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P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 26

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 = 26
目标最小化 容器半径
已知最好（未证明）84.978191068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 ↗
169.956382136y0
0x169.956382136
已验证构造r = 84.978191068

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

84.978191068

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": [
    [
      "143.724901888",
      "27.835003987"
    ],
    [
      "70.520199019",
      "126.884742687"
    ],
    [
      "56.612377668",
      "53.090979427"
    ],
    [
      "4.924801008",
      "83.953592431"
    ],
    [
      "163.170780154",
      "68.762236204"
    ],
    [
      "14.839931815",
      "121.270269719"
    ],
    [
      "135.54009609",
      "25.614127505"
    ],
    [
      "125.617773123",
      "150.354534423"
    ],
    [
      "61.822724885",
      "157.341924414"
    ],
    [
      "74.171477708",
      "114.846423319"
    ],
    [
      "54.698837109",
      "17.48053572"
    ],
    [
      "77.135800411",
      "12.422603549"
    ],
    [
      "125.786507486",
      "86.27414257"
    ],
    [
      "15.714515163",
      "69.498840404"
    ],
    [
      "154.842015632",
      "88.228177082"
    ],
    [
      "17.580521433",
      "99.441103108"
    ],
    [
      "73.021330439",
      "41.129242796"
    ],
    [
      "98.839849832",
      "101.599860868"
    ],
    [
      "100.293803159",
      "64.628439204"
    ],
    [
      "36.037847404",
      "42.234676046"
    ],
    [
      "107.542520317",
      "25.245683701"
    ],
    [
      "139.964918426",
      "54.274569709"
    ],
    [
      "135.446443106",
      "120.953900987"
    ],
    [
      "43.638421168",
      "129.804112257"
    ],
    [
      "93.191598359",
      "144.227284615"
    ],
    [
      "58.856178078",
      "82.176180591"
    ]
  ],
  "radius": "84.978191068"
}
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": 26
}

当前第一名的答案

{
  "centers": [
    [
      "143.724901888",
      "27.835003987"
    ],
    [
      "70.520199019",
      "126.884742687"
    ],
    [
      "56.612377668",
      "53.090979427"
    ],
    [
      "4.924801008",
      "83.953592431"
    ],
    [
      "163.170780154",
      "68.762236204"
    ],
    [
      "14.839931815",
      "121.270269719"
    ],
    [
      "135.54009609",
      "25.614127505"
    ],
    [
      "125.617773123",
      "150.354534423"
    ],
    [
      "61.822724885",
      "157.341924414"
    ],
    [
      "74.171477708",
      "114.846423319"
    ],
    [
      "54.698837109",
      "17.48053572"
    ],
    [
      "77.135800411",
      "12.422603549"
    ],
    [
      "125.786507486",
      "86.27414257"
    ],
    [
      "15.714515163",
      "69.498840404"
    ],
    [
      "154.842015632",
      "88.228177082"
    ],
    [
      "17.580521433",
      "99.441103108"
    ],
    [
      "73.021330439",
      "41.129242796"
    ],
    [
      "98.839849832",
      "101.599860868"
    ],
    [
      "100.293803159",
      "64.628439204"
    ],
    [
      "36.037847404",
      "42.234676046"
    ],
    [
      "107.542520317",
      "25.245683701"
    ],
    [
      "139.964918426",
      "54.274569709"
    ],
    [
      "135.446443106",
      "120.953900987"
    ],
    [
      "43.638421168",
      "129.804112257"
    ],
    [
      "93.191598359",
      "144.227284615"
    ],
    [
      "58.856178078",
      "82.176180591"
    ]
  ],
  "radius": "84.978191068"
}

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