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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 = 41

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 = 41
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[0.1377408129253445436683331609755, 0.1377408129253445436683331609765]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 = 0.137740812

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 = 41 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": [
    [
      "1.055686357",
      "0.139540858"
    ],
    [
      "0.781468019",
      "0.165892885"
    ],
    [
      "1.324220638",
      "0.201018221"
    ],
    [
      "0.529555831",
      "0.27738448"
    ],
    [
      "1.559660842",
      "0.344049813"
    ],
    [
      "0.962990028",
      "0.398958507"
    ],
    [
      "1.231524308",
      "0.46043587"
    ],
    [
      "0.325663151",
      "0.462635394"
    ],
    [
      "0.703376577",
      "0.491105033"
    ],
    [
      "1.737974952",
      "0.554036014"
    ],
    [
      "1.466964512",
      "0.603467462"
    ],
    [
      "1.012880751",
      "0.669884773"
    ],
    [
      "0.499483896",
      "0.676355947"
    ],
    [
      "0.190601856",
      "0.702736565"
    ],
    [
      "0.757441776",
      "0.773040269"
    ],
    [
      "1.261985408",
      "0.787515535"
    ],
    [
      "1.840961963",
      "0.809542965"
    ],
    [
      "1.566465505",
      "0.87045109"
    ],
    [
      "0.54322557",
      "0.948342697"
    ],
    [
      "0.138158016",
      "0.973180219"
    ],
    [
      "1.070081046",
      "0.985158753"
    ],
    [
      "0.80118345",
      "1.045027019"
    ],
    [
      "1.337197275",
      "1.052531197"
    ],
    [
      "1.858109714",
      "1.084490379"
    ],
    [
      "0.352095094",
      "1.146734398"
    ],
    [
      "1.592246098",
      "1.156647595"
    ],
    [
      "0.610052974",
      "1.243418721"
    ],
    [
      "1.159210822",
      "1.286544152"
    ],
    [
      "0.883089182",
      "1.312674962"
    ],
    [
      "1.787667887",
      "1.350813633"
    ],
    [
      "1.441404881",
      "1.387162352"
    ],
    [
      "0.233809494",
      "1.395528777"
    ],
    [
      "0.687765416",
      "1.507711934"
    ],
    [
      "1.225561045",
      "1.55833935"
    ],
    [
      "1.636826671",
      "1.58132839"
    ],
    [
      "0.951578471",
      "1.587039095"
    ],
    [
      "0.39765709",
      "1.616987783"
    ],
    [
      "1.420982834",
      "1.752505388"
    ],
    [
      "0.622987455",
      "1.775469178"
    ],
    [
      "1.162168142",
      "1.846872127"
    ],
    [
      "0.88680051",
      "1.854796339"
    ]
  ],
  "radius": "0.137740812"
}
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": 41
}

起步布局答案

{
  "centers": [
    [
      "1.055686357",
      "0.139540858"
    ],
    [
      "0.781468019",
      "0.165892885"
    ],
    [
      "1.324220638",
      "0.201018221"
    ],
    [
      "0.529555831",
      "0.27738448"
    ],
    [
      "1.559660842",
      "0.344049813"
    ],
    [
      "0.962990028",
      "0.398958507"
    ],
    [
      "1.231524308",
      "0.46043587"
    ],
    [
      "0.325663151",
      "0.462635394"
    ],
    [
      "0.703376577",
      "0.491105033"
    ],
    [
      "1.737974952",
      "0.554036014"
    ],
    [
      "1.466964512",
      "0.603467462"
    ],
    [
      "1.012880751",
      "0.669884773"
    ],
    [
      "0.499483896",
      "0.676355947"
    ],
    [
      "0.190601856",
      "0.702736565"
    ],
    [
      "0.757441776",
      "0.773040269"
    ],
    [
      "1.261985408",
      "0.787515535"
    ],
    [
      "1.840961963",
      "0.809542965"
    ],
    [
      "1.566465505",
      "0.87045109"
    ],
    [
      "0.54322557",
      "0.948342697"
    ],
    [
      "0.138158016",
      "0.973180219"
    ],
    [
      "1.070081046",
      "0.985158753"
    ],
    [
      "0.80118345",
      "1.045027019"
    ],
    [
      "1.337197275",
      "1.052531197"
    ],
    [
      "1.858109714",
      "1.084490379"
    ],
    [
      "0.352095094",
      "1.146734398"
    ],
    [
      "1.592246098",
      "1.156647595"
    ],
    [
      "0.610052974",
      "1.243418721"
    ],
    [
      "1.159210822",
      "1.286544152"
    ],
    [
      "0.883089182",
      "1.312674962"
    ],
    [
      "1.787667887",
      "1.350813633"
    ],
    [
      "1.441404881",
      "1.387162352"
    ],
    [
      "0.233809494",
      "1.395528777"
    ],
    [
      "0.687765416",
      "1.507711934"
    ],
    [
      "1.225561045",
      "1.55833935"
    ],
    [
      "1.636826671",
      "1.58132839"
    ],
    [
      "0.951578471",
      "1.587039095"
    ],
    [
      "0.39765709",
      "1.616987783"
    ],
    [
      "1.420982834",
      "1.752505388"
    ],
    [
      "0.622987455",
      "1.775469178"
    ],
    [
      "1.162168142",
      "1.846872127"
    ],
    [
      "0.88680051",
      "1.854796339"
    ]
  ],
  "radius": "0.137740812"
}

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

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