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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 = 96

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 = 96
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[0.0522754254695, 0.0522754254705]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 = 0.052275425

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 = 96 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": [
    [
      "0.052275425",
      "0.052275425"
    ],
    [
      "0.233362811",
      "0.052275425"
    ],
    [
      "0.414450197",
      "0.052275425"
    ],
    [
      "0.585549803",
      "0.052275425"
    ],
    [
      "0.766637189",
      "0.052275425"
    ],
    [
      "0.947724575",
      "0.052275425"
    ],
    [
      "0.142819118",
      "0.104550851"
    ],
    [
      "0.323906504",
      "0.104550851"
    ],
    [
      "0.676093496",
      "0.104550851"
    ],
    [
      "0.857180882",
      "0.104550851"
    ],
    [
      "0.5",
      "0.112376271"
    ],
    [
      "0.052275425",
      "0.156826276"
    ],
    [
      "0.233362811",
      "0.156826276"
    ],
    [
      "0.766637189",
      "0.156826276"
    ],
    [
      "0.947724575",
      "0.156826276"
    ],
    [
      "0.409456307",
      "0.164651697"
    ],
    [
      "0.590543693",
      "0.164651697"
    ],
    [
      "0.142819118",
      "0.209101702"
    ],
    [
      "0.857180882",
      "0.209101702"
    ],
    [
      "0.318912614",
      "0.216927122"
    ],
    [
      "0.5",
      "0.216927122"
    ],
    [
      "0.681087386",
      "0.216927122"
    ],
    [
      "0.052275425",
      "0.261377127"
    ],
    [
      "0.947724575",
      "0.261377127"
    ],
    [
      "0.228368921",
      "0.269202548"
    ],
    [
      "0.409456307",
      "0.269202548"
    ],
    [
      "0.590543693",
      "0.269202548"
    ],
    [
      "0.771631079",
      "0.269202548"
    ],
    [
      "0.137825228",
      "0.321477973"
    ],
    [
      "0.318912614",
      "0.321477973"
    ],
    [
      "0.5",
      "0.321477973"
    ],
    [
      "0.681087386",
      "0.321477973"
    ],
    [
      "0.862174772",
      "0.321477973"
    ],
    [
      "0.228368921",
      "0.373753399"
    ],
    [
      "0.409456307",
      "0.373753399"
    ],
    [
      "0.590543693",
      "0.373753399"
    ],
    [
      "0.771631079",
      "0.373753399"
    ],
    [
      "0.052275425",
      "0.381578819"
    ],
    [
      "0.947724575",
      "0.381578819"
    ],
    [
      "0.5",
      "0.426028824"
    ],
    [
      "0.318912614",
      "0.426028824"
    ],
    [
      "0.681087386",
      "0.426028824"
    ],
    [
      "0.142819118",
      "0.433854244"
    ],
    [
      "0.857180882",
      "0.433854244"
    ],
    [
      "0.409456307",
      "0.478304249"
    ],
    [
      "0.590543693",
      "0.478304249"
    ],
    [
      "0.052275425",
      "0.48612967"
    ],
    [
      "0.233362811",
      "0.48612967"
    ],
    [
      "0.766637189",
      "0.48612967"
    ],
    [
      "0.947724575",
      "0.48612967"
    ],
    [
      "0.5",
      "0.530579675"
    ],
    [
      "0.142819118",
      "0.538405095"
    ],
    [
      "0.323906504",
      "0.538405095"
    ],
    [
      "0.676093496",
      "0.538405095"
    ],
    [
      "0.857180882",
      "0.538405095"
    ],
    [
      "0.052275425",
      "0.590680521"
    ],
    [
      "0.233362811",
      "0.590680521"
    ],
    [
      "0.414450197",
      "0.590680521"
    ],
    [
      "0.585549803",
      "0.590680521"
    ],
    [
      "0.766637189",
      "0.590680521"
    ],
    [
      "0.947724575",
      "0.590680521"
    ],
    [
      "0.142819118",
      "0.642955946"
    ],
    [
      "0.323906504",
      "0.642955946"
    ],
    [
      "0.676093496",
      "0.642955946"
    ],
    [
      "0.857180882",
      "0.642955946"
    ],
    [
      "0.5",
      "0.650781366"
    ],
    [
      "0.052275425",
      "0.695231372"
    ],
    [
      "0.947724575",
      "0.695231372"
    ],
    [
      "0.233362811",
      "0.695231372"
    ],
    [
      "0.766637189",
      "0.695231372"
    ],
    [
      "0.409456307",
      "0.703056792"
    ],
    [
      "0.590543693",
      "0.703056792"
    ],
    [
      "0.143524201",
      "0.748747335"
    ],
    [
      "0.856475799",
      "0.748747335"
    ],
    [
      "0.318912614",
      "0.755332217"
    ],
    [
      "0.5",
      "0.755332217"
    ],
    [
      "0.681087386",
      "0.755332217"
    ],
    [
      "0.052275425",
      "0.799782222"
    ],
    [
      "0.947724575",
      "0.799782222"
    ],
    [
      "0.24347302",
      "0.827718321"
    ],
    [
      "0.75652698",
      "0.827718321"
    ],
    [
      "0.447724575",
      "0.84587591"
    ],
    [
      "0.552275425",
      "0.84587591"
    ],
    [
      "0.142097249",
      "0.853288488"
    ],
    [
      "0.857902751",
      "0.853288488"
    ],
    [
      "0.343786718",
      "0.857180882"
    ],
    [
      "0.656213282",
      "0.857180882"
    ],
    [
      "0.5",
      "0.936419603"
    ],
    [
      "0.061229917",
      "0.938770083"
    ],
    [
      "0.938770083",
      "0.938770083"
    ],
    [
      "0.186960441",
      "0.947724575"
    ],
    [
      "0.291511292",
      "0.947724575"
    ],
    [
      "0.396062143",
      "0.947724575"
    ],
    [
      "0.603937857",
      "0.947724575"
    ],
    [
      "0.708488708",
      "0.947724575"
    ],
    [
      "0.813039559",
      "0.947724575"
    ]
  ],
  "radius": "0.052275425"
}
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": 96
}

起步布局答案

{
  "centers": [
    [
      "0.052275425",
      "0.052275425"
    ],
    [
      "0.233362811",
      "0.052275425"
    ],
    [
      "0.414450197",
      "0.052275425"
    ],
    [
      "0.585549803",
      "0.052275425"
    ],
    [
      "0.766637189",
      "0.052275425"
    ],
    [
      "0.947724575",
      "0.052275425"
    ],
    [
      "0.142819118",
      "0.104550851"
    ],
    [
      "0.323906504",
      "0.104550851"
    ],
    [
      "0.676093496",
      "0.104550851"
    ],
    [
      "0.857180882",
      "0.104550851"
    ],
    [
      "0.5",
      "0.112376271"
    ],
    [
      "0.052275425",
      "0.156826276"
    ],
    [
      "0.233362811",
      "0.156826276"
    ],
    [
      "0.766637189",
      "0.156826276"
    ],
    [
      "0.947724575",
      "0.156826276"
    ],
    [
      "0.409456307",
      "0.164651697"
    ],
    [
      "0.590543693",
      "0.164651697"
    ],
    [
      "0.142819118",
      "0.209101702"
    ],
    [
      "0.857180882",
      "0.209101702"
    ],
    [
      "0.318912614",
      "0.216927122"
    ],
    [
      "0.5",
      "0.216927122"
    ],
    [
      "0.681087386",
      "0.216927122"
    ],
    [
      "0.052275425",
      "0.261377127"
    ],
    [
      "0.947724575",
      "0.261377127"
    ],
    [
      "0.228368921",
      "0.269202548"
    ],
    [
      "0.409456307",
      "0.269202548"
    ],
    [
      "0.590543693",
      "0.269202548"
    ],
    [
      "0.771631079",
      "0.269202548"
    ],
    [
      "0.137825228",
      "0.321477973"
    ],
    [
      "0.318912614",
      "0.321477973"
    ],
    [
      "0.5",
      "0.321477973"
    ],
    [
      "0.681087386",
      "0.321477973"
    ],
    [
      "0.862174772",
      "0.321477973"
    ],
    [
      "0.228368921",
      "0.373753399"
    ],
    [
      "0.409456307",
      "0.373753399"
    ],
    [
      "0.590543693",
      "0.373753399"
    ],
    [
      "0.771631079",
      "0.373753399"
    ],
    [
      "0.052275425",
      "0.381578819"
    ],
    [
      "0.947724575",
      "0.381578819"
    ],
    [
      "0.5",
      "0.426028824"
    ],
    [
      "0.318912614",
      "0.426028824"
    ],
    [
      "0.681087386",
      "0.426028824"
    ],
    [
      "0.142819118",
      "0.433854244"
    ],
    [
      "0.857180882",
      "0.433854244"
    ],
    [
      "0.409456307",
      "0.478304249"
    ],
    [
      "0.590543693",
      "0.478304249"
    ],
    [
      "0.052275425",
      "0.48612967"
    ],
    [
      "0.233362811",
      "0.48612967"
    ],
    [
      "0.766637189",
      "0.48612967"
    ],
    [
      "0.947724575",
      "0.48612967"
    ],
    [
      "0.5",
      "0.530579675"
    ],
    [
      "0.142819118",
      "0.538405095"
    ],
    [
      "0.323906504",
      "0.538405095"
    ],
    [
      "0.676093496",
      "0.538405095"
    ],
    [
      "0.857180882",
      "0.538405095"
    ],
    [
      "0.052275425",
      "0.590680521"
    ],
    [
      "0.233362811",
      "0.590680521"
    ],
    [
      "0.414450197",
      "0.590680521"
    ],
    [
      "0.585549803",
      "0.590680521"
    ],
    [
      "0.766637189",
      "0.590680521"
    ],
    [
      "0.947724575",
      "0.590680521"
    ],
    [
      "0.142819118",
      "0.642955946"
    ],
    [
      "0.323906504",
      "0.642955946"
    ],
    [
      "0.676093496",
      "0.642955946"
    ],
    [
      "0.857180882",
      "0.642955946"
    ],
    [
      "0.5",
      "0.650781366"
    ],
    [
      "0.052275425",
      "0.695231372"
    ],
    [
      "0.947724575",
      "0.695231372"
    ],
    [
      "0.233362811",
      "0.695231372"
    ],
    [
      "0.766637189",
      "0.695231372"
    ],
    [
      "0.409456307",
      "0.703056792"
    ],
    [
      "0.590543693",
      "0.703056792"
    ],
    [
      "0.143524201",
      "0.748747335"
    ],
    [
      "0.856475799",
      "0.748747335"
    ],
    [
      "0.318912614",
      "0.755332217"
    ],
    [
      "0.5",
      "0.755332217"
    ],
    [
      "0.681087386",
      "0.755332217"
    ],
    [
      "0.052275425",
      "0.799782222"
    ],
    [
      "0.947724575",
      "0.799782222"
    ],
    [
      "0.24347302",
      "0.827718321"
    ],
    [
      "0.75652698",
      "0.827718321"
    ],
    [
      "0.447724575",
      "0.84587591"
    ],
    [
      "0.552275425",
      "0.84587591"
    ],
    [
      "0.142097249",
      "0.853288488"
    ],
    [
      "0.857902751",
      "0.853288488"
    ],
    [
      "0.343786718",
      "0.857180882"
    ],
    [
      "0.656213282",
      "0.857180882"
    ],
    [
      "0.5",
      "0.936419603"
    ],
    [
      "0.061229917",
      "0.938770083"
    ],
    [
      "0.938770083",
      "0.938770083"
    ],
    [
      "0.186960441",
      "0.947724575"
    ],
    [
      "0.291511292",
      "0.947724575"
    ],
    [
      "0.396062143",
      "0.947724575"
    ],
    [
      "0.603937857",
      "0.947724575"
    ],
    [
      "0.708488708",
      "0.947724575"
    ],
    [
      "0.813039559",
      "0.947724575"
    ]
  ],
  "radius": "0.052275425"
}

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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