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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 = 42

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 = 42
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[0.1361137487156978691032825206055, 0.1361137487156978691032825206065]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 = 0.136113748

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 = 42 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": [
    [
      "0.873627599",
      "0.145406839"
    ],
    [
      "1.145837381",
      "0.148512529"
    ],
    [
      "0.613966644",
      "0.227162564"
    ],
    [
      "1.403565457",
      "0.236171238"
    ],
    [
      "0.392638943",
      "0.38566133"
    ],
    [
      "1.621219334",
      "0.399678424"
    ],
    [
      "1.19878675",
      "0.415540944"
    ],
    [
      "0.902084111",
      "0.419301856"
    ],
    [
      "0.640566752",
      "0.498087359"
    ],
    [
      "1.416440627",
      "0.579048131"
    ],
    [
      "0.231622418",
      "0.605164153"
    ],
    [
      "1.777185903",
      "0.622797764"
    ],
    [
      "1.083615077",
      "0.662205283"
    ],
    [
      "0.815099047",
      "0.7070043"
    ],
    [
      "0.479798088",
      "0.718013911"
    ],
    [
      "1.572407196",
      "0.80216747"
    ],
    [
      "1.301206987",
      "0.825794926"
    ],
    [
      "0.146906069",
      "0.86387432"
    ],
    [
      "1.833954751",
      "0.893471572"
    ],
    [
      "1.050738246",
      "0.932440226"
    ],
    [
      "0.662727522",
      "0.932594035"
    ],
    [
      "0.394244582",
      "0.977590943"
    ],
    [
      "1.593230472",
      "1.07359739"
    ],
    [
      "1.322030262",
      "1.097224846"
    ],
    [
      "0.146902262",
      "1.136101817"
    ],
    [
      "1.846090919",
      "1.174440857"
    ],
    [
      "0.933646289",
      "1.178198815"
    ],
    [
      "0.662723714",
      "1.204821533"
    ],
    [
      "0.394240775",
      "1.24981844"
    ],
    [
      "1.16588984",
      "1.320222557"
    ],
    [
      "1.49693877",
      "1.32822598"
    ],
    [
      "1.749799217",
      "1.429069446"
    ],
    [
      "0.838373284",
      "1.433210315"
    ],
    [
      "0.56745071",
      "1.459833032"
    ],
    [
      "0.29896777",
      "1.50482994"
    ],
    [
      "1.32619173",
      "1.540247815"
    ],
    [
      "1.059048721",
      "1.592615966"
    ],
    [
      "1.579052177",
      "1.641091282"
    ],
    [
      "0.809862311",
      "1.703940688"
    ],
    [
      "0.538939737",
      "1.730563405"
    ],
    [
      "1.297680758",
      "1.810978188"
    ],
    [
      "1.030537749",
      "1.863346339"
    ]
  ],
  "radius": "0.136113748"
}
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": 42
}

起步布局答案

{
  "centers": [
    [
      "0.873627599",
      "0.145406839"
    ],
    [
      "1.145837381",
      "0.148512529"
    ],
    [
      "0.613966644",
      "0.227162564"
    ],
    [
      "1.403565457",
      "0.236171238"
    ],
    [
      "0.392638943",
      "0.38566133"
    ],
    [
      "1.621219334",
      "0.399678424"
    ],
    [
      "1.19878675",
      "0.415540944"
    ],
    [
      "0.902084111",
      "0.419301856"
    ],
    [
      "0.640566752",
      "0.498087359"
    ],
    [
      "1.416440627",
      "0.579048131"
    ],
    [
      "0.231622418",
      "0.605164153"
    ],
    [
      "1.777185903",
      "0.622797764"
    ],
    [
      "1.083615077",
      "0.662205283"
    ],
    [
      "0.815099047",
      "0.7070043"
    ],
    [
      "0.479798088",
      "0.718013911"
    ],
    [
      "1.572407196",
      "0.80216747"
    ],
    [
      "1.301206987",
      "0.825794926"
    ],
    [
      "0.146906069",
      "0.86387432"
    ],
    [
      "1.833954751",
      "0.893471572"
    ],
    [
      "1.050738246",
      "0.932440226"
    ],
    [
      "0.662727522",
      "0.932594035"
    ],
    [
      "0.394244582",
      "0.977590943"
    ],
    [
      "1.593230472",
      "1.07359739"
    ],
    [
      "1.322030262",
      "1.097224846"
    ],
    [
      "0.146902262",
      "1.136101817"
    ],
    [
      "1.846090919",
      "1.174440857"
    ],
    [
      "0.933646289",
      "1.178198815"
    ],
    [
      "0.662723714",
      "1.204821533"
    ],
    [
      "0.394240775",
      "1.24981844"
    ],
    [
      "1.16588984",
      "1.320222557"
    ],
    [
      "1.49693877",
      "1.32822598"
    ],
    [
      "1.749799217",
      "1.429069446"
    ],
    [
      "0.838373284",
      "1.433210315"
    ],
    [
      "0.56745071",
      "1.459833032"
    ],
    [
      "0.29896777",
      "1.50482994"
    ],
    [
      "1.32619173",
      "1.540247815"
    ],
    [
      "1.059048721",
      "1.592615966"
    ],
    [
      "1.579052177",
      "1.641091282"
    ],
    [
      "0.809862311",
      "1.703940688"
    ],
    [
      "0.538939737",
      "1.730563405"
    ],
    [
      "1.297680758",
      "1.810978188"
    ],
    [
      "1.030537749",
      "1.863346339"
    ]
  ],
  "radius": "0.136113748"
}

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

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