P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 37

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 = 37

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	[0.0820897664285, 0.0820897664295]	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

12345678910111213141516171819202122232425262728293031323334353637

0x1

已验证构造r = 0.082089766

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": [
    [
      "0.082131745",
      "0.082089766"
    ],
    [
      "0.368910715",
      "0.082089766"
    ],
    [
      "0.69363852",
      "0.082089766"
    ],
    [
      "0.857818053",
      "0.082089766"
    ],
    [
      "0.531274617",
      "0.106438938"
    ],
    [
      "0.22552123",
      "0.162054601"
    ],
    [
      "0.63354357",
      "0.234875617"
    ],
    [
      "0.917910234",
      "0.234876706"
    ],
    [
      "0.36646755",
      "0.24625112"
    ],
    [
      "0.082089766",
      "0.246269294"
    ],
    [
      "0.775726587",
      "0.316966472"
    ],
    [
      "0.224283904",
      "0.328340886"
    ],
    [
      "0.504062776",
      "0.335819378"
    ],
    [
      "0.917910234",
      "0.399056239"
    ],
    [
      "0.082089766",
      "0.410448827"
    ],
    [
      "0.36187913",
      "0.417909144"
    ],
    [
      "0.646245793",
      "0.417910234"
    ],
    [
      "0.219696112",
      "0.5"
    ],
    [
      "0.504062147",
      "0.5"
    ],
    [
      "0.78842944",
      "0.5"
    ],
    [
      "0.646245793",
      "0.582089766"
    ],
    [
      "0.36187913",
      "0.582090856"
    ],
    [
      "0.082089766",
      "0.589551173"
    ],
    [
      "0.917910234",
      "0.600943761"
    ],
    [
      "0.504062776",
      "0.664180622"
    ],
    [
      "0.224283904",
      "0.671659114"
    ],
    [
      "0.775726587",
      "0.683033528"
    ],
    [
      "0.082089766",
      "0.753730706"
    ],
    [
      "0.36646755",
      "0.75374888"
    ],
    [
      "0.917910234",
      "0.765123294"
    ],
    [
      "0.63354357",
      "0.765124383"
    ],
    [
      "0.22552123",
      "0.837945399"
    ],
    [
      "0.531274617",
      "0.893561062"
    ],
    [
      "0.082131745",
      "0.917910234"
    ],
    [
      "0.368910715",
      "0.917910234"
    ],
    [
      "0.69363852",
      "0.917910234"
    ],
    [
      "0.857818053",
      "0.917910234"
    ]
  ],
  "radius": "0.082089766"
}

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": 37
}

起步布局答案

{
  "centers": [
    [
      "0.082131745",
      "0.082089766"
    ],
    [
      "0.368910715",
      "0.082089766"
    ],
    [
      "0.69363852",
      "0.082089766"
    ],
    [
      "0.857818053",
      "0.082089766"
    ],
    [
      "0.531274617",
      "0.106438938"
    ],
    [
      "0.22552123",
      "0.162054601"
    ],
    [
      "0.63354357",
      "0.234875617"
    ],
    [
      "0.917910234",
      "0.234876706"
    ],
    [
      "0.36646755",
      "0.24625112"
    ],
    [
      "0.082089766",
      "0.246269294"
    ],
    [
      "0.775726587",
      "0.316966472"
    ],
    [
      "0.224283904",
      "0.328340886"
    ],
    [
      "0.504062776",
      "0.335819378"
    ],
    [
      "0.917910234",
      "0.399056239"
    ],
    [
      "0.082089766",
      "0.410448827"
    ],
    [
      "0.36187913",
      "0.417909144"
    ],
    [
      "0.646245793",
      "0.417910234"
    ],
    [
      "0.219696112",
      "0.5"
    ],
    [
      "0.504062147",
      "0.5"
    ],
    [
      "0.78842944",
      "0.5"
    ],
    [
      "0.646245793",
      "0.582089766"
    ],
    [
      "0.36187913",
      "0.582090856"
    ],
    [
      "0.082089766",
      "0.589551173"
    ],
    [
      "0.917910234",
      "0.600943761"
    ],
    [
      "0.504062776",
      "0.664180622"
    ],
    [
      "0.224283904",
      "0.671659114"
    ],
    [
      "0.775726587",
      "0.683033528"
    ],
    [
      "0.082089766",
      "0.753730706"
    ],
    [
      "0.36646755",
      "0.75374888"
    ],
    [
      "0.917910234",
      "0.765123294"
    ],
    [
      "0.63354357",
      "0.765124383"
    ],
    [
      "0.22552123",
      "0.837945399"
    ],
    [
      "0.531274617",
      "0.893561062"
    ],
    [
      "0.082131745",
      "0.917910234"
    ],
    [
      "0.368910715",
      "0.917910234"
    ],
    [
      "0.69363852",
      "0.917910234"
    ],
    [
      "0.857818053",
      "0.917910234"
    ]
  ],
  "radius": "0.082089766"
}

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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