P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 23

把半径分别为 1, 2, …, n 的 n 个圆互不重叠地放进一个圆里，使容器的半径尽可能小。

子题n = 23

目标最小化容器半径

已知最好（未证明）71.19946161值由 Packomania（Eckard Specht）ccin 坐标文件反推（表页只印四位小数，文件带十二位半径），该表未把这一项标为已证明最优

严格定义

容器圆心在 (radius, radius)、半径为 radius 的圆，radius 由你给出，它就是分数；单位取最小圆的半径
提交radius 与 centers，centers 按半径 1, 2, …, n 的顺序排列
约束第 i 个圆的半径恰好是 i；两两内部不重叠；每个圆完整落在容器圆内
目标让容器半径尽可能小

放大来摆，然后提交 ↗

142.39892322y0

1234567891011121314151617181920212223

0x142.39892322

已验证构造r = 71.19946161

帮助理解

哪里有优化空间

大圆定骨架、小圆填缝：n 每加一，新来的最大圆都可能颠覆上一轮的整个布局。

前沿在哪里

半径 1..n 装入最小圆正是 2005 年 Al Zimmermann 程序设计竞赛的赛题（n = 5..50），全部最好结果收录在 Packomania 的 ccin 表，但证明一个也没有。本站尚未录入这些值。

查看来源

当前第一名

71.19946161

容器半径

已追平已知最好

纪录保持者创始基准

解题方式人工

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

单位就是最小那个圆的半径：第 i 个圆的半径正好是 i。容器是你自己给出的圆，圆心在 (radius, radius)，所以坐标范围是 0 到 2·radius，radius 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "7.5"，最多九位小数。

提交 radius 与 centers，centers 按半径 1,2,…,n 的顺序排列。容器圆心在 (radius, radius)。每个数写成十进制字符串，例如 "7.5"。

当前第一名的答案

{
  "centers": [
    [
      "6.360296909",
      "97.994280619"
    ],
    [
      "11.477457967",
      "36.30774167"
    ],
    [
      "94.151211311",
      "135.033534209"
    ],
    [
      "138.326831894",
      "70.14597142"
    ],
    [
      "43.21144529",
      "11.221435567"
    ],
    [
      "115.35977741",
      "23.232428112"
    ],
    [
      "41.707998644",
      "128.206175605"
    ],
    [
      "8.076661552",
      "68.087539284"
    ],
    [
      "10.529824125",
      "84.909607926"
    ],
    [
      "13.460597982",
      "50.911590153"
    ],
    [
      "119.777871597",
      "106.753832104"
    ],
    [
      "102.225673093",
      "121.61722094"
    ],
    [
      "127.932513706",
      "84.181685283"
    ],
    [
      "59.385056237",
      "87.77231061"
    ],
    [
      "62.927054049",
      "15.612172071"
    ],
    [
      "93.504500009",
      "20.713132627"
    ],
    [
      "34.174521589",
      "68.395422606"
    ],
    [
      "34.618933395",
      "32.572428705"
    ],
    [
      "30.787206456",
      "104.239337907"
    ],
    [
      "94.352297765",
      "88.955333869"
    ],
    [
      "69.226674449",
      "121.360143989"
    ],
    [
      "116.159974921",
      "51.220991562"
    ],
    [
      "71.18442378",
      "52.704163729"
    ]
  ],
  "radius": "71.19946161"
}

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": 23
}

当前第一名的答案

{
  "centers": [
    [
      "6.360296909",
      "97.994280619"
    ],
    [
      "11.477457967",
      "36.30774167"
    ],
    [
      "94.151211311",
      "135.033534209"
    ],
    [
      "138.326831894",
      "70.14597142"
    ],
    [
      "43.21144529",
      "11.221435567"
    ],
    [
      "115.35977741",
      "23.232428112"
    ],
    [
      "41.707998644",
      "128.206175605"
    ],
    [
      "8.076661552",
      "68.087539284"
    ],
    [
      "10.529824125",
      "84.909607926"
    ],
    [
      "13.460597982",
      "50.911590153"
    ],
    [
      "119.777871597",
      "106.753832104"
    ],
    [
      "102.225673093",
      "121.61722094"
    ],
    [
      "127.932513706",
      "84.181685283"
    ],
    [
      "59.385056237",
      "87.77231061"
    ],
    [
      "62.927054049",
      "15.612172071"
    ],
    [
      "93.504500009",
      "20.713132627"
    ],
    [
      "34.174521589",
      "68.395422606"
    ],
    [
      "34.618933395",
      "32.572428705"
    ],
    [
      "30.787206456",
      "104.239337907"
    ],
    [
      "94.352297765",
      "88.955333869"
    ],
    [
      "69.226674449",
      "121.360143989"
    ],
    [
      "116.159974921",
      "51.220991562"
    ],
    [
      "71.18442378",
      "52.704163729"
    ]
  ],
  "radius": "71.19946161"
}

提交 radius 与 centers，centers 按半径 1,2,…,n 的顺序排列。容器圆心在 (radius, radius)。每个数写成十进制字符串，例如 "7.5"。 · 验证器 v1.0.0