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 = 23

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 = 23
目标最小化 正方形边长
已知最好（未证明）126.003553139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 ↗
126.003553139y0
0x126.003553139
已验证构造side = 126.003553139

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

126.003553139

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": [
    [
      "124.177383643",
      "124.178013661"
    ],
    [
      "123.955490782",
      "37.193473863"
    ],
    [
      "58.828827527",
      "122.931459604"
    ],
    [
      "96.778492975",
      "121.967866472"
    ],
    [
      "65.129161354",
      "5.024387579"
    ],
    [
      "42.752382562",
      "44.638829299"
    ],
    [
      "46.287531429",
      "99.841922558"
    ],
    [
      "118.002077799",
      "27.893848286"
    ],
    [
      "47.17923954",
      "116.9486136"
    ],
    [
      "116.003553139",
      "10"
    ],
    [
      "61.448438643",
      "44.427532092"
    ],
    [
      "12.0103384",
      "74.961872387"
    ],
    [
      "13.043616935",
      "13.035446635"
    ],
    [
      "112.003553139",
      "112.003553139"
    ],
    [
      "111.003553139",
      "49.805530127"
    ],
    [
      "109.873407207",
      "81.696331232"
    ],
    [
      "82.036805906",
      "63.404337236"
    ],
    [
      "18",
      "43.721954959"
    ],
    [
      "45.528620435",
      "19"
    ],
    [
      "20",
      "106.003553139"
    ],
    [
      "44.99223285",
      "71.87189929"
    ],
    [
      "86.406239133",
      "22.165484112"
    ],
    [
      "76.12452977",
      "102.964986762"
    ]
  ],
  "side": "126.003553139"
}
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": 23
}

当前第一名的答案

{
  "centers": [
    [
      "124.177383643",
      "124.178013661"
    ],
    [
      "123.955490782",
      "37.193473863"
    ],
    [
      "58.828827527",
      "122.931459604"
    ],
    [
      "96.778492975",
      "121.967866472"
    ],
    [
      "65.129161354",
      "5.024387579"
    ],
    [
      "42.752382562",
      "44.638829299"
    ],
    [
      "46.287531429",
      "99.841922558"
    ],
    [
      "118.002077799",
      "27.893848286"
    ],
    [
      "47.17923954",
      "116.9486136"
    ],
    [
      "116.003553139",
      "10"
    ],
    [
      "61.448438643",
      "44.427532092"
    ],
    [
      "12.0103384",
      "74.961872387"
    ],
    [
      "13.043616935",
      "13.035446635"
    ],
    [
      "112.003553139",
      "112.003553139"
    ],
    [
      "111.003553139",
      "49.805530127"
    ],
    [
      "109.873407207",
      "81.696331232"
    ],
    [
      "82.036805906",
      "63.404337236"
    ],
    [
      "18",
      "43.721954959"
    ],
    [
      "45.528620435",
      "19"
    ],
    [
      "20",
      "106.003553139"
    ],
    [
      "44.99223285",
      "71.87189929"
    ],
    [
      "86.406239133",
      "22.165484112"
    ],
    [
      "76.12452977",
      "102.964986762"
    ]
  ],
  "side": "126.003553139"
}

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