P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 40

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 = 40

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	[0.0791867525165, 0.0791867525175]	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

12345678910111213141516171819202122232425262728293031323334353637383940

0x1

已验证构造r = 0.079186752

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

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": [
    [
      "0.079186753",
      "0.079186753"
    ],
    [
      "0.35349771",
      "0.079186753"
    ],
    [
      "0.63431503",
      "0.079186753"
    ],
    [
      "0.920813247",
      "0.079186753"
    ],
    [
      "0.777564139",
      "0.146728295"
    ],
    [
      "0.49390637",
      "0.15245049"
    ],
    [
      "0.216342231",
      "0.158373505"
    ],
    [
      "0.079186753",
      "0.237560258"
    ],
    [
      "0.63431503",
      "0.237560258"
    ],
    [
      "0.920813247",
      "0.237560258"
    ],
    [
      "0.35349771",
      "0.237560258"
    ],
    [
      "0.777564139",
      "0.3051018"
    ],
    [
      "0.49390637",
      "0.310823995"
    ],
    [
      "0.216342231",
      "0.31674701"
    ],
    [
      "0.079186753",
      "0.395933763"
    ],
    [
      "0.35349771",
      "0.395933763"
    ],
    [
      "0.63431503",
      "0.395933763"
    ],
    [
      "0.920813247",
      "0.395933763"
    ],
    [
      "0.777564139",
      "0.463475305"
    ],
    [
      "0.49390637",
      "0.4691975"
    ],
    [
      "0.216342231",
      "0.475120515"
    ],
    [
      "0.63431503",
      "0.554307268"
    ],
    [
      "0.920813247",
      "0.554307268"
    ],
    [
      "0.079186753",
      "0.554307268"
    ],
    [
      "0.35349771",
      "0.554307268"
    ],
    [
      "0.777564139",
      "0.62184881"
    ],
    [
      "0.49390637",
      "0.627571005"
    ],
    [
      "0.22617562",
      "0.648494595"
    ],
    [
      "0.081385219",
      "0.712665513"
    ],
    [
      "0.920813247",
      "0.712680773"
    ],
    [
      "0.622035842",
      "0.728922407"
    ],
    [
      "0.777564139",
      "0.780222315"
    ],
    [
      "0.308717025",
      "0.783657769"
    ],
    [
      "0.46709053",
      "0.783657769"
    ],
    [
      "0.079186753",
      "0.871023758"
    ],
    [
      "0.903980718",
      "0.903980718"
    ],
    [
      "0.229530272",
      "0.920813247"
    ],
    [
      "0.387903777",
      "0.920813247"
    ],
    [
      "0.546277282",
      "0.920813247"
    ],
    [
      "0.704650787",
      "0.920813247"
    ]
  ],
  "radius": "0.079186752"
}

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": 40
}

起步布局答案

{
  "centers": [
    [
      "0.079186753",
      "0.079186753"
    ],
    [
      "0.35349771",
      "0.079186753"
    ],
    [
      "0.63431503",
      "0.079186753"
    ],
    [
      "0.920813247",
      "0.079186753"
    ],
    [
      "0.777564139",
      "0.146728295"
    ],
    [
      "0.49390637",
      "0.15245049"
    ],
    [
      "0.216342231",
      "0.158373505"
    ],
    [
      "0.079186753",
      "0.237560258"
    ],
    [
      "0.63431503",
      "0.237560258"
    ],
    [
      "0.920813247",
      "0.237560258"
    ],
    [
      "0.35349771",
      "0.237560258"
    ],
    [
      "0.777564139",
      "0.3051018"
    ],
    [
      "0.49390637",
      "0.310823995"
    ],
    [
      "0.216342231",
      "0.31674701"
    ],
    [
      "0.079186753",
      "0.395933763"
    ],
    [
      "0.35349771",
      "0.395933763"
    ],
    [
      "0.63431503",
      "0.395933763"
    ],
    [
      "0.920813247",
      "0.395933763"
    ],
    [
      "0.777564139",
      "0.463475305"
    ],
    [
      "0.49390637",
      "0.4691975"
    ],
    [
      "0.216342231",
      "0.475120515"
    ],
    [
      "0.63431503",
      "0.554307268"
    ],
    [
      "0.920813247",
      "0.554307268"
    ],
    [
      "0.079186753",
      "0.554307268"
    ],
    [
      "0.35349771",
      "0.554307268"
    ],
    [
      "0.777564139",
      "0.62184881"
    ],
    [
      "0.49390637",
      "0.627571005"
    ],
    [
      "0.22617562",
      "0.648494595"
    ],
    [
      "0.081385219",
      "0.712665513"
    ],
    [
      "0.920813247",
      "0.712680773"
    ],
    [
      "0.622035842",
      "0.728922407"
    ],
    [
      "0.777564139",
      "0.780222315"
    ],
    [
      "0.308717025",
      "0.783657769"
    ],
    [
      "0.46709053",
      "0.783657769"
    ],
    [
      "0.079186753",
      "0.871023758"
    ],
    [
      "0.903980718",
      "0.903980718"
    ],
    [
      "0.229530272",
      "0.920813247"
    ],
    [
      "0.387903777",
      "0.920813247"
    ],
    [
      "0.546277282",
      "0.920813247"
    ],
    [
      "0.704650787",
      "0.920813247"
    ]
  ],
  "radius": "0.079186752"
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

啊？？！NUE_13#310 条回复18 小时前 空悲切/.lird#282 条回复15 天前