P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 31

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 = 31

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	[0.0893383333505, 0.0893383333515]	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

12345678910111213141516171819202122232425262728293031

0x1

已验证构造r = 0.089338333

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": [
    [
      "0.089338333",
      "0.089338333"
    ],
    [
      "0.268015",
      "0.089338333"
    ],
    [
      "0.446691667",
      "0.089338333"
    ],
    [
      "0.754662534",
      "0.089338333"
    ],
    [
      "0.910661667",
      "0.176456768"
    ],
    [
      "0.6006771",
      "0.179968557"
    ],
    [
      "0.757215322",
      "0.267996763"
    ],
    [
      "0.089338333",
      "0.268015"
    ],
    [
      "0.268015",
      "0.268015"
    ],
    [
      "0.44667343",
      "0.270567789"
    ],
    [
      "0.603229889",
      "0.358626987"
    ],
    [
      "0.910661667",
      "0.359536758"
    ],
    [
      "0.270567789",
      "0.44667343"
    ],
    [
      "0.089338333",
      "0.446691667"
    ],
    [
      "0.449226218",
      "0.449226218"
    ],
    [
      "0.756676233",
      "0.450166982"
    ],
    [
      "0.910661667",
      "0.540797206"
    ],
    [
      "0.57884704",
      "0.57884704"
    ],
    [
      "0.179968557",
      "0.6006771"
    ],
    [
      "0.358626987",
      "0.603229889"
    ],
    [
      "0.755180874",
      "0.628837391"
    ],
    [
      "0.905733727",
      "0.72668524"
    ],
    [
      "0.089338333",
      "0.754662534"
    ],
    [
      "0.628837391",
      "0.755180874"
    ],
    [
      "0.450166982",
      "0.756676233"
    ],
    [
      "0.267996763",
      "0.757215322"
    ],
    [
      "0.905733727",
      "0.905733727"
    ],
    [
      "0.72668524",
      "0.905733727"
    ],
    [
      "0.176456768",
      "0.910661667"
    ],
    [
      "0.359536758",
      "0.910661667"
    ],
    [
      "0.540797206",
      "0.910661667"
    ]
  ],
  "radius": "0.089338333"
}

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": 31
}

起步布局答案

{
  "centers": [
    [
      "0.089338333",
      "0.089338333"
    ],
    [
      "0.268015",
      "0.089338333"
    ],
    [
      "0.446691667",
      "0.089338333"
    ],
    [
      "0.754662534",
      "0.089338333"
    ],
    [
      "0.910661667",
      "0.176456768"
    ],
    [
      "0.6006771",
      "0.179968557"
    ],
    [
      "0.757215322",
      "0.267996763"
    ],
    [
      "0.089338333",
      "0.268015"
    ],
    [
      "0.268015",
      "0.268015"
    ],
    [
      "0.44667343",
      "0.270567789"
    ],
    [
      "0.603229889",
      "0.358626987"
    ],
    [
      "0.910661667",
      "0.359536758"
    ],
    [
      "0.270567789",
      "0.44667343"
    ],
    [
      "0.089338333",
      "0.446691667"
    ],
    [
      "0.449226218",
      "0.449226218"
    ],
    [
      "0.756676233",
      "0.450166982"
    ],
    [
      "0.910661667",
      "0.540797206"
    ],
    [
      "0.57884704",
      "0.57884704"
    ],
    [
      "0.179968557",
      "0.6006771"
    ],
    [
      "0.358626987",
      "0.603229889"
    ],
    [
      "0.755180874",
      "0.628837391"
    ],
    [
      "0.905733727",
      "0.72668524"
    ],
    [
      "0.089338333",
      "0.754662534"
    ],
    [
      "0.628837391",
      "0.755180874"
    ],
    [
      "0.450166982",
      "0.756676233"
    ],
    [
      "0.267996763",
      "0.757215322"
    ],
    [
      "0.905733727",
      "0.905733727"
    ],
    [
      "0.72668524",
      "0.905733727"
    ],
    [
      "0.176456768",
      "0.910661667"
    ],
    [
      "0.359536758",
      "0.910661667"
    ],
    [
      "0.540797206",
      "0.910661667"
    ]
  ],
  "radius": "0.089338333"
}

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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