P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 22

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 = 22

目标最小化容器半径

已知最好（未证明）66.760286243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 ↗

133.520572486y0

12345678910111213141516171819202122

0x133.520572486

已验证构造r = 66.760286243

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

66.760286243

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": [
    [
      "69.954238826",
      "132.427598424"
    ],
    [
      "127.02973635",
      "85.645952167"
    ],
    [
      "37.595764489",
      "122.985209374"
    ],
    [
      "14.046039388",
      "100.669376542"
    ],
    [
      "124.962577039",
      "46.175492176"
    ],
    [
      "7.83967703",
      "52.490187171"
    ],
    [
      "38.983257972",
      "13.847825402"
    ],
    [
      "96.051352313",
      "15.821060809"
    ],
    [
      "101.82344648",
      "112.644156151"
    ],
    [
      "84.559659357",
      "120.643906931"
    ],
    [
      "113.841247438",
      "96.605376671"
    ],
    [
      "49.725630445",
      "90.249738646"
    ],
    [
      "31.400899795",
      "107.255744839"
    ],
    [
      "85.503657632",
      "53.7810404"
    ],
    [
      "107.996419228",
      "35.475967993"
    ],
    [
      "58.755413602",
      "116.885416402"
    ],
    [
      "55.570836136",
      "61.844922672"
    ],
    [
      "115.513194753",
      "67.608476232"
    ],
    [
      "28.803357507",
      "37.772073072"
    ],
    [
      "20.915119783",
      "75.965995437"
    ],
    [
      "82.687680703",
      "88.667574679"
    ],
    [
      "66.744562395",
      "22.232929861"
    ]
  ],
  "radius": "66.760286243"
}

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": 22
}

当前第一名的答案

{
  "centers": [
    [
      "69.954238826",
      "132.427598424"
    ],
    [
      "127.02973635",
      "85.645952167"
    ],
    [
      "37.595764489",
      "122.985209374"
    ],
    [
      "14.046039388",
      "100.669376542"
    ],
    [
      "124.962577039",
      "46.175492176"
    ],
    [
      "7.83967703",
      "52.490187171"
    ],
    [
      "38.983257972",
      "13.847825402"
    ],
    [
      "96.051352313",
      "15.821060809"
    ],
    [
      "101.82344648",
      "112.644156151"
    ],
    [
      "84.559659357",
      "120.643906931"
    ],
    [
      "113.841247438",
      "96.605376671"
    ],
    [
      "49.725630445",
      "90.249738646"
    ],
    [
      "31.400899795",
      "107.255744839"
    ],
    [
      "85.503657632",
      "53.7810404"
    ],
    [
      "107.996419228",
      "35.475967993"
    ],
    [
      "58.755413602",
      "116.885416402"
    ],
    [
      "55.570836136",
      "61.844922672"
    ],
    [
      "115.513194753",
      "67.608476232"
    ],
    [
      "28.803357507",
      "37.772073072"
    ],
    [
      "20.915119783",
      "75.965995437"
    ],
    [
      "82.687680703",
      "88.667574679"
    ],
    [
      "66.744562395",
      "22.232929861"
    ]
  ],
  "radius": "66.760286243"
}

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