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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 48

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 = 48
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[0.0724322912795, 0.0724322912805]见来源署名packomania.com ↗
本站纪录0.072432291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
0x1
已验证构造r = 0.072432291

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
单位正方形内的等圆装箱 n = 48 的当前纪录构型，0.072432291
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）
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.072432290→0.072432291
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": [
    [
      "0.072432291",
      "0.072432291"
    ],
    [
      "0.323345108",
      "0.072432291"
    ],
    [
      "0.574257926",
      "0.072432291"
    ],
    [
      "0.786745240",
      "0.072432291"
    ],
    [
      "0.927402175",
      "0.117005266"
    ],
    [
      "0.197888700",
      "0.144864583"
    ],
    [
      "0.448801517",
      "0.144864581"
    ],
    [
      "0.680501583",
      "0.170910880"
    ],
    [
      "0.818908372",
      "0.213681292"
    ],
    [
      "0.072432291",
      "0.217296873"
    ],
    [
      "0.323345109",
      "0.217296873"
    ],
    [
      "0.555045175",
      "0.243343172"
    ],
    [
      "0.197888699",
      "0.289729165"
    ],
    [
      "0.927567709",
      "0.309487846"
    ],
    [
      "0.429588766",
      "0.315775462"
    ],
    [
      "0.680501583",
      "0.315775464"
    ],
    [
      "0.072432291",
      "0.362161457"
    ],
    [
      "0.304132356",
      "0.388207754"
    ],
    [
      "0.555045174",
      "0.388207754"
    ],
    [
      "0.805957992",
      "0.388207754"
    ],
    [
      "0.178675948",
      "0.460640046"
    ],
    [
      "0.429588765",
      "0.460640044"
    ],
    [
      "0.680501583",
      "0.460640046"
    ],
    [
      "0.927567709",
      "0.466927663"
    ],
    [
      "0.304132357",
      "0.533072336"
    ],
    [
      "0.555045174",
      "0.533072337"
    ],
    [
      "0.802111300",
      "0.539359954"
    ],
    [
      "0.072432291",
      "0.559118635"
    ],
    [
      "0.429588764",
      "0.605504629"
    ],
    [
      "0.676654891",
      "0.611792245"
    ],
    [
      "0.927567709",
      "0.611792245"
    ],
    [
      "0.197888700",
      "0.631550926"
    ],
    [
      "0.551198482",
      "0.684224536"
    ],
    [
      "0.802111300",
      "0.684224537"
    ],
    [
      "0.072432291",
      "0.703983217"
    ],
    [
      "0.323345108",
      "0.703983219"
    ],
    [
      "0.676654891",
      "0.756656828"
    ],
    [
      "0.927567709",
      "0.756656828"
    ],
    [
      "0.197888699",
      "0.776415509"
    ],
    [
      "0.444954825",
      "0.782703127"
    ],
    [
      "0.802111300",
      "0.829089119"
    ],
    [
      "0.072432291",
      "0.848847801"
    ],
    [
      "0.319498416",
      "0.855135417"
    ],
    [
      "0.570411234",
      "0.855135417"
    ],
    [
      "0.910828974",
      "0.927260602"
    ],
    [
      "0.194042008",
      "0.927567709"
    ],
    [
      "0.444954825",
      "0.927567709"
    ],
    [
      "0.695867643",
      "0.927567709"
    ]
  ],
  "radius": "0.072432291"
}
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": 48
}

当前第一名的答案

{
  "centers": [
    [
      "0.072432291",
      "0.072432291"
    ],
    [
      "0.323345108",
      "0.072432291"
    ],
    [
      "0.574257926",
      "0.072432291"
    ],
    [
      "0.786745240",
      "0.072432291"
    ],
    [
      "0.927402175",
      "0.117005266"
    ],
    [
      "0.197888700",
      "0.144864583"
    ],
    [
      "0.448801517",
      "0.144864581"
    ],
    [
      "0.680501583",
      "0.170910880"
    ],
    [
      "0.818908372",
      "0.213681292"
    ],
    [
      "0.072432291",
      "0.217296873"
    ],
    [
      "0.323345109",
      "0.217296873"
    ],
    [
      "0.555045175",
      "0.243343172"
    ],
    [
      "0.197888699",
      "0.289729165"
    ],
    [
      "0.927567709",
      "0.309487846"
    ],
    [
      "0.429588766",
      "0.315775462"
    ],
    [
      "0.680501583",
      "0.315775464"
    ],
    [
      "0.072432291",
      "0.362161457"
    ],
    [
      "0.304132356",
      "0.388207754"
    ],
    [
      "0.555045174",
      "0.388207754"
    ],
    [
      "0.805957992",
      "0.388207754"
    ],
    [
      "0.178675948",
      "0.460640046"
    ],
    [
      "0.429588765",
      "0.460640044"
    ],
    [
      "0.680501583",
      "0.460640046"
    ],
    [
      "0.927567709",
      "0.466927663"
    ],
    [
      "0.304132357",
      "0.533072336"
    ],
    [
      "0.555045174",
      "0.533072337"
    ],
    [
      "0.802111300",
      "0.539359954"
    ],
    [
      "0.072432291",
      "0.559118635"
    ],
    [
      "0.429588764",
      "0.605504629"
    ],
    [
      "0.676654891",
      "0.611792245"
    ],
    [
      "0.927567709",
      "0.611792245"
    ],
    [
      "0.197888700",
      "0.631550926"
    ],
    [
      "0.551198482",
      "0.684224536"
    ],
    [
      "0.802111300",
      "0.684224537"
    ],
    [
      "0.072432291",
      "0.703983217"
    ],
    [
      "0.323345108",
      "0.703983219"
    ],
    [
      "0.676654891",
      "0.756656828"
    ],
    [
      "0.927567709",
      "0.756656828"
    ],
    [
      "0.197888699",
      "0.776415509"
    ],
    [
      "0.444954825",
      "0.782703127"
    ],
    [
      "0.802111300",
      "0.829089119"
    ],
    [
      "0.072432291",
      "0.848847801"
    ],
    [
      "0.319498416",
      "0.855135417"
    ],
    [
      "0.570411234",
      "0.855135417"
    ],
    [
      "0.910828974",
      "0.927260602"
    ],
    [
      "0.194042008",
      "0.927567709"
    ],
    [
      "0.444954825",
      "0.927567709"
    ],
    [
      "0.695867643",
      "0.927567709"
    ]
  ],
  "radius": "0.072432291"
}

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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