P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 21

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 = 21

目标最小化容器半径

已知最好（未证明）62.558877096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 ↗

125.117754192y0

123456789101112131415161718192021

0x125.117754192

已验证构造r = 62.558877096

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

62.558877096

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": [
    [
      "11.451036857",
      "93.838656129"
    ],
    [
      "72.360586834",
      "120.947310929"
    ],
    [
      "114.561355032",
      "91.502250506"
    ],
    [
      "34.869867571",
      "114.09076702"
    ],
    [
      "5.046228055",
      "62.482242651"
    ],
    [
      "117.487593728",
      "49.077680285"
    ],
    [
      "88.938202268",
      "13.661842139"
    ],
    [
      "51.314927121",
      "49.872265188"
    ],
    [
      "33.030026424",
      "17.875543584"
    ],
    [
      "80.686356943",
      "111.89274007"
    ],
    [
      "97.998769651",
      "100.006598296"
    ],
    [
      "28.085901237",
      "99.490598094"
    ],
    [
      "15.39559282",
      "77.513920947"
    ],
    [
      "76.570127321",
      "87.129747441"
    ],
    [
      "74.216372245",
      "57.155431476"
    ],
    [
      "54.910610402",
      "108.485264424"
    ],
    [
      "98.938071736",
      "35.479326198"
    ],
    [
      "46.768132052",
      "75.474659364"
    ],
    [
      "105.309520966",
      "70.911015895"
    ],
    [
      "23.649292469",
      "45.316422406"
    ],
    [
      "62.573339128",
      "23.090203165"
    ]
  ],
  "radius": "62.558877096"
}

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": 21
}

当前第一名的答案

{
  "centers": [
    [
      "11.451036857",
      "93.838656129"
    ],
    [
      "72.360586834",
      "120.947310929"
    ],
    [
      "114.561355032",
      "91.502250506"
    ],
    [
      "34.869867571",
      "114.09076702"
    ],
    [
      "5.046228055",
      "62.482242651"
    ],
    [
      "117.487593728",
      "49.077680285"
    ],
    [
      "88.938202268",
      "13.661842139"
    ],
    [
      "51.314927121",
      "49.872265188"
    ],
    [
      "33.030026424",
      "17.875543584"
    ],
    [
      "80.686356943",
      "111.89274007"
    ],
    [
      "97.998769651",
      "100.006598296"
    ],
    [
      "28.085901237",
      "99.490598094"
    ],
    [
      "15.39559282",
      "77.513920947"
    ],
    [
      "76.570127321",
      "87.129747441"
    ],
    [
      "74.216372245",
      "57.155431476"
    ],
    [
      "54.910610402",
      "108.485264424"
    ],
    [
      "98.938071736",
      "35.479326198"
    ],
    [
      "46.768132052",
      "75.474659364"
    ],
    [
      "105.309520966",
      "70.911015895"
    ],
    [
      "23.649292469",
      "45.316422406"
    ],
    [
      "62.573339128",
      "23.090203165"
    ]
  ],
  "radius": "62.558877096"
}

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