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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 = 65

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 = 65
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[0.0632039570715, 0.0632039570725]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 = 0.063203956

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 = 65 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": [
    [
      "0.063203957",
      "0.063203957"
    ],
    [
      "0.282148887",
      "0.063203957"
    ],
    [
      "0.5",
      "0.063203957"
    ],
    [
      "0.717851113",
      "0.063203957"
    ],
    [
      "0.936796043",
      "0.063203957"
    ],
    [
      "0.172676422",
      "0.126407914"
    ],
    [
      "0.827323578",
      "0.126407914"
    ],
    [
      "0.391074443",
      "0.127345861"
    ],
    [
      "0.608925557",
      "0.127345861"
    ],
    [
      "0.063203957",
      "0.189611871"
    ],
    [
      "0.936796043",
      "0.189611871"
    ],
    [
      "0.281601979",
      "0.190549818"
    ],
    [
      "0.718398021",
      "0.190549818"
    ],
    [
      "0.5",
      "0.191487766"
    ],
    [
      "0.172129514",
      "0.253753775"
    ],
    [
      "0.827870486",
      "0.253753775"
    ],
    [
      "0.390527535",
      "0.254691723"
    ],
    [
      "0.609472465",
      "0.254691723"
    ],
    [
      "0.28105507",
      "0.31789568"
    ],
    [
      "0.5",
      "0.31789568"
    ],
    [
      "0.71894493",
      "0.31789568"
    ],
    [
      "0.063203957",
      "0.31789568"
    ],
    [
      "0.936796043",
      "0.31789568"
    ],
    [
      "0.390527535",
      "0.381099637"
    ],
    [
      "0.609472465",
      "0.381099637"
    ],
    [
      "0.172129514",
      "0.382037584"
    ],
    [
      "0.827870486",
      "0.382037584"
    ],
    [
      "0.5",
      "0.444303594"
    ],
    [
      "0.281601979",
      "0.445241541"
    ],
    [
      "0.718398021",
      "0.445241541"
    ],
    [
      "0.063203957",
      "0.446179488"
    ],
    [
      "0.936796043",
      "0.446179488"
    ],
    [
      "0.391074443",
      "0.508445498"
    ],
    [
      "0.608925557",
      "0.508445498"
    ],
    [
      "0.172676422",
      "0.509383445"
    ],
    [
      "0.827323578",
      "0.509383445"
    ],
    [
      "0.063203957",
      "0.572587402"
    ],
    [
      "0.282148887",
      "0.572587402"
    ],
    [
      "0.717851113",
      "0.572587402"
    ],
    [
      "0.936796043",
      "0.572587402"
    ],
    [
      "0.5",
      "0.572587402"
    ],
    [
      "0.172676422",
      "0.635791359"
    ],
    [
      "0.827323578",
      "0.635791359"
    ],
    [
      "0.391074443",
      "0.636729307"
    ],
    [
      "0.608925557",
      "0.636729307"
    ],
    [
      "0.063203957",
      "0.698995317"
    ],
    [
      "0.936796043",
      "0.698995317"
    ],
    [
      "0.281601979",
      "0.699933264"
    ],
    [
      "0.718398021",
      "0.699933264"
    ],
    [
      "0.5",
      "0.700871211"
    ],
    [
      "0.815482187",
      "0.789612965"
    ],
    [
      "0.184517811",
      "0.789612979"
    ],
    [
      "0.431546779",
      "0.80714029"
    ],
    [
      "0.568453221",
      "0.80714029"
    ],
    [
      "0.30625664",
      "0.823913538"
    ],
    [
      "0.69374336",
      "0.823913538"
    ],
    [
      "0.063203957",
      "0.825403231"
    ],
    [
      "0.936796043",
      "0.825403231"
    ],
    [
      "0.5",
      "0.91340937"
    ],
    [
      "0.122958473",
      "0.936796043"
    ],
    [
      "0.249366387",
      "0.936796043"
    ],
    [
      "0.375774301",
      "0.936796043"
    ],
    [
      "0.624225699",
      "0.936796043"
    ],
    [
      "0.750633613",
      "0.936796043"
    ],
    [
      "0.877041527",
      "0.936796043"
    ]
  ],
  "radius": "0.063203956"
}
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": 65
}

起步布局答案

{
  "centers": [
    [
      "0.063203957",
      "0.063203957"
    ],
    [
      "0.282148887",
      "0.063203957"
    ],
    [
      "0.5",
      "0.063203957"
    ],
    [
      "0.717851113",
      "0.063203957"
    ],
    [
      "0.936796043",
      "0.063203957"
    ],
    [
      "0.172676422",
      "0.126407914"
    ],
    [
      "0.827323578",
      "0.126407914"
    ],
    [
      "0.391074443",
      "0.127345861"
    ],
    [
      "0.608925557",
      "0.127345861"
    ],
    [
      "0.063203957",
      "0.189611871"
    ],
    [
      "0.936796043",
      "0.189611871"
    ],
    [
      "0.281601979",
      "0.190549818"
    ],
    [
      "0.718398021",
      "0.190549818"
    ],
    [
      "0.5",
      "0.191487766"
    ],
    [
      "0.172129514",
      "0.253753775"
    ],
    [
      "0.827870486",
      "0.253753775"
    ],
    [
      "0.390527535",
      "0.254691723"
    ],
    [
      "0.609472465",
      "0.254691723"
    ],
    [
      "0.28105507",
      "0.31789568"
    ],
    [
      "0.5",
      "0.31789568"
    ],
    [
      "0.71894493",
      "0.31789568"
    ],
    [
      "0.063203957",
      "0.31789568"
    ],
    [
      "0.936796043",
      "0.31789568"
    ],
    [
      "0.390527535",
      "0.381099637"
    ],
    [
      "0.609472465",
      "0.381099637"
    ],
    [
      "0.172129514",
      "0.382037584"
    ],
    [
      "0.827870486",
      "0.382037584"
    ],
    [
      "0.5",
      "0.444303594"
    ],
    [
      "0.281601979",
      "0.445241541"
    ],
    [
      "0.718398021",
      "0.445241541"
    ],
    [
      "0.063203957",
      "0.446179488"
    ],
    [
      "0.936796043",
      "0.446179488"
    ],
    [
      "0.391074443",
      "0.508445498"
    ],
    [
      "0.608925557",
      "0.508445498"
    ],
    [
      "0.172676422",
      "0.509383445"
    ],
    [
      "0.827323578",
      "0.509383445"
    ],
    [
      "0.063203957",
      "0.572587402"
    ],
    [
      "0.282148887",
      "0.572587402"
    ],
    [
      "0.717851113",
      "0.572587402"
    ],
    [
      "0.936796043",
      "0.572587402"
    ],
    [
      "0.5",
      "0.572587402"
    ],
    [
      "0.172676422",
      "0.635791359"
    ],
    [
      "0.827323578",
      "0.635791359"
    ],
    [
      "0.391074443",
      "0.636729307"
    ],
    [
      "0.608925557",
      "0.636729307"
    ],
    [
      "0.063203957",
      "0.698995317"
    ],
    [
      "0.936796043",
      "0.698995317"
    ],
    [
      "0.281601979",
      "0.699933264"
    ],
    [
      "0.718398021",
      "0.699933264"
    ],
    [
      "0.5",
      "0.700871211"
    ],
    [
      "0.815482187",
      "0.789612965"
    ],
    [
      "0.184517811",
      "0.789612979"
    ],
    [
      "0.431546779",
      "0.80714029"
    ],
    [
      "0.568453221",
      "0.80714029"
    ],
    [
      "0.30625664",
      "0.823913538"
    ],
    [
      "0.69374336",
      "0.823913538"
    ],
    [
      "0.063203957",
      "0.825403231"
    ],
    [
      "0.936796043",
      "0.825403231"
    ],
    [
      "0.5",
      "0.91340937"
    ],
    [
      "0.122958473",
      "0.936796043"
    ],
    [
      "0.249366387",
      "0.936796043"
    ],
    [
      "0.375774301",
      "0.936796043"
    ],
    [
      "0.624225699",
      "0.936796043"
    ],
    [
      "0.750633613",
      "0.936796043"
    ],
    [
      "0.877041527",
      "0.936796043"
    ]
  ],
  "radius": "0.063203956"
}

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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