极值竞技场
中/EN
P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 21

把半径分别为 1, 2, …, n 的 n 个圆互不重叠地放进一个正方形，使正方形边长尽可能小。

子题n = 21
目标最小化 正方形边长
已知最好（未证明）110.560297198值取自 Packomania（Eckard Specht）的 csqn 汇总表，该表未把这一项标为已证明最优

严格定义

  • 容器边长 side 的正方形，左下角是原点，side 由你给出，它就是分数；单位取最小圆的半径
  • 提交side 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；每个圆完整落在正方形内
  • 目标让正方形边长尽可能小
放大来摆，然后提交 ↗
110.560297198y0
0x110.560297198
已验证构造side = 110.560297198

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

n ≤ 4 由圆心距的初等下界证明（见各子题）。圆容器版是 Zimmermann 竞赛的赛题（见 Packomania），正方形容器版未见文献表，其余 n 开放。

查看来源
当前第一名

110.560297198

正方形边长

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正方形，左下角是原点 (0, 0)，右上角是 (side, side)，side 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "3.5"，最多九位小数。

提交 side 与 centers，centers 按半径 1,2,…,n 的顺序排列。每个数写成十进制字符串，例如 "3.5"。

当前第一名的答案

{
  "centers": [
    [
      "6.288438811",
      "1.113913211"
    ],
    [
      "40.518642607",
      "20.429448833"
    ],
    [
      "35.339064893",
      "3.055240234"
    ],
    [
      "106.511721271",
      "27.75521986"
    ],
    [
      "5.016297609",
      "38.61118889"
    ],
    [
      "76.885270951",
      "85.955954706"
    ],
    [
      "34.826520073",
      "81.348100485"
    ],
    [
      "84.738434854",
      "74.322192045"
    ],
    [
      "101.557045289",
      "71.789441563"
    ],
    [
      "46.792774643",
      "10.039273346"
    ],
    [
      "66.691063509",
      "99.560297198"
    ],
    [
      "43.712812922",
      "98.560297198"
    ],
    [
      "97.560297198",
      "13"
    ],
    [
      "70.578822071",
      "14"
    ],
    [
      "95.560297198",
      "95.560297198"
    ],
    [
      "16",
      "94.560297198"
    ],
    [
      "50.112833769",
      "37.283971373"
    ],
    [
      "57.776931735",
      "71.434543878"
    ],
    [
      "19.009616584",
      "19.009754041"
    ],
    [
      "20",
      "58.783209557"
    ],
    [
      "87.238245281",
      "45.414785818"
    ]
  ],
  "side": "110.560297198"
}
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": [
    [
      "6.288438811",
      "1.113913211"
    ],
    [
      "40.518642607",
      "20.429448833"
    ],
    [
      "35.339064893",
      "3.055240234"
    ],
    [
      "106.511721271",
      "27.75521986"
    ],
    [
      "5.016297609",
      "38.61118889"
    ],
    [
      "76.885270951",
      "85.955954706"
    ],
    [
      "34.826520073",
      "81.348100485"
    ],
    [
      "84.738434854",
      "74.322192045"
    ],
    [
      "101.557045289",
      "71.789441563"
    ],
    [
      "46.792774643",
      "10.039273346"
    ],
    [
      "66.691063509",
      "99.560297198"
    ],
    [
      "43.712812922",
      "98.560297198"
    ],
    [
      "97.560297198",
      "13"
    ],
    [
      "70.578822071",
      "14"
    ],
    [
      "95.560297198",
      "95.560297198"
    ],
    [
      "16",
      "94.560297198"
    ],
    [
      "50.112833769",
      "37.283971373"
    ],
    [
      "57.776931735",
      "71.434543878"
    ],
    [
      "19.009616584",
      "19.009754041"
    ],
    [
      "20",
      "58.783209557"
    ],
    [
      "87.238245281",
      "45.414785818"
    ]
  ],
  "side": "110.560297198"
}

提交 side 与 centers，centers 按半径 1,2,…,n 的顺序排列。每个数写成十进制字符串，例如 "3.5"。 · 验证器 v1.0.0