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 = 131

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 = 131
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[0.0448872323675, 0.0448872323685]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 = 0.044887232

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 = 131 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": [
    [
      "0.044887232",
      "0.044887232"
    ],
    [
      "0.200381167",
      "0.044887232"
    ],
    [
      "0.355875101",
      "0.044887232"
    ],
    [
      "0.511369035",
      "0.044887232"
    ],
    [
      "0.666862969",
      "0.044887232"
    ],
    [
      "0.822356903",
      "0.044887232"
    ],
    [
      "0.945999208",
      "0.053984271"
    ],
    [
      "0.122634199",
      "0.089774465"
    ],
    [
      "0.278128134",
      "0.089774465"
    ],
    [
      "0.433622068",
      "0.089774465"
    ],
    [
      "0.589116002",
      "0.089774465"
    ],
    [
      "0.744609936",
      "0.089774465"
    ],
    [
      "0.877365801",
      "0.115834225"
    ],
    [
      "0.044887232",
      "0.134661697"
    ],
    [
      "0.200381167",
      "0.134661697"
    ],
    [
      "0.355875101",
      "0.134661697"
    ],
    [
      "0.511369035",
      "0.134661697"
    ],
    [
      "0.666862969",
      "0.134661697"
    ],
    [
      "0.799618833",
      "0.160721457"
    ],
    [
      "0.955112768",
      "0.160721457"
    ],
    [
      "0.122634199",
      "0.179548929"
    ],
    [
      "0.278128134",
      "0.179548929"
    ],
    [
      "0.433622068",
      "0.179548929"
    ],
    [
      "0.589116002",
      "0.179548929"
    ],
    [
      "0.721871866",
      "0.205608689"
    ],
    [
      "0.877365801",
      "0.205608689"
    ],
    [
      "0.044887232",
      "0.224436162"
    ],
    [
      "0.200381167",
      "0.224436162"
    ],
    [
      "0.355875101",
      "0.224436162"
    ],
    [
      "0.511369035",
      "0.224436162"
    ],
    [
      "0.644124899",
      "0.250495922"
    ],
    [
      "0.799618833",
      "0.250495922"
    ],
    [
      "0.955112768",
      "0.250495922"
    ],
    [
      "0.122634199",
      "0.269323394"
    ],
    [
      "0.278128134",
      "0.269323394"
    ],
    [
      "0.433622068",
      "0.269323394"
    ],
    [
      "0.566377932",
      "0.295383154"
    ],
    [
      "0.721871866",
      "0.295383154"
    ],
    [
      "0.877365801",
      "0.295383154"
    ],
    [
      "0.044887232",
      "0.314210627"
    ],
    [
      "0.200381167",
      "0.314210627"
    ],
    [
      "0.355875101",
      "0.314210627"
    ],
    [
      "0.488630965",
      "0.340270387"
    ],
    [
      "0.644124899",
      "0.340270387"
    ],
    [
      "0.799618833",
      "0.340270387"
    ],
    [
      "0.955112768",
      "0.340270387"
    ],
    [
      "0.122634199",
      "0.359097859"
    ],
    [
      "0.278128134",
      "0.359097859"
    ],
    [
      "0.410883998",
      "0.385157619"
    ],
    [
      "0.566377932",
      "0.385157619"
    ],
    [
      "0.721871866",
      "0.385157619"
    ],
    [
      "0.877365801",
      "0.385157619"
    ],
    [
      "0.044887232",
      "0.403985091"
    ],
    [
      "0.200381167",
      "0.403985091"
    ],
    [
      "0.333137031",
      "0.430044851"
    ],
    [
      "0.488630965",
      "0.430044851"
    ],
    [
      "0.644124899",
      "0.430044851"
    ],
    [
      "0.799618833",
      "0.430044851"
    ],
    [
      "0.955112768",
      "0.430044851"
    ],
    [
      "0.122634199",
      "0.448872324"
    ],
    [
      "0.255390064",
      "0.474932084"
    ],
    [
      "0.410883998",
      "0.474932084"
    ],
    [
      "0.566377932",
      "0.474932084"
    ],
    [
      "0.721871866",
      "0.474932084"
    ],
    [
      "0.877365801",
      "0.474932084"
    ],
    [
      "0.044887232",
      "0.493759556"
    ],
    [
      "0.177643097",
      "0.519819316"
    ],
    [
      "0.333137031",
      "0.519819316"
    ],
    [
      "0.488630965",
      "0.519819316"
    ],
    [
      "0.644124899",
      "0.519819316"
    ],
    [
      "0.799618833",
      "0.519819316"
    ],
    [
      "0.955112768",
      "0.519819316"
    ],
    [
      "0.255390064",
      "0.564706548"
    ],
    [
      "0.410883998",
      "0.564706548"
    ],
    [
      "0.566377932",
      "0.564706548"
    ],
    [
      "0.721871866",
      "0.564706548"
    ],
    [
      "0.877365801",
      "0.564706548"
    ],
    [
      "0.092152238",
      "0.570084347"
    ],
    [
      "0.172828453",
      "0.609464582"
    ],
    [
      "0.333137031",
      "0.609593781"
    ],
    [
      "0.488630965",
      "0.609593781"
    ],
    [
      "0.644124899",
      "0.609593781"
    ],
    [
      "0.799618833",
      "0.609593781"
    ],
    [
      "0.955112768",
      "0.609593781"
    ],
    [
      "0.044887232",
      "0.646409139"
    ],
    [
      "0.25057542",
      "0.654351814"
    ],
    [
      "0.410883998",
      "0.654481013"
    ],
    [
      "0.566377932",
      "0.654481013"
    ],
    [
      "0.721871866",
      "0.654481013"
    ],
    [
      "0.877365801",
      "0.654481013"
    ],
    [
      "0.125563447",
      "0.685789373"
    ],
    [
      "0.328322387",
      "0.699239047"
    ],
    [
      "0.488630965",
      "0.699368245"
    ],
    [
      "0.644124899",
      "0.699368245"
    ],
    [
      "0.799618833",
      "0.699368245"
    ],
    [
      "0.955112768",
      "0.699368245"
    ],
    [
      "0.203310414",
      "0.730676606"
    ],
    [
      "0.044887232",
      "0.736183604"
    ],
    [
      "0.406069354",
      "0.744126279"
    ],
    [
      "0.566377932",
      "0.744255478"
    ],
    [
      "0.721871866",
      "0.744255478"
    ],
    [
      "0.877365801",
      "0.744255478"
    ],
    [
      "0.125563447",
      "0.775563838"
    ],
    [
      "0.281057381",
      "0.775563838"
    ],
    [
      "0.483816321",
      "0.789013511"
    ],
    [
      "0.644124899",
      "0.78914271"
    ],
    [
      "0.799618833",
      "0.78914271"
    ],
    [
      "0.955112768",
      "0.78914271"
    ],
    [
      "0.203310414",
      "0.820451071"
    ],
    [
      "0.358804348",
      "0.820451071"
    ],
    [
      "0.044887232",
      "0.825958068"
    ],
    [
      "0.561563288",
      "0.833900744"
    ],
    [
      "0.729051903",
      "0.851057438"
    ],
    [
      "0.125563447",
      "0.865338303"
    ],
    [
      "0.281057381",
      "0.865338303"
    ],
    [
      "0.436551316",
      "0.865338303"
    ],
    [
      "0.909053914",
      "0.866201367"
    ],
    [
      "0.819883731",
      "0.876600062"
    ],
    [
      "0.639310255",
      "0.878787976"
    ],
    [
      "0.203310414",
      "0.910225535"
    ],
    [
      "0.358804348",
      "0.910225535"
    ],
    [
      "0.514298283",
      "0.910225535"
    ],
    [
      "0.044887232",
      "0.915732533"
    ],
    [
      "0.955112768",
      "0.943260023"
    ],
    [
      "0.125563447",
      "0.955112768"
    ],
    [
      "0.281057381",
      "0.955112768"
    ],
    [
      "0.436551316",
      "0.955112768"
    ],
    [
      "0.59204525",
      "0.955112768"
    ],
    [
      "0.68657526",
      "0.955112768"
    ],
    [
      "0.776349725",
      "0.955112768"
    ],
    [
      "0.86612419",
      "0.955112768"
    ]
  ],
  "radius": "0.044887232"
}
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": 131
}

起步布局答案

{
  "centers": [
    [
      "0.044887232",
      "0.044887232"
    ],
    [
      "0.200381167",
      "0.044887232"
    ],
    [
      "0.355875101",
      "0.044887232"
    ],
    [
      "0.511369035",
      "0.044887232"
    ],
    [
      "0.666862969",
      "0.044887232"
    ],
    [
      "0.822356903",
      "0.044887232"
    ],
    [
      "0.945999208",
      "0.053984271"
    ],
    [
      "0.122634199",
      "0.089774465"
    ],
    [
      "0.278128134",
      "0.089774465"
    ],
    [
      "0.433622068",
      "0.089774465"
    ],
    [
      "0.589116002",
      "0.089774465"
    ],
    [
      "0.744609936",
      "0.089774465"
    ],
    [
      "0.877365801",
      "0.115834225"
    ],
    [
      "0.044887232",
      "0.134661697"
    ],
    [
      "0.200381167",
      "0.134661697"
    ],
    [
      "0.355875101",
      "0.134661697"
    ],
    [
      "0.511369035",
      "0.134661697"
    ],
    [
      "0.666862969",
      "0.134661697"
    ],
    [
      "0.799618833",
      "0.160721457"
    ],
    [
      "0.955112768",
      "0.160721457"
    ],
    [
      "0.122634199",
      "0.179548929"
    ],
    [
      "0.278128134",
      "0.179548929"
    ],
    [
      "0.433622068",
      "0.179548929"
    ],
    [
      "0.589116002",
      "0.179548929"
    ],
    [
      "0.721871866",
      "0.205608689"
    ],
    [
      "0.877365801",
      "0.205608689"
    ],
    [
      "0.044887232",
      "0.224436162"
    ],
    [
      "0.200381167",
      "0.224436162"
    ],
    [
      "0.355875101",
      "0.224436162"
    ],
    [
      "0.511369035",
      "0.224436162"
    ],
    [
      "0.644124899",
      "0.250495922"
    ],
    [
      "0.799618833",
      "0.250495922"
    ],
    [
      "0.955112768",
      "0.250495922"
    ],
    [
      "0.122634199",
      "0.269323394"
    ],
    [
      "0.278128134",
      "0.269323394"
    ],
    [
      "0.433622068",
      "0.269323394"
    ],
    [
      "0.566377932",
      "0.295383154"
    ],
    [
      "0.721871866",
      "0.295383154"
    ],
    [
      "0.877365801",
      "0.295383154"
    ],
    [
      "0.044887232",
      "0.314210627"
    ],
    [
      "0.200381167",
      "0.314210627"
    ],
    [
      "0.355875101",
      "0.314210627"
    ],
    [
      "0.488630965",
      "0.340270387"
    ],
    [
      "0.644124899",
      "0.340270387"
    ],
    [
      "0.799618833",
      "0.340270387"
    ],
    [
      "0.955112768",
      "0.340270387"
    ],
    [
      "0.122634199",
      "0.359097859"
    ],
    [
      "0.278128134",
      "0.359097859"
    ],
    [
      "0.410883998",
      "0.385157619"
    ],
    [
      "0.566377932",
      "0.385157619"
    ],
    [
      "0.721871866",
      "0.385157619"
    ],
    [
      "0.877365801",
      "0.385157619"
    ],
    [
      "0.044887232",
      "0.403985091"
    ],
    [
      "0.200381167",
      "0.403985091"
    ],
    [
      "0.333137031",
      "0.430044851"
    ],
    [
      "0.488630965",
      "0.430044851"
    ],
    [
      "0.644124899",
      "0.430044851"
    ],
    [
      "0.799618833",
      "0.430044851"
    ],
    [
      "0.955112768",
      "0.430044851"
    ],
    [
      "0.122634199",
      "0.448872324"
    ],
    [
      "0.255390064",
      "0.474932084"
    ],
    [
      "0.410883998",
      "0.474932084"
    ],
    [
      "0.566377932",
      "0.474932084"
    ],
    [
      "0.721871866",
      "0.474932084"
    ],
    [
      "0.877365801",
      "0.474932084"
    ],
    [
      "0.044887232",
      "0.493759556"
    ],
    [
      "0.177643097",
      "0.519819316"
    ],
    [
      "0.333137031",
      "0.519819316"
    ],
    [
      "0.488630965",
      "0.519819316"
    ],
    [
      "0.644124899",
      "0.519819316"
    ],
    [
      "0.799618833",
      "0.519819316"
    ],
    [
      "0.955112768",
      "0.519819316"
    ],
    [
      "0.255390064",
      "0.564706548"
    ],
    [
      "0.410883998",
      "0.564706548"
    ],
    [
      "0.566377932",
      "0.564706548"
    ],
    [
      "0.721871866",
      "0.564706548"
    ],
    [
      "0.877365801",
      "0.564706548"
    ],
    [
      "0.092152238",
      "0.570084347"
    ],
    [
      "0.172828453",
      "0.609464582"
    ],
    [
      "0.333137031",
      "0.609593781"
    ],
    [
      "0.488630965",
      "0.609593781"
    ],
    [
      "0.644124899",
      "0.609593781"
    ],
    [
      "0.799618833",
      "0.609593781"
    ],
    [
      "0.955112768",
      "0.609593781"
    ],
    [
      "0.044887232",
      "0.646409139"
    ],
    [
      "0.25057542",
      "0.654351814"
    ],
    [
      "0.410883998",
      "0.654481013"
    ],
    [
      "0.566377932",
      "0.654481013"
    ],
    [
      "0.721871866",
      "0.654481013"
    ],
    [
      "0.877365801",
      "0.654481013"
    ],
    [
      "0.125563447",
      "0.685789373"
    ],
    [
      "0.328322387",
      "0.699239047"
    ],
    [
      "0.488630965",
      "0.699368245"
    ],
    [
      "0.644124899",
      "0.699368245"
    ],
    [
      "0.799618833",
      "0.699368245"
    ],
    [
      "0.955112768",
      "0.699368245"
    ],
    [
      "0.203310414",
      "0.730676606"
    ],
    [
      "0.044887232",
      "0.736183604"
    ],
    [
      "0.406069354",
      "0.744126279"
    ],
    [
      "0.566377932",
      "0.744255478"
    ],
    [
      "0.721871866",
      "0.744255478"
    ],
    [
      "0.877365801",
      "0.744255478"
    ],
    [
      "0.125563447",
      "0.775563838"
    ],
    [
      "0.281057381",
      "0.775563838"
    ],
    [
      "0.483816321",
      "0.789013511"
    ],
    [
      "0.644124899",
      "0.78914271"
    ],
    [
      "0.799618833",
      "0.78914271"
    ],
    [
      "0.955112768",
      "0.78914271"
    ],
    [
      "0.203310414",
      "0.820451071"
    ],
    [
      "0.358804348",
      "0.820451071"
    ],
    [
      "0.044887232",
      "0.825958068"
    ],
    [
      "0.561563288",
      "0.833900744"
    ],
    [
      "0.729051903",
      "0.851057438"
    ],
    [
      "0.125563447",
      "0.865338303"
    ],
    [
      "0.281057381",
      "0.865338303"
    ],
    [
      "0.436551316",
      "0.865338303"
    ],
    [
      "0.909053914",
      "0.866201367"
    ],
    [
      "0.819883731",
      "0.876600062"
    ],
    [
      "0.639310255",
      "0.878787976"
    ],
    [
      "0.203310414",
      "0.910225535"
    ],
    [
      "0.358804348",
      "0.910225535"
    ],
    [
      "0.514298283",
      "0.910225535"
    ],
    [
      "0.044887232",
      "0.915732533"
    ],
    [
      "0.955112768",
      "0.943260023"
    ],
    [
      "0.125563447",
      "0.955112768"
    ],
    [
      "0.281057381",
      "0.955112768"
    ],
    [
      "0.436551316",
      "0.955112768"
    ],
    [
      "0.59204525",
      "0.955112768"
    ],
    [
      "0.68657526",
      "0.955112768"
    ],
    [
      "0.776349725",
      "0.955112768"
    ],
    [
      "0.86612419",
      "0.955112768"
    ]
  ],
  "radius": "0.044887232"
}

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