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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 = 61

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 = 61
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[0.0646662689055, 0.0646662689065]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 = 0.064666268

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 = 61 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": [
    [
      "0.064666269",
      "0.064666269"
    ],
    [
      "0.193998807",
      "0.064666269"
    ],
    [
      "0.323331345",
      "0.064666269"
    ],
    [
      "0.452663882",
      "0.064666269"
    ],
    [
      "0.58199642",
      "0.064666269"
    ],
    [
      "0.711328958",
      "0.064666269"
    ],
    [
      "0.935333731",
      "0.064666269"
    ],
    [
      "0.823331345",
      "0.12933752"
    ],
    [
      "0.064666269",
      "0.193998807"
    ],
    [
      "0.193998807",
      "0.193998807"
    ],
    [
      "0.323331345",
      "0.193998807"
    ],
    [
      "0.452663882",
      "0.193998807"
    ],
    [
      "0.58199642",
      "0.193998807"
    ],
    [
      "0.711328958",
      "0.194008771"
    ],
    [
      "0.935333731",
      "0.194008771"
    ],
    [
      "0.823331344",
      "0.258680022"
    ],
    [
      "0.064666269",
      "0.323331345"
    ],
    [
      "0.193998807",
      "0.323331345"
    ],
    [
      "0.323331345",
      "0.323331345"
    ],
    [
      "0.452663882",
      "0.323331345"
    ],
    [
      "0.58199642",
      "0.323331345"
    ],
    [
      "0.711328956",
      "0.323351271"
    ],
    [
      "0.935333731",
      "0.323351273"
    ],
    [
      "0.823331343",
      "0.388022522"
    ],
    [
      "0.064666269",
      "0.452663882"
    ],
    [
      "0.193998807",
      "0.452663882"
    ],
    [
      "0.323331345",
      "0.452663882"
    ],
    [
      "0.452663882",
      "0.452663882"
    ],
    [
      "0.58199642",
      "0.452663882"
    ],
    [
      "0.711328955",
      "0.452693769"
    ],
    [
      "0.935333731",
      "0.452693771"
    ],
    [
      "0.823331342",
      "0.517365019"
    ],
    [
      "0.193998807",
      "0.58199642"
    ],
    [
      "0.452663882",
      "0.58199642"
    ],
    [
      "0.064666269",
      "0.58199642"
    ],
    [
      "0.323331345",
      "0.58199642"
    ],
    [
      "0.58199642",
      "0.58199642"
    ],
    [
      "0.711328952",
      "0.582036263"
    ],
    [
      "0.935333731",
      "0.582036266"
    ],
    [
      "0.825257999",
      "0.649935004"
    ],
    [
      "0.582036263",
      "0.711328952"
    ],
    [
      "0.452693769",
      "0.711328955"
    ],
    [
      "0.323351271",
      "0.711328956"
    ],
    [
      "0.194008771",
      "0.711328958"
    ],
    [
      "0.064666269",
      "0.711328958"
    ],
    [
      "0.711447628",
      "0.711368747"
    ],
    [
      "0.935333731",
      "0.717833742"
    ],
    [
      "0.804733616",
      "0.804733619"
    ],
    [
      "0.64670751",
      "0.823331341"
    ],
    [
      "0.517365019",
      "0.823331342"
    ],
    [
      "0.388022522",
      "0.823331343"
    ],
    [
      "0.258680022",
      "0.823331344"
    ],
    [
      "0.12933752",
      "0.823331345"
    ],
    [
      "0.935333731",
      "0.847166279"
    ],
    [
      "0.064666269",
      "0.935333731"
    ],
    [
      "0.194008771",
      "0.935333731"
    ],
    [
      "0.323351273",
      "0.935333731"
    ],
    [
      "0.452693771",
      "0.935333731"
    ],
    [
      "0.582036266",
      "0.935333731"
    ],
    [
      "0.711378755",
      "0.935333731"
    ],
    [
      "0.840711293",
      "0.935333731"
    ]
  ],
  "radius": "0.064666268"
}
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": 61
}

起步布局答案

{
  "centers": [
    [
      "0.064666269",
      "0.064666269"
    ],
    [
      "0.193998807",
      "0.064666269"
    ],
    [
      "0.323331345",
      "0.064666269"
    ],
    [
      "0.452663882",
      "0.064666269"
    ],
    [
      "0.58199642",
      "0.064666269"
    ],
    [
      "0.711328958",
      "0.064666269"
    ],
    [
      "0.935333731",
      "0.064666269"
    ],
    [
      "0.823331345",
      "0.12933752"
    ],
    [
      "0.064666269",
      "0.193998807"
    ],
    [
      "0.193998807",
      "0.193998807"
    ],
    [
      "0.323331345",
      "0.193998807"
    ],
    [
      "0.452663882",
      "0.193998807"
    ],
    [
      "0.58199642",
      "0.193998807"
    ],
    [
      "0.711328958",
      "0.194008771"
    ],
    [
      "0.935333731",
      "0.194008771"
    ],
    [
      "0.823331344",
      "0.258680022"
    ],
    [
      "0.064666269",
      "0.323331345"
    ],
    [
      "0.193998807",
      "0.323331345"
    ],
    [
      "0.323331345",
      "0.323331345"
    ],
    [
      "0.452663882",
      "0.323331345"
    ],
    [
      "0.58199642",
      "0.323331345"
    ],
    [
      "0.711328956",
      "0.323351271"
    ],
    [
      "0.935333731",
      "0.323351273"
    ],
    [
      "0.823331343",
      "0.388022522"
    ],
    [
      "0.064666269",
      "0.452663882"
    ],
    [
      "0.193998807",
      "0.452663882"
    ],
    [
      "0.323331345",
      "0.452663882"
    ],
    [
      "0.452663882",
      "0.452663882"
    ],
    [
      "0.58199642",
      "0.452663882"
    ],
    [
      "0.711328955",
      "0.452693769"
    ],
    [
      "0.935333731",
      "0.452693771"
    ],
    [
      "0.823331342",
      "0.517365019"
    ],
    [
      "0.193998807",
      "0.58199642"
    ],
    [
      "0.452663882",
      "0.58199642"
    ],
    [
      "0.064666269",
      "0.58199642"
    ],
    [
      "0.323331345",
      "0.58199642"
    ],
    [
      "0.58199642",
      "0.58199642"
    ],
    [
      "0.711328952",
      "0.582036263"
    ],
    [
      "0.935333731",
      "0.582036266"
    ],
    [
      "0.825257999",
      "0.649935004"
    ],
    [
      "0.582036263",
      "0.711328952"
    ],
    [
      "0.452693769",
      "0.711328955"
    ],
    [
      "0.323351271",
      "0.711328956"
    ],
    [
      "0.194008771",
      "0.711328958"
    ],
    [
      "0.064666269",
      "0.711328958"
    ],
    [
      "0.711447628",
      "0.711368747"
    ],
    [
      "0.935333731",
      "0.717833742"
    ],
    [
      "0.804733616",
      "0.804733619"
    ],
    [
      "0.64670751",
      "0.823331341"
    ],
    [
      "0.517365019",
      "0.823331342"
    ],
    [
      "0.388022522",
      "0.823331343"
    ],
    [
      "0.258680022",
      "0.823331344"
    ],
    [
      "0.12933752",
      "0.823331345"
    ],
    [
      "0.935333731",
      "0.847166279"
    ],
    [
      "0.064666269",
      "0.935333731"
    ],
    [
      "0.194008771",
      "0.935333731"
    ],
    [
      "0.323351273",
      "0.935333731"
    ],
    [
      "0.452693771",
      "0.935333731"
    ],
    [
      "0.582036266",
      "0.935333731"
    ],
    [
      "0.711378755",
      "0.935333731"
    ],
    [
      "0.840711293",
      "0.935333731"
    ]
  ],
  "radius": "0.064666268"
}

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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