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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 = 22

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 = 22
目标最小化 正方形边长
已知最好（未证明）118.363865189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 ↗
118.363865189y0
0x118.363865189
已验证构造side = 118.363865189

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

118.363865189

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": [
    [
      "36.708265902",
      "97.651192266"
    ],
    [
      "2.813546574",
      "115.548304651"
    ],
    [
      "114.362456362",
      "67.185933886"
    ],
    [
      "4.376588711",
      "31.567282732"
    ],
    [
      "32.320508076",
      "5"
    ],
    [
      "22.630738548",
      "79.063757937"
    ],
    [
      "86.057509881",
      "71.291663288"
    ],
    [
      "8.045386851",
      "77.807839712"
    ],
    [
      "78.627837772",
      "9"
    ],
    [
      "80.264338691",
      "30.55398921"
    ],
    [
      "44.505258283",
      "107.363865189"
    ],
    [
      "106.363865189",
      "106.363865189"
    ],
    [
      "105.363865189",
      "81.383873195"
    ],
    [
      "42.340655008",
      "82.457751628"
    ],
    [
      "15",
      "15"
    ],
    [
      "102.363865189",
      "16.848630615"
    ],
    [
      "17",
      "101.363865189"
    ],
    [
      "100.363865189",
      "50.789756112"
    ],
    [
      "63.986450152",
      "57.548726103"
    ],
    [
      "51.980763552",
      "20.442614891"
    ],
    [
      "74.748193786",
      "96.906100028"
    ],
    [
      "23.241637393",
      "51.070422969"
    ]
  ],
  "side": "118.363865189"
}
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": 22
}

当前第一名的答案

{
  "centers": [
    [
      "36.708265902",
      "97.651192266"
    ],
    [
      "2.813546574",
      "115.548304651"
    ],
    [
      "114.362456362",
      "67.185933886"
    ],
    [
      "4.376588711",
      "31.567282732"
    ],
    [
      "32.320508076",
      "5"
    ],
    [
      "22.630738548",
      "79.063757937"
    ],
    [
      "86.057509881",
      "71.291663288"
    ],
    [
      "8.045386851",
      "77.807839712"
    ],
    [
      "78.627837772",
      "9"
    ],
    [
      "80.264338691",
      "30.55398921"
    ],
    [
      "44.505258283",
      "107.363865189"
    ],
    [
      "106.363865189",
      "106.363865189"
    ],
    [
      "105.363865189",
      "81.383873195"
    ],
    [
      "42.340655008",
      "82.457751628"
    ],
    [
      "15",
      "15"
    ],
    [
      "102.363865189",
      "16.848630615"
    ],
    [
      "17",
      "101.363865189"
    ],
    [
      "100.363865189",
      "50.789756112"
    ],
    [
      "63.986450152",
      "57.548726103"
    ],
    [
      "51.980763552",
      "20.442614891"
    ],
    [
      "74.748193786",
      "96.906100028"
    ],
    [
      "23.241637393",
      "51.070422969"
    ]
  ],
  "side": "118.363865189"
}

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