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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 = 73

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 = 73
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[0.0593660505825, 0.0593660505835]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 = 0.059366050

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 = 73 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": [
    [
      "0.068621522",
      "0.059366051"
    ],
    [
      "0.187353623",
      "0.059366051"
    ],
    [
      "0.306085724",
      "0.059366051"
    ],
    [
      "0.424817825",
      "0.059366051"
    ],
    [
      "0.630467857",
      "0.059366051"
    ],
    [
      "0.836117889",
      "0.059366051"
    ],
    [
      "0.940633949",
      "0.115701696"
    ],
    [
      "0.527642841",
      "0.118732101"
    ],
    [
      "0.733292873",
      "0.118732101"
    ],
    [
      "0.059366051",
      "0.177736858"
    ],
    [
      "0.178098152",
      "0.177736858"
    ],
    [
      "0.296830253",
      "0.177736858"
    ],
    [
      "0.415562354",
      "0.177736858"
    ],
    [
      "0.630467857",
      "0.178098152"
    ],
    [
      "0.836117889",
      "0.178098152"
    ],
    [
      "0.940633949",
      "0.234433797"
    ],
    [
      "0.527642841",
      "0.237464202"
    ],
    [
      "0.733292873",
      "0.237464202"
    ],
    [
      "0.068224948",
      "0.296138006"
    ],
    [
      "0.186957049",
      "0.296138006"
    ],
    [
      "0.30568915",
      "0.296138006"
    ],
    [
      "0.424421251",
      "0.296138006"
    ],
    [
      "0.630467857",
      "0.296830253"
    ],
    [
      "0.836117889",
      "0.296830253"
    ],
    [
      "0.940633949",
      "0.353165898"
    ],
    [
      "0.733292873",
      "0.356196303"
    ],
    [
      "0.526842307",
      "0.356198803"
    ],
    [
      "0.178098152",
      "0.414539154"
    ],
    [
      "0.296830253",
      "0.414539154"
    ],
    [
      "0.059366051",
      "0.414539154"
    ],
    [
      "0.415562354",
      "0.414539154"
    ],
    [
      "0.630467857",
      "0.415562354"
    ],
    [
      "0.836117889",
      "0.415562354"
    ],
    [
      "0.940633949",
      "0.471897999"
    ],
    [
      "0.527642841",
      "0.474928405"
    ],
    [
      "0.733292873",
      "0.474928405"
    ],
    [
      "0.424062533",
      "0.532966595"
    ],
    [
      "0.303223369",
      "0.533099012"
    ],
    [
      "0.182324201",
      "0.533196022"
    ],
    [
      "0.059366051",
      "0.533271255"
    ],
    [
      "0.630467857",
      "0.534294455"
    ],
    [
      "0.836117889",
      "0.534294455"
    ],
    [
      "0.940633949",
      "0.5906301"
    ],
    [
      "0.733292873",
      "0.593660506"
    ],
    [
      "0.511696558",
      "0.614667495"
    ],
    [
      "0.120907276",
      "0.634809373"
    ],
    [
      "0.363754953",
      "0.635242241"
    ],
    [
      "0.242855784",
      "0.635339251"
    ],
    [
      "0.630467857",
      "0.653026556"
    ],
    [
      "0.836117889",
      "0.653026556"
    ],
    [
      "0.940633949",
      "0.709362201"
    ],
    [
      "0.733292873",
      "0.712392607"
    ],
    [
      "0.544004384",
      "0.734398417"
    ],
    [
      "0.059366051",
      "0.736347491"
    ],
    [
      "0.425296178",
      "0.736780359"
    ],
    [
      "0.181438859",
      "0.736952602"
    ],
    [
      "0.303387368",
      "0.73748248"
    ],
    [
      "0.831716413",
      "0.780825901"
    ],
    [
      "0.940633949",
      "0.828094303"
    ],
    [
      "0.723995016",
      "0.830760093"
    ],
    [
      "0.60543954",
      "0.837233968"
    ],
    [
      "0.486713103",
      "0.83839371"
    ],
    [
      "0.119897634",
      "0.83849072"
    ],
    [
      "0.364928593",
      "0.839020598"
    ],
    [
      "0.241970443",
      "0.839095831"
    ],
    [
      "0.784301558",
      "0.93303635"
    ],
    [
      "0.059366051",
      "0.940633949"
    ],
    [
      "0.180429218",
      "0.940633949"
    ],
    [
      "0.303511668",
      "0.940633949"
    ],
    [
      "0.426345518",
      "0.940633949"
    ],
    [
      "0.547080688",
      "0.940633949"
    ],
    [
      "0.66581279",
      "0.940633949"
    ],
    [
      "0.902790327",
      "0.940633949"
    ]
  ],
  "radius": "0.05936605"
}
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": 73
}

起步布局答案

{
  "centers": [
    [
      "0.068621522",
      "0.059366051"
    ],
    [
      "0.187353623",
      "0.059366051"
    ],
    [
      "0.306085724",
      "0.059366051"
    ],
    [
      "0.424817825",
      "0.059366051"
    ],
    [
      "0.630467857",
      "0.059366051"
    ],
    [
      "0.836117889",
      "0.059366051"
    ],
    [
      "0.940633949",
      "0.115701696"
    ],
    [
      "0.527642841",
      "0.118732101"
    ],
    [
      "0.733292873",
      "0.118732101"
    ],
    [
      "0.059366051",
      "0.177736858"
    ],
    [
      "0.178098152",
      "0.177736858"
    ],
    [
      "0.296830253",
      "0.177736858"
    ],
    [
      "0.415562354",
      "0.177736858"
    ],
    [
      "0.630467857",
      "0.178098152"
    ],
    [
      "0.836117889",
      "0.178098152"
    ],
    [
      "0.940633949",
      "0.234433797"
    ],
    [
      "0.527642841",
      "0.237464202"
    ],
    [
      "0.733292873",
      "0.237464202"
    ],
    [
      "0.068224948",
      "0.296138006"
    ],
    [
      "0.186957049",
      "0.296138006"
    ],
    [
      "0.30568915",
      "0.296138006"
    ],
    [
      "0.424421251",
      "0.296138006"
    ],
    [
      "0.630467857",
      "0.296830253"
    ],
    [
      "0.836117889",
      "0.296830253"
    ],
    [
      "0.940633949",
      "0.353165898"
    ],
    [
      "0.733292873",
      "0.356196303"
    ],
    [
      "0.526842307",
      "0.356198803"
    ],
    [
      "0.178098152",
      "0.414539154"
    ],
    [
      "0.296830253",
      "0.414539154"
    ],
    [
      "0.059366051",
      "0.414539154"
    ],
    [
      "0.415562354",
      "0.414539154"
    ],
    [
      "0.630467857",
      "0.415562354"
    ],
    [
      "0.836117889",
      "0.415562354"
    ],
    [
      "0.940633949",
      "0.471897999"
    ],
    [
      "0.527642841",
      "0.474928405"
    ],
    [
      "0.733292873",
      "0.474928405"
    ],
    [
      "0.424062533",
      "0.532966595"
    ],
    [
      "0.303223369",
      "0.533099012"
    ],
    [
      "0.182324201",
      "0.533196022"
    ],
    [
      "0.059366051",
      "0.533271255"
    ],
    [
      "0.630467857",
      "0.534294455"
    ],
    [
      "0.836117889",
      "0.534294455"
    ],
    [
      "0.940633949",
      "0.5906301"
    ],
    [
      "0.733292873",
      "0.593660506"
    ],
    [
      "0.511696558",
      "0.614667495"
    ],
    [
      "0.120907276",
      "0.634809373"
    ],
    [
      "0.363754953",
      "0.635242241"
    ],
    [
      "0.242855784",
      "0.635339251"
    ],
    [
      "0.630467857",
      "0.653026556"
    ],
    [
      "0.836117889",
      "0.653026556"
    ],
    [
      "0.940633949",
      "0.709362201"
    ],
    [
      "0.733292873",
      "0.712392607"
    ],
    [
      "0.544004384",
      "0.734398417"
    ],
    [
      "0.059366051",
      "0.736347491"
    ],
    [
      "0.425296178",
      "0.736780359"
    ],
    [
      "0.181438859",
      "0.736952602"
    ],
    [
      "0.303387368",
      "0.73748248"
    ],
    [
      "0.831716413",
      "0.780825901"
    ],
    [
      "0.940633949",
      "0.828094303"
    ],
    [
      "0.723995016",
      "0.830760093"
    ],
    [
      "0.60543954",
      "0.837233968"
    ],
    [
      "0.486713103",
      "0.83839371"
    ],
    [
      "0.119897634",
      "0.83849072"
    ],
    [
      "0.364928593",
      "0.839020598"
    ],
    [
      "0.241970443",
      "0.839095831"
    ],
    [
      "0.784301558",
      "0.93303635"
    ],
    [
      "0.059366051",
      "0.940633949"
    ],
    [
      "0.180429218",
      "0.940633949"
    ],
    [
      "0.303511668",
      "0.940633949"
    ],
    [
      "0.426345518",
      "0.940633949"
    ],
    [
      "0.547080688",
      "0.940633949"
    ],
    [
      "0.66581279",
      "0.940633949"
    ],
    [
      "0.902790327",
      "0.940633949"
    ]
  ],
  "radius": "0.05936605"
}

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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