",
      "0.2185069"
    ],
    [
      "0.44258216",
      "0.326600842"
    ],
    [
      "1.598123027",
      "0.362481436"
    ],
    [
      "0.834960128",
      "0.374149229"
    ],
    [
      "1.086487563",
      "0.38203369"
    ],
    [
      "1.325346785",
      "0.461245351"
    ],
    [
      "0.620028695",
      "0.505041147"
    ],
    [
      "0.273844329",
      "0.513297815"
    ],
    [
      "1.754952737",
      "0.559287577"
    ],
    [
      "1.023775023",
      "0.625745308"
    ],
    [
      "1.513096087",
      "0.628811502"
    ],
    [
      "0.451290864",
      "0.69173812"
    ],
    [
      "0.782489269",
      "0.697225374"
    ],
    [
      "1.262634245",
      "0.704956969"
    ],
    [
      "0.164572302",
      "0.745101915"
    ],
    [
      "1.694233897",
      "0.803503543"
    ],
    [
      "0.973480567",
      "0.872319197"
    ],
    [
      "1.450383547",
      "0.872523119"
    ],
    [
      "0.613751438",
      "0.883922348"
    ],
    [
      "0.366084728",
      "0.928525218"
    ],
    [
      "1.211897665",
      "0.952851814"
    ],
    [
      "1.873938079",
      "0.979670004"
    ],
    [
      "0.125832119",
      "1.003404447"
    ],
    [
      "1.631521357",
      "1.04721516"
    ],
    [
      "0.799286978",
      "1.053936402"
    ],
    [
      "1.393035475",
      "1.127543855"
    ],
    [
      "0.558472448",
      "1.129426825"
    ],
    [
      "1.037704077",
      "1.134469019"
    ],
    [
      "0.310805738",
      "1.174029696"
    ],
    [
      "1.835149932",
      "1.238966012"
    ],
    [
      "1.58428976",
      "1.294394026"
    ],
    [
      "0.744007988",
      "1.299440879"
    ],
    [
      "1.218841886",
      "1.30916106"
    ],
    [
      "0.485217787",
      "1.372532552"
    ],
    [
      "0.979720221",
      "1.387576902"
    ],
    [
      "0.229011606",
      "1.412016954"
    ],
    [
      "1.408545656",
      "1.47451128"
    ],
    [
      "1.707040461",
      "1.514076707"
    ],
    [
      "0.704449703",
      "1.547963227"
    ],
    [
      "1.188772654",
      "1.597100199"
    ],
    [
      "0.378331135",
      "1.614580261"
    ],
    [
      "0.940161936",
      "1.63609925"
    ],
    [
      "1.531296357",
      "1.694193961"
    ],
    [
      "0.579168794",
      "1.766212876"
    ],
    [
      "1.311523356",
      "1.81678288"
    ],
    [
      "0.814881027",
      "1.854348899"
    ],
    [
      "1.065934198",
      "1.871684436"
    ]
  ],
  "radius": "0.125825489"
}
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": 50
}

起步布局答案

{
  "centers": [
    [
      "0.90133979",
      "0.131410777"
    ],
    [
      "1.152867225",
      "0.139295239"
    ],
    [
      "0.657513593",
      "0.195708924"
    ],
    [
      "1.391726447",
      "0.2185069"
    ],
    [
      "0.44258216",
      "0.326600842"
    ],
    [
      "1.598123027",
      "0.362481436"
    ],
    [
      "0.834960128",
      "0.374149229"
    ],
    [
      "1.086487563",
      "0.38203369"
    ],
    [
      "1.325346785",
      "0.461245351"
    ],
    [
      "0.620028695",
      "0.505041147"
    ],
    [
      "0.273844329",
      "0.513297815"
    ],
    [
      "1.754952737",
      "0.559287577"
    ],
    [
      "1.023775023",
      "0.625745308"
    ],
    [
      "1.513096087",
      "0.628811502"
    ],
    [
      "0.451290864",
      "0.69173812"
    ],
    [
      "0.782489269",
      "0.697225374"
    ],
    [
      "1.262634245",
      "0.704956969"
    ],
    [
      "0.164572302",
      "0.745101915"
    ],
    [
      "1.694233897",
      "0.803503543"
    ],
    [
      "0.973480567",
      "0.872319197"
    ],
    [
      "1.450383547",
      "0.872523119"
    ],
    [
      "0.613751438",
      "0.883922348"
    ],
    [
      "0.366084728",
      "0.928525218"
    ],
    [
      "1.211897665",
      "0.952851814"
    ],
    [
      "1.873938079",
      "0.979670004"
    ],
    [
      "0.125832119",
      "1.003404447"
    ],
    [
      "1.631521357",
      "1.04721516"
    ],
    [
      "0.799286978",
      "1.053936402"
    ],
    [
      "1.393035475",
      "1.127543855"
    ],
    [
      "0.558472448",
      "1.129426825"
    ],
    [
      "1.037704077",
      "1.134469019"
    ],
    [
      "0.310805738",
      "1.174029696"
    ],
    [
      "1.835149932",
      "1.238966012"
    ],
    [
      "1.58428976",
      "1.294394026"
    ],
    [
      "0.744007988",
      "1.299440879"
    ],
    [
      "1.218841886",
      "1.30916106"
    ],
    [
      "0.485217787",
      "1.372532552"
    ],
    [
      "0.979720221",
      "1.387576902"
    ],
    [
      "0.229011606",
      "1.412016954"
    ],
    [
      "1.408545656",
      "1.47451128"
    ],
    [
      "1.707040461",
      "1.514076707"
    ],
    [
      "0.704449703",
      "1.547963227"
    ],
    [
      "1.188772654",
      "1.597100199"
    ],
    [
      "0.378331135",
      "1.614580261"
    ],
    [
      "0.940161936",
      "1.63609925"
    ],
    [
      "1.531296357",
      "1.694193961"
    ],
    [
      "0.579168794",
      "1.766212876"
    ],
    [
      "1.311523356",
      "1.81678288"
    ],
    [
      "0.814881027",
      "1.854348899"
    ],
    [
      "1.065934198",
      "1.871684436"
    ]
  ],
  "radius": "0.125825489"
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？