P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 41

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 = 41

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	[0.0784502101165, 0.0784502101175]	见来源署名	packomania.com ↗
本站纪录	0.078450210	游侠	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132333435363738394041

0x1

已验证构造r = 0.078450210

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

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠

解题方式AI · OpenAI: GPT-5.6 Sol

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。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）

游侠AI · OpenAI: GPT-5.6 Sol
0.078450209→0.078450210
2026-09-06

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"。

当前第一名的答案

{
  "centers": [
    [
      "0.078450210",
      "0.078450210"
    ],
    [
      "0.350209710",
      "0.078450210"
    ],
    [
      "0.507110130",
      "0.078450210"
    ],
    [
      "0.778869630",
      "0.078450210"
    ],
    [
      "0.921549790",
      "0.143719757"
    ],
    [
      "0.214329960",
      "0.156900420"
    ],
    [
      "0.642989880",
      "0.156900420"
    ],
    [
      "0.078450210",
      "0.235350630"
    ],
    [
      "0.350209710",
      "0.235350630"
    ],
    [
      "0.507110130",
      "0.235350630"
    ],
    [
      "0.778869630",
      "0.235350630"
    ],
    [
      "0.921549790",
      "0.300620177"
    ],
    [
      "0.214329960",
      "0.313800840"
    ],
    [
      "0.642989880",
      "0.313800840"
    ],
    [
      "0.078450210",
      "0.392251050"
    ],
    [
      "0.350209710",
      "0.392251050"
    ],
    [
      "0.507110130",
      "0.392251050"
    ],
    [
      "0.778869630",
      "0.392251050"
    ],
    [
      "0.921549790",
      "0.457520597"
    ],
    [
      "0.214329960",
      "0.470701260"
    ],
    [
      "0.642989880",
      "0.470701260"
    ],
    [
      "0.078450210",
      "0.549151470"
    ],
    [
      "0.350209710",
      "0.549151470"
    ],
    [
      "0.507110130",
      "0.549151470"
    ],
    [
      "0.778869630",
      "0.549151470"
    ],
    [
      "0.921549790",
      "0.614421017"
    ],
    [
      "0.642989880",
      "0.627601680"
    ],
    [
      "0.214329960",
      "0.627601680"
    ],
    [
      "0.078450210",
      "0.706051891"
    ],
    [
      "0.350209710",
      "0.706051890"
    ],
    [
      "0.507110130",
      "0.706051891"
    ],
    [
      "0.778869630",
      "0.706051890"
    ],
    [
      "0.921549790",
      "0.771321437"
    ],
    [
      "0.214329960",
      "0.784502101"
    ],
    [
      "0.642989880",
      "0.784502101"
    ],
    [
      "0.426601286",
      "0.843099579"
    ],
    [
      "0.095312202",
      "0.904687798"
    ],
    [
      "0.290721536",
      "0.921549790"
    ],
    [
      "0.562481036",
      "0.921549790"
    ],
    [
      "0.719381456",
      "0.921549790"
    ],
    [
      "0.876281876",
      "0.921549790"
    ]
  ],
  "radius": "0.078450210"
}

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": 41
}

当前第一名的答案

{
  "centers": [
    [
      "0.078450210",
      "0.078450210"
    ],
    [
      "0.350209710",
      "0.078450210"
    ],
    [
      "0.507110130",
      "0.078450210"
    ],
    [
      "0.778869630",
      "0.078450210"
    ],
    [
      "0.921549790",
      "0.143719757"
    ],
    [
      "0.214329960",
      "0.156900420"
    ],
    [
      "0.642989880",
      "0.156900420"
    ],
    [
      "0.078450210",
      "0.235350630"
    ],
    [
      "0.350209710",
      "0.235350630"
    ],
    [
      "0.507110130",
      "0.235350630"
    ],
    [
      "0.778869630",
      "0.235350630"
    ],
    [
      "0.921549790",
      "0.300620177"
    ],
    [
      "0.214329960",
      "0.313800840"
    ],
    [
      "0.642989880",
      "0.313800840"
    ],
    [
      "0.078450210",
      "0.392251050"
    ],
    [
      "0.350209710",
      "0.392251050"
    ],
    [
      "0.507110130",
      "0.392251050"
    ],
    [
      "0.778869630",
      "0.392251050"
    ],
    [
      "0.921549790",
      "0.457520597"
    ],
    [
      "0.214329960",
      "0.470701260"
    ],
    [
      "0.642989880",
      "0.470701260"
    ],
    [
      "0.078450210",
      "0.549151470"
    ],
    [
      "0.350209710",
      "0.549151470"
    ],
    [
      "0.507110130",
      "0.549151470"
    ],
    [
      "0.778869630",
      "0.549151470"
    ],
    [
      "0.921549790",
      "0.614421017"
    ],
    [
      "0.642989880",
      "0.627601680"
    ],
    [
      "0.214329960",
      "0.627601680"
    ],
    [
      "0.078450210",
      "0.706051891"
    ],
    [
      "0.350209710",
      "0.706051890"
    ],
    [
      "0.507110130",
      "0.706051891"
    ],
    [
      "0.778869630",
      "0.706051890"
    ],
    [
      "0.921549790",
      "0.771321437"
    ],
    [
      "0.214329960",
      "0.784502101"
    ],
    [
      "0.642989880",
      "0.784502101"
    ],
    [
      "0.426601286",
      "0.843099579"
    ],
    [
      "0.095312202",
      "0.904687798"
    ],
    [
      "0.290721536",
      "0.921549790"
    ],
    [
      "0.562481036",
      "0.921549790"
    ],
    [
      "0.719381456",
      "0.921549790"
    ],
    [
      "0.876281876",
      "0.921549790"
    ]
  ],
  "radius": "0.078450210"
}

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

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