极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 66

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 = 66
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[0.0628622569025, 0.0628622569035]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 = 0.062862256

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 = 66 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": [
    [
      "0.062960426",
      "0.062862257"
    ],
    [
      "0.284095865",
      "0.062862257"
    ],
    [
      "0.524219242",
      "0.062862257"
    ],
    [
      "0.77367956",
      "0.062862257"
    ],
    [
      "0.899404074",
      "0.062862257"
    ],
    [
      "0.648949401",
      "0.078643284"
    ],
    [
      "0.404157553",
      "0.100169687"
    ],
    [
      "0.173528145",
      "0.12270733"
    ],
    [
      "0.719376497",
      "0.182790667"
    ],
    [
      "0.937137743",
      "0.182790667"
    ],
    [
      "0.49649758",
      "0.185492444"
    ],
    [
      "0.280617042",
      "0.188575544"
    ],
    [
      "0.062862257",
      "0.188586732"
    ],
    [
      "0.608751704",
      "0.244779237"
    ],
    [
      "0.82825712",
      "0.245652924"
    ],
    [
      "0.17174288",
      "0.251448989"
    ],
    [
      "0.389477591",
      "0.251472557"
    ],
    [
      "0.719376497",
      "0.308515181"
    ],
    [
      "0.937137743",
      "0.308515181"
    ],
    [
      "0.500128039",
      "0.31116453"
    ],
    [
      "0.062862257",
      "0.314311246"
    ],
    [
      "0.280603429",
      "0.314346003"
    ],
    [
      "0.610495875",
      "0.371377438"
    ],
    [
      "0.82825712",
      "0.371377438"
    ],
    [
      "0.171722806",
      "0.37720826"
    ],
    [
      "0.389310104",
      "0.377508587"
    ],
    [
      "0.719376497",
      "0.434239695"
    ],
    [
      "0.937137743",
      "0.434239695"
    ],
    [
      "0.500128039",
      "0.436889044"
    ],
    [
      "0.062862257",
      "0.440105273"
    ],
    [
      "0.280429481",
      "0.440370844"
    ],
    [
      "0.610495875",
      "0.497101952"
    ],
    [
      "0.82825712",
      "0.497101952"
    ],
    [
      "0.389310104",
      "0.503233101"
    ],
    [
      "0.171568932",
      "0.503267857"
    ],
    [
      "0.719376497",
      "0.559964208"
    ],
    [
      "0.937137743",
      "0.559964208"
    ],
    [
      "0.500128039",
      "0.562613558"
    ],
    [
      "0.280449555",
      "0.566130114"
    ],
    [
      "0.062862257",
      "0.566430442"
    ],
    [
      "0.610495875",
      "0.622826465"
    ],
    [
      "0.82825712",
      "0.622826465"
    ],
    [
      "0.389350241",
      "0.628957608"
    ],
    [
      "0.17174288",
      "0.629292699"
    ],
    [
      "0.719376497",
      "0.685688722"
    ],
    [
      "0.937137743",
      "0.685688722"
    ],
    [
      "0.500168176",
      "0.688338065"
    ],
    [
      "0.280643566",
      "0.692120193"
    ],
    [
      "0.062862257",
      "0.692154956"
    ],
    [
      "0.82825712",
      "0.748550979"
    ],
    [
      "0.610516509",
      "0.748586706"
    ],
    [
      "0.17174288",
      "0.755017212"
    ],
    [
      "0.38999105",
      "0.755051035"
    ],
    [
      "0.937137743",
      "0.811413236"
    ],
    [
      "0.719397132",
      "0.811448963"
    ],
    [
      "0.503165489",
      "0.814026845"
    ],
    [
      "0.280643566",
      "0.817844707"
    ],
    [
      "0.062862257",
      "0.817879469"
    ],
    [
      "0.826748152",
      "0.876889102"
    ],
    [
      "0.597031574",
      "0.897667813"
    ],
    [
      "0.353025536",
      "0.920643071"
    ],
    [
      "0.102663231",
      "0.937137743"
    ],
    [
      "0.228387745",
      "0.937137743"
    ],
    [
      "0.477663328",
      "0.937137743"
    ],
    [
      "0.716399819",
      "0.937137743"
    ],
    [
      "0.937096485",
      "0.937137743"
    ]
  ],
  "radius": "0.062862256"
}
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": 66
}

起步布局答案

{
  "centers": [
    [
      "0.062960426",
      "0.062862257"
    ],
    [
      "0.284095865",
      "0.062862257"
    ],
    [
      "0.524219242",
      "0.062862257"
    ],
    [
      "0.77367956",
      "0.062862257"
    ],
    [
      "0.899404074",
      "0.062862257"
    ],
    [
      "0.648949401",
      "0.078643284"
    ],
    [
      "0.404157553",
      "0.100169687"
    ],
    [
      "0.173528145",
      "0.12270733"
    ],
    [
      "0.719376497",
      "0.182790667"
    ],
    [
      "0.937137743",
      "0.182790667"
    ],
    [
      "0.49649758",
      "0.185492444"
    ],
    [
      "0.280617042",
      "0.188575544"
    ],
    [
      "0.062862257",
      "0.188586732"
    ],
    [
      "0.608751704",
      "0.244779237"
    ],
    [
      "0.82825712",
      "0.245652924"
    ],
    [
      "0.17174288",
      "0.251448989"
    ],
    [
      "0.389477591",
      "0.251472557"
    ],
    [
      "0.719376497",
      "0.308515181"
    ],
    [
      "0.937137743",
      "0.308515181"
    ],
    [
      "0.500128039",
      "0.31116453"
    ],
    [
      "0.062862257",
      "0.314311246"
    ],
    [
      "0.280603429",
      "0.314346003"
    ],
    [
      "0.610495875",
      "0.371377438"
    ],
    [
      "0.82825712",
      "0.371377438"
    ],
    [
      "0.171722806",
      "0.37720826"
    ],
    [
      "0.389310104",
      "0.377508587"
    ],
    [
      "0.719376497",
      "0.434239695"
    ],
    [
      "0.937137743",
      "0.434239695"
    ],
    [
      "0.500128039",
      "0.436889044"
    ],
    [
      "0.062862257",
      "0.440105273"
    ],
    [
      "0.280429481",
      "0.440370844"
    ],
    [
      "0.610495875",
      "0.497101952"
    ],
    [
      "0.82825712",
      "0.497101952"
    ],
    [
      "0.389310104",
      "0.503233101"
    ],
    [
      "0.171568932",
      "0.503267857"
    ],
    [
      "0.719376497",
      "0.559964208"
    ],
    [
      "0.937137743",
      "0.559964208"
    ],
    [
      "0.500128039",
      "0.562613558"
    ],
    [
      "0.280449555",
      "0.566130114"
    ],
    [
      "0.062862257",
      "0.566430442"
    ],
    [
      "0.610495875",
      "0.622826465"
    ],
    [
      "0.82825712",
      "0.622826465"
    ],
    [
      "0.389350241",
      "0.628957608"
    ],
    [
      "0.17174288",
      "0.629292699"
    ],
    [
      "0.719376497",
      "0.685688722"
    ],
    [
      "0.937137743",
      "0.685688722"
    ],
    [
      "0.500168176",
      "0.688338065"
    ],
    [
      "0.280643566",
      "0.692120193"
    ],
    [
      "0.062862257",
      "0.692154956"
    ],
    [
      "0.82825712",
      "0.748550979"
    ],
    [
      "0.610516509",
      "0.748586706"
    ],
    [
      "0.17174288",
      "0.755017212"
    ],
    [
      "0.38999105",
      "0.755051035"
    ],
    [
      "0.937137743",
      "0.811413236"
    ],
    [
      "0.719397132",
      "0.811448963"
    ],
    [
      "0.503165489",
      "0.814026845"
    ],
    [
      "0.280643566",
      "0.817844707"
    ],
    [
      "0.062862257",
      "0.817879469"
    ],
    [
      "0.826748152",
      "0.876889102"
    ],
    [
      "0.597031574",
      "0.897667813"
    ],
    [
      "0.353025536",
      "0.920643071"
    ],
    [
      "0.102663231",
      "0.937137743"
    ],
    [
      "0.228387745",
      "0.937137743"
    ],
    [
      "0.477663328",
      "0.937137743"
    ],
    [
      "0.716399819",
      "0.937137743"
    ],
    [
      "0.937096485",
      "0.937137743"
    ]
  ],
  "radius": "0.062862256"
}

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

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复