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 = 28

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 = 28
目标最小化 容器半径
已知最好（未证明）94.525877104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 ↗
189.051754208y0
0x189.051754208
已验证构造r = 94.525877104

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

94.525877104

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": [
    [
      "129.467740948",
      "91.312391582"
    ],
    [
      "78.104931616",
      "113.644615153"
    ],
    [
      "184.415474039",
      "111.75005881"
    ],
    [
      "60.486320245",
      "10.644370566"
    ],
    [
      "74.352027104",
      "181.743345939"
    ],
    [
      "6.023865624",
      "96.393931876"
    ],
    [
      "19.178260829",
      "49.992596298"
    ],
    [
      "133.309922535",
      "171.870758486"
    ],
    [
      "74.518892718",
      "11.374765019"
    ],
    [
      "178.250372935",
      "100.301894581"
    ],
    [
      "173.760486336",
      "120.83431356"
    ],
    [
      "163.017091603",
      "48.488132501"
    ],
    [
      "35.549444004",
      "150.805000873"
    ],
    [
      "143.731425982",
      "30.785718016"
    ],
    [
      "59.573311424",
      "165.958945647"
    ],
    [
      "170.691004864",
      "75.416018703"
    ],
    [
      "150.907708675",
      "101.828538309"
    ],
    [
      "91.830656717",
      "67.594102583"
    ],
    [
      "21.229263109",
      "76.31147376"
    ],
    [
      "65.080177313",
      "131.394882979"
    ],
    [
      "60.386136819",
      "90.664476843"
    ],
    [
      "25.59554795",
      "117.078317601"
    ],
    [
      "132.805535211",
      "66.159065142"
    ],
    [
      "146.170821842",
      "142.553983711"
    ],
    [
      "104.955671849",
      "26.541710348"
    ],
    [
      "100.45105769",
      "162.795109323"
    ],
    [
      "106.922118161",
      "109.988061959"
    ],
    [
      "53.508686429",
      "42.149523723"
    ]
  ],
  "radius": "94.525877104"
}
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": 28
}

当前第一名的答案

{
  "centers": [
    [
      "129.467740948",
      "91.312391582"
    ],
    [
      "78.104931616",
      "113.644615153"
    ],
    [
      "184.415474039",
      "111.75005881"
    ],
    [
      "60.486320245",
      "10.644370566"
    ],
    [
      "74.352027104",
      "181.743345939"
    ],
    [
      "6.023865624",
      "96.393931876"
    ],
    [
      "19.178260829",
      "49.992596298"
    ],
    [
      "133.309922535",
      "171.870758486"
    ],
    [
      "74.518892718",
      "11.374765019"
    ],
    [
      "178.250372935",
      "100.301894581"
    ],
    [
      "173.760486336",
      "120.83431356"
    ],
    [
      "163.017091603",
      "48.488132501"
    ],
    [
      "35.549444004",
      "150.805000873"
    ],
    [
      "143.731425982",
      "30.785718016"
    ],
    [
      "59.573311424",
      "165.958945647"
    ],
    [
      "170.691004864",
      "75.416018703"
    ],
    [
      "150.907708675",
      "101.828538309"
    ],
    [
      "91.830656717",
      "67.594102583"
    ],
    [
      "21.229263109",
      "76.31147376"
    ],
    [
      "65.080177313",
      "131.394882979"
    ],
    [
      "60.386136819",
      "90.664476843"
    ],
    [
      "25.59554795",
      "117.078317601"
    ],
    [
      "132.805535211",
      "66.159065142"
    ],
    [
      "146.170821842",
      "142.553983711"
    ],
    [
      "104.955671849",
      "26.541710348"
    ],
    [
      "100.45105769",
      "162.795109323"
    ],
    [
      "106.922118161",
      "109.988061959"
    ],
    [
      "53.508686429",
      "42.149523723"
    ]
  ],
  "radius": "94.525877104"
}

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