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P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 25

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 = 25
目标最小化 正方形边长
已知最好（未证明）142.220544222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 ↗
142.220544222y0
0x142.220544222
已验证构造side = 142.220544222

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

142.220544222

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": [
    [
      "137.752543605",
      "100.226230806"
    ],
    [
      "39.939510993",
      "46.317391298"
    ],
    [
      "72.836395841",
      "88.333421893"
    ],
    [
      "4.031029631",
      "31.093439927"
    ],
    [
      "32.320508076",
      "5"
    ],
    [
      "43.452907217",
      "136.22017019"
    ],
    [
      "92.533062011",
      "135.220544222"
    ],
    [
      "134.220407881",
      "60.040116702"
    ],
    [
      "38.736735692",
      "97.025333807"
    ],
    [
      "115.09362906",
      "61.111910942"
    ],
    [
      "131.220544222",
      "11"
    ],
    [
      "82.747554815",
      "12"
    ],
    [
      "107.562446571",
      "15.036634185"
    ],
    [
      "128.219261254",
      "81.205820744"
    ],
    [
      "15",
      "15"
    ],
    [
      "125.9915912",
      "37.494467939"
    ],
    [
      "17",
      "82.759489231"
    ],
    [
      "18.644435605",
      "47.798141551"
    ],
    [
      "94.724353869",
      "81.85387539"
    ],
    [
      "90.223581635",
      "43.114450389"
    ],
    [
      "21",
      "121.220544222"
    ],
    [
      "51.092085244",
      "24.40690317"
    ],
    [
      "54.655217137",
      "69.265615715"
    ],
    [
      "118.220544222",
      "117.866776397"
    ],
    [
      "66.958435453",
      "115.986759455"
    ]
  ],
  "side": "142.220544222"
}
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": [
    [
      "137.752543605",
      "100.226230806"
    ],
    [
      "39.939510993",
      "46.317391298"
    ],
    [
      "72.836395841",
      "88.333421893"
    ],
    [
      "4.031029631",
      "31.093439927"
    ],
    [
      "32.320508076",
      "5"
    ],
    [
      "43.452907217",
      "136.22017019"
    ],
    [
      "92.533062011",
      "135.220544222"
    ],
    [
      "134.220407881",
      "60.040116702"
    ],
    [
      "38.736735692",
      "97.025333807"
    ],
    [
      "115.09362906",
      "61.111910942"
    ],
    [
      "131.220544222",
      "11"
    ],
    [
      "82.747554815",
      "12"
    ],
    [
      "107.562446571",
      "15.036634185"
    ],
    [
      "128.219261254",
      "81.205820744"
    ],
    [
      "15",
      "15"
    ],
    [
      "125.9915912",
      "37.494467939"
    ],
    [
      "17",
      "82.759489231"
    ],
    [
      "18.644435605",
      "47.798141551"
    ],
    [
      "94.724353869",
      "81.85387539"
    ],
    [
      "90.223581635",
      "43.114450389"
    ],
    [
      "21",
      "121.220544222"
    ],
    [
      "51.092085244",
      "24.40690317"
    ],
    [
      "54.655217137",
      "69.265615715"
    ],
    [
      "118.220544222",
      "117.866776397"
    ],
    [
      "66.958435453",
      "115.986759455"
    ]
  ],
  "side": "142.220544222"
}

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