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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 = 60

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 = 60
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[0.0650304126475, 0.0650304126485]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
0x1
已验证构造r = 0.065030412

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 = 60 的当前构型
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": [
    [
      "0.065030413",
      "0.065030413"
    ],
    [
      "0.29030237",
      "0.065030413"
    ],
    [
      "0.515574328",
      "0.065030413"
    ],
    [
      "0.645635153",
      "0.065030413"
    ],
    [
      "0.775695978",
      "0.065030413"
    ],
    [
      "0.917570802",
      "0.082429198"
    ],
    [
      "0.177666391",
      "0.130060825"
    ],
    [
      "0.402938349",
      "0.130060825"
    ],
    [
      "0.596817942",
      "0.185582054"
    ],
    [
      "0.822333609",
      "0.186441902"
    ],
    [
      "0.065030413",
      "0.195091238"
    ],
    [
      "0.29030237",
      "0.195091238"
    ],
    [
      "0.477263266",
      "0.236792381"
    ],
    [
      "0.709328641",
      "0.250828978"
    ],
    [
      "0.934969587",
      "0.251472314"
    ],
    [
      "0.177666391",
      "0.260121651"
    ],
    [
      "0.364627287",
      "0.301822794"
    ],
    [
      "0.596692662",
      "0.31585939"
    ],
    [
      "0.822333609",
      "0.316502727"
    ],
    [
      "0.065030413",
      "0.325152063"
    ],
    [
      "0.251991308",
      "0.366853206"
    ],
    [
      "0.477137986",
      "0.367069717"
    ],
    [
      "0.709328641",
      "0.380889803"
    ],
    [
      "0.934969587",
      "0.38153314"
    ],
    [
      "0.13935533",
      "0.431883619"
    ],
    [
      "0.364502007",
      "0.43210013"
    ],
    [
      "0.596692662",
      "0.445920216"
    ],
    [
      "0.822333609",
      "0.446563552"
    ],
    [
      "0.251866028",
      "0.497130543"
    ],
    [
      "0.477137986",
      "0.497130543"
    ],
    [
      "0.709328641",
      "0.510950628"
    ],
    [
      "0.934969587",
      "0.511593965"
    ],
    [
      "0.065030413",
      "0.538615175"
    ],
    [
      "0.364502007",
      "0.562160955"
    ],
    [
      "0.596692662",
      "0.575981041"
    ],
    [
      "0.822333609",
      "0.576624378"
    ],
    [
      "0.177541111",
      "0.603862098"
    ],
    [
      "0.477137986",
      "0.627191368"
    ],
    [
      "0.709328641",
      "0.641011454"
    ],
    [
      "0.934969587",
      "0.64165479"
    ],
    [
      "0.29017709",
      "0.668892511"
    ],
    [
      "0.065030413",
      "0.669109022"
    ],
    [
      "0.596692662",
      "0.706041866"
    ],
    [
      "0.822333609",
      "0.706685203"
    ],
    [
      "0.402813068",
      "0.733922924"
    ],
    [
      "0.177666391",
      "0.734139435"
    ],
    [
      "0.709328641",
      "0.771072279"
    ],
    [
      "0.934969587",
      "0.771715616"
    ],
    [
      "0.065030413",
      "0.799169847"
    ],
    [
      "0.29030237",
      "0.799169847"
    ],
    [
      "0.511934655",
      "0.804692251"
    ],
    [
      "0.822333609",
      "0.836746028"
    ],
    [
      "0.177666391",
      "0.86420026"
    ],
    [
      "0.624570633",
      "0.869722664"
    ],
    [
      "0.399423956",
      "0.869939175"
    ],
    [
      "0.93061989",
      "0.930619895"
    ],
    [
      "0.065900351",
      "0.93409965"
    ],
    [
      "0.286787978",
      "0.934969587"
    ],
    [
      "0.512059935",
      "0.934969587"
    ],
    [
      "0.737081332",
      "0.934969587"
    ]
  ],
  "radius": "0.065030412"
}
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": 60
}

起步布局答案

{
  "centers": [
    [
      "0.065030413",
      "0.065030413"
    ],
    [
      "0.29030237",
      "0.065030413"
    ],
    [
      "0.515574328",
      "0.065030413"
    ],
    [
      "0.645635153",
      "0.065030413"
    ],
    [
      "0.775695978",
      "0.065030413"
    ],
    [
      "0.917570802",
      "0.082429198"
    ],
    [
      "0.177666391",
      "0.130060825"
    ],
    [
      "0.402938349",
      "0.130060825"
    ],
    [
      "0.596817942",
      "0.185582054"
    ],
    [
      "0.822333609",
      "0.186441902"
    ],
    [
      "0.065030413",
      "0.195091238"
    ],
    [
      "0.29030237",
      "0.195091238"
    ],
    [
      "0.477263266",
      "0.236792381"
    ],
    [
      "0.709328641",
      "0.250828978"
    ],
    [
      "0.934969587",
      "0.251472314"
    ],
    [
      "0.177666391",
      "0.260121651"
    ],
    [
      "0.364627287",
      "0.301822794"
    ],
    [
      "0.596692662",
      "0.31585939"
    ],
    [
      "0.822333609",
      "0.316502727"
    ],
    [
      "0.065030413",
      "0.325152063"
    ],
    [
      "0.251991308",
      "0.366853206"
    ],
    [
      "0.477137986",
      "0.367069717"
    ],
    [
      "0.709328641",
      "0.380889803"
    ],
    [
      "0.934969587",
      "0.38153314"
    ],
    [
      "0.13935533",
      "0.431883619"
    ],
    [
      "0.364502007",
      "0.43210013"
    ],
    [
      "0.596692662",
      "0.445920216"
    ],
    [
      "0.822333609",
      "0.446563552"
    ],
    [
      "0.251866028",
      "0.497130543"
    ],
    [
      "0.477137986",
      "0.497130543"
    ],
    [
      "0.709328641",
      "0.510950628"
    ],
    [
      "0.934969587",
      "0.511593965"
    ],
    [
      "0.065030413",
      "0.538615175"
    ],
    [
      "0.364502007",
      "0.562160955"
    ],
    [
      "0.596692662",
      "0.575981041"
    ],
    [
      "0.822333609",
      "0.576624378"
    ],
    [
      "0.177541111",
      "0.603862098"
    ],
    [
      "0.477137986",
      "0.627191368"
    ],
    [
      "0.709328641",
      "0.641011454"
    ],
    [
      "0.934969587",
      "0.64165479"
    ],
    [
      "0.29017709",
      "0.668892511"
    ],
    [
      "0.065030413",
      "0.669109022"
    ],
    [
      "0.596692662",
      "0.706041866"
    ],
    [
      "0.822333609",
      "0.706685203"
    ],
    [
      "0.402813068",
      "0.733922924"
    ],
    [
      "0.177666391",
      "0.734139435"
    ],
    [
      "0.709328641",
      "0.771072279"
    ],
    [
      "0.934969587",
      "0.771715616"
    ],
    [
      "0.065030413",
      "0.799169847"
    ],
    [
      "0.29030237",
      "0.799169847"
    ],
    [
      "0.511934655",
      "0.804692251"
    ],
    [
      "0.822333609",
      "0.836746028"
    ],
    [
      "0.177666391",
      "0.86420026"
    ],
    [
      "0.624570633",
      "0.869722664"
    ],
    [
      "0.399423956",
      "0.869939175"
    ],
    [
      "0.93061989",
      "0.930619895"
    ],
    [
      "0.065900351",
      "0.93409965"
    ],
    [
      "0.286787978",
      "0.934969587"
    ],
    [
      "0.512059935",
      "0.934969587"
    ],
    [
      "0.737081332",
      "0.934969587"
    ]
  ],
  "radius": "0.065030412"
}

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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