P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 32

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 = 32

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	[0.0878581570875, 0.0878581570885]	见来源署名	packomania.com ↗
本站纪录	0.087858157	邱仲普	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

1234567891011121314151617181920212223242526272829303132

0x1

已验证构造r = 0.087858157

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。

纪录保持者邱仲普

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公开方法与可复现线索（点击展开）＋

从公开的 n=32 圆心构型与半径开始，在整数网格上把共同半径增加一个单位；用所有边界约束和圆对距离的线性化不等式求整数圆心位移，迭代修复后再以精确平方距离复核全部 32 个圆。得到的构型保持单位正方形内且无重叠，半径由 0.087858156 提高到 0.087858157；服务器精确验证通过。

在讨论区查看与回复 →

历史纪录（1 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.087858156→0.087858157
2026-09-04

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": [
    [
      "0.087858157",
      "0.087858157"
    ],
    [
      "0.263574471",
      "0.087858157"
    ],
    [
      "0.439290785",
      "0.087858157"
    ],
    [
      "0.772814609",
      "0.087858157"
    ],
    [
      "0.606052697",
      "0.143235840"
    ],
    [
      "0.912141843",
      "0.194928905"
    ],
    [
      "0.123280631",
      "0.259967051"
    ],
    [
      "0.298996945",
      "0.259967051"
    ],
    [
      "0.474713259",
      "0.259967051"
    ],
    [
      "0.747275106",
      "0.269190568"
    ],
    [
      "0.912141843",
      "0.370645220"
    ],
    [
      "0.606094052",
      "0.376651713"
    ],
    [
      "0.263574471",
      "0.432075945"
    ],
    [
      "0.087858157",
      "0.432075945"
    ],
    [
      "0.439290785",
      "0.432075945"
    ],
    [
      "0.760811709",
      "0.459950391"
    ],
    [
      "0.912141843",
      "0.549255562"
    ],
    [
      "0.606037229",
      "0.552368018"
    ],
    [
      "0.087858157",
      "0.607792259"
    ],
    [
      "0.263574471",
      "0.607792259"
    ],
    [
      "0.439290785",
      "0.607792259"
    ],
    [
      "0.759967050",
      "0.637113717"
    ],
    [
      "0.912141843",
      "0.724971876"
    ],
    [
      "0.574175532",
      "0.734910850"
    ],
    [
      "0.175716314",
      "0.759967051"
    ],
    [
      "0.351432628",
      "0.759967051"
    ],
    [
      "0.759967050",
      "0.812830031"
    ],
    [
      "0.908010424",
      "0.908010424"
    ],
    [
      "0.087858157",
      "0.912141843"
    ],
    [
      "0.263574471",
      "0.912141843"
    ],
    [
      "0.439290785",
      "0.912141843"
    ],
    [
      "0.615007100",
      "0.912141843"
    ]
  ],
  "radius": "0.087858157"
}

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": 32
}

当前第一名的答案

{
  "centers": [
    [
      "0.087858157",
      "0.087858157"
    ],
    [
      "0.263574471",
      "0.087858157"
    ],
    [
      "0.439290785",
      "0.087858157"
    ],
    [
      "0.772814609",
      "0.087858157"
    ],
    [
      "0.606052697",
      "0.143235840"
    ],
    [
      "0.912141843",
      "0.194928905"
    ],
    [
      "0.123280631",
      "0.259967051"
    ],
    [
      "0.298996945",
      "0.259967051"
    ],
    [
      "0.474713259",
      "0.259967051"
    ],
    [
      "0.747275106",
      "0.269190568"
    ],
    [
      "0.912141843",
      "0.370645220"
    ],
    [
      "0.606094052",
      "0.376651713"
    ],
    [
      "0.263574471",
      "0.432075945"
    ],
    [
      "0.087858157",
      "0.432075945"
    ],
    [
      "0.439290785",
      "0.432075945"
    ],
    [
      "0.760811709",
      "0.459950391"
    ],
    [
      "0.912141843",
      "0.549255562"
    ],
    [
      "0.606037229",
      "0.552368018"
    ],
    [
      "0.087858157",
      "0.607792259"
    ],
    [
      "0.263574471",
      "0.607792259"
    ],
    [
      "0.439290785",
      "0.607792259"
    ],
    [
      "0.759967050",
      "0.637113717"
    ],
    [
      "0.912141843",
      "0.724971876"
    ],
    [
      "0.574175532",
      "0.734910850"
    ],
    [
      "0.175716314",
      "0.759967051"
    ],
    [
      "0.351432628",
      "0.759967051"
    ],
    [
      "0.759967050",
      "0.812830031"
    ],
    [
      "0.908010424",
      "0.908010424"
    ],
    [
      "0.087858157",
      "0.912141843"
    ],
    [
      "0.263574471",
      "0.912141843"
    ],
    [
      "0.439290785",
      "0.912141843"
    ],
    [
      "0.615007100",
      "0.912141843"
    ]
  ],
  "radius": "0.087858157"
}

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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