极值竞技场
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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 = 25

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 = 25
目标最小化 容器半径
已知最好（未证明）80.285864442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 ↗
160.571728884y0
0x160.571728884
已验证构造r = 80.285864442

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

80.285864442

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": [
    [
      "123.422612405",
      "32.557565682"
    ],
    [
      "104.985967042",
      "152.995604199"
    ],
    [
      "78.998473417",
      "40.396583515"
    ],
    [
      "24.27663079",
      "29.33263025"
    ],
    [
      "64.021710871",
      "153.736971854"
    ],
    [
      "22.575085344",
      "126.954287121"
    ],
    [
      "129.111074181",
      "25.737654805"
    ],
    [
      "75.251699019",
      "8.175508899"
    ],
    [
      "146.863765685",
      "105.762472051"
    ],
    [
      "136.899071098",
      "121.939766689"
    ],
    [
      "149.34254394",
      "85.916675055"
    ],
    [
      "20.628386528",
      "47.06573743"
    ],
    [
      "13.630649296",
      "71.095116513"
    ],
    [
      "55.771334071",
      "96.696495403"
    ],
    [
      "82.950210449",
      "145.517339638"
    ],
    [
      "42.599109751",
      "69.742970467"
    ],
    [
      "112.983049355",
      "134.470684612"
    ],
    [
      "84.222574346",
      "112.541877698"
    ],
    [
      "22.874477559",
      "101.730906679"
    ],
    [
      "119.08737856",
      "97.427466455"
    ],
    [
      "49.918152437",
      "131.203599959"
    ],
    [
      "99.831713512",
      "25.375011437"
    ],
    [
      "132.37797352",
      "56.451384448"
    ],
    [
      "52.923652814",
      "31.098387268"
    ],
    [
      "83.767008688",
      "69.544291021"
    ]
  ],
  "radius": "80.285864442"
}
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": 25
}

当前第一名的答案

{
  "centers": [
    [
      "123.422612405",
      "32.557565682"
    ],
    [
      "104.985967042",
      "152.995604199"
    ],
    [
      "78.998473417",
      "40.396583515"
    ],
    [
      "24.27663079",
      "29.33263025"
    ],
    [
      "64.021710871",
      "153.736971854"
    ],
    [
      "22.575085344",
      "126.954287121"
    ],
    [
      "129.111074181",
      "25.737654805"
    ],
    [
      "75.251699019",
      "8.175508899"
    ],
    [
      "146.863765685",
      "105.762472051"
    ],
    [
      "136.899071098",
      "121.939766689"
    ],
    [
      "149.34254394",
      "85.916675055"
    ],
    [
      "20.628386528",
      "47.06573743"
    ],
    [
      "13.630649296",
      "71.095116513"
    ],
    [
      "55.771334071",
      "96.696495403"
    ],
    [
      "82.950210449",
      "145.517339638"
    ],
    [
      "42.599109751",
      "69.742970467"
    ],
    [
      "112.983049355",
      "134.470684612"
    ],
    [
      "84.222574346",
      "112.541877698"
    ],
    [
      "22.874477559",
      "101.730906679"
    ],
    [
      "119.08737856",
      "97.427466455"
    ],
    [
      "49.918152437",
      "131.203599959"
    ],
    [
      "99.831713512",
      "25.375011437"
    ],
    [
      "132.37797352",
      "56.451384448"
    ],
    [
      "52.923652814",
      "31.098387268"
    ],
    [
      "83.767008688",
      "69.544291021"
    ]
  ],
  "radius": "80.285864442"
}

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