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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 43

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 43
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[0.0763398105795, 0.0763398105805]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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 ↗

当前展示：起步布局

1y0
0x1
已验证构造r = 0.076339810

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 = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 43 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

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.076376207",
      "0.076339811"
    ],
    [
      "0.229055828",
      "0.076339811"
    ],
    [
      "0.381735449",
      "0.076339811"
    ],
    [
      "0.646828289",
      "0.076339811"
    ],
    [
      "0.923660189",
      "0.076339811"
    ],
    [
      "0.785244239",
      "0.140776535"
    ],
    [
      "0.514281869",
      "0.152119183"
    ],
    [
      "0.910256054",
      "0.228429902"
    ],
    [
      "0.64618194",
      "0.229018064"
    ],
    [
      "0.076339811",
      "0.229019427"
    ],
    [
      "0.229019432",
      "0.229019427"
    ],
    [
      "0.381699053",
      "0.229019427"
    ],
    [
      "0.51394089",
      "0.305329081"
    ],
    [
      "0.778389745",
      "0.305386664"
    ],
    [
      "0.923660189",
      "0.380519993"
    ],
    [
      "0.646148695",
      "0.381697681"
    ],
    [
      "0.229054252",
      "0.381699045"
    ],
    [
      "0.381733873",
      "0.381699045"
    ],
    [
      "0.076339811",
      "0.381699049"
    ],
    [
      "0.791044853",
      "0.457540911"
    ],
    [
      "0.513941678",
      "0.458067645"
    ],
    [
      "0.152697035",
      "0.513913424"
    ],
    [
      "0.320857495",
      "0.521717377"
    ],
    [
      "0.923660189",
      "0.533199614"
    ],
    [
      "0.658803803",
      "0.533851928"
    ],
    [
      "0.791044853",
      "0.610220532"
    ],
    [
      "0.526560216",
      "0.610224929"
    ],
    [
      "0.230872072",
      "0.645061166"
    ],
    [
      "0.076339811",
      "0.646127799"
    ],
    [
      "0.382683638",
      "0.661318934"
    ],
    [
      "0.923660189",
      "0.685879236"
    ],
    [
      "0.658803803",
      "0.68653155"
    ],
    [
      "0.505441777",
      "0.768939077"
    ],
    [
      "0.79646605",
      "0.770334654"
    ],
    [
      "0.292698216",
      "0.784662723"
    ],
    [
      "0.14001877",
      "0.784893994"
    ],
    [
      "0.660201698",
      "0.839204771"
    ],
    [
      "0.923660189",
      "0.854790072"
    ],
    [
      "0.381013495",
      "0.909207807"
    ],
    [
      "0.076339811",
      "0.923660189"
    ],
    [
      "0.229019432",
      "0.923660189"
    ],
    [
      "0.533007559",
      "0.923660189"
    ],
    [
      "0.787395837",
      "0.923660189"
    ]
  ],
  "radius": "0.07633981"
}
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": 43
}

起步布局答案

{
  "centers": [
    [
      "0.076376207",
      "0.076339811"
    ],
    [
      "0.229055828",
      "0.076339811"
    ],
    [
      "0.381735449",
      "0.076339811"
    ],
    [
      "0.646828289",
      "0.076339811"
    ],
    [
      "0.923660189",
      "0.076339811"
    ],
    [
      "0.785244239",
      "0.140776535"
    ],
    [
      "0.514281869",
      "0.152119183"
    ],
    [
      "0.910256054",
      "0.228429902"
    ],
    [
      "0.64618194",
      "0.229018064"
    ],
    [
      "0.076339811",
      "0.229019427"
    ],
    [
      "0.229019432",
      "0.229019427"
    ],
    [
      "0.381699053",
      "0.229019427"
    ],
    [
      "0.51394089",
      "0.305329081"
    ],
    [
      "0.778389745",
      "0.305386664"
    ],
    [
      "0.923660189",
      "0.380519993"
    ],
    [
      "0.646148695",
      "0.381697681"
    ],
    [
      "0.229054252",
      "0.381699045"
    ],
    [
      "0.381733873",
      "0.381699045"
    ],
    [
      "0.076339811",
      "0.381699049"
    ],
    [
      "0.791044853",
      "0.457540911"
    ],
    [
      "0.513941678",
      "0.458067645"
    ],
    [
      "0.152697035",
      "0.513913424"
    ],
    [
      "0.320857495",
      "0.521717377"
    ],
    [
      "0.923660189",
      "0.533199614"
    ],
    [
      "0.658803803",
      "0.533851928"
    ],
    [
      "0.791044853",
      "0.610220532"
    ],
    [
      "0.526560216",
      "0.610224929"
    ],
    [
      "0.230872072",
      "0.645061166"
    ],
    [
      "0.076339811",
      "0.646127799"
    ],
    [
      "0.382683638",
      "0.661318934"
    ],
    [
      "0.923660189",
      "0.685879236"
    ],
    [
      "0.658803803",
      "0.68653155"
    ],
    [
      "0.505441777",
      "0.768939077"
    ],
    [
      "0.79646605",
      "0.770334654"
    ],
    [
      "0.292698216",
      "0.784662723"
    ],
    [
      "0.14001877",
      "0.784893994"
    ],
    [
      "0.660201698",
      "0.839204771"
    ],
    [
      "0.923660189",
      "0.854790072"
    ],
    [
      "0.381013495",
      "0.909207807"
    ],
    [
      "0.076339811",
      "0.923660189"
    ],
    [
      "0.229019432",
      "0.923660189"
    ],
    [
      "0.533007559",
      "0.923660189"
    ],
    [
      "0.787395837",
      "0.923660189"
    ]
  ],
  "radius": "0.07633981"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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