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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 = 28

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 = 28
目标最小化 正方形边长
已知最好（未证明）167.820337893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 ↗
167.820337893y0
0x167.820337893
已验证构造side = 167.820337893

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

167.820337893

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": [
    [
      "43.428492588",
      "68.831770644"
    ],
    [
      "72.4645074",
      "109.826130328"
    ],
    [
      "95.602063207",
      "161.801255201"
    ],
    [
      "133.507033246",
      "4.443612475"
    ],
    [
      "161.708520307",
      "122.820803055"
    ],
    [
      "81.145230698",
      "6.009261812"
    ],
    [
      "49.685667095",
      "160.820337893"
    ],
    [
      "98.267177254",
      "62.877839783"
    ],
    [
      "9.024512801",
      "93.080448401"
    ],
    [
      "10.694503901",
      "112.083422918"
    ],
    [
      "11",
      "45.631435515"
    ],
    [
      "118.106851324",
      "67.285794376"
    ],
    [
      "111.952030151",
      "154.806922971"
    ],
    [
      "14",
      "70.450782808"
    ],
    [
      "114.223717844",
      "126.567971764"
    ],
    [
      "37.333276994",
      "51.594532259"
    ],
    [
      "150.820337893",
      "17"
    ],
    [
      "19.80362263",
      "18"
    ],
    [
      "71.560536651",
      "58.908539124"
    ],
    [
      "147.820337893",
      "53.87817783"
    ],
    [
      "58.688066821",
      "21"
    ],
    [
      "145.820337893",
      "145.820337893"
    ],
    [
      "95.069768117",
      "93.712504932"
    ],
    [
      "24",
      "143.463883592"
    ],
    [
      "142.820337893",
      "98.599537381"
    ],
    [
      "75.643027663",
      "140.443502212"
    ],
    [
      "45.182695033",
      "97.071067001"
    ],
    [
      "107.185489434",
      "28"
    ]
  ],
  "side": "167.820337893"
}
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": [
    [
      "43.428492588",
      "68.831770644"
    ],
    [
      "72.4645074",
      "109.826130328"
    ],
    [
      "95.602063207",
      "161.801255201"
    ],
    [
      "133.507033246",
      "4.443612475"
    ],
    [
      "161.708520307",
      "122.820803055"
    ],
    [
      "81.145230698",
      "6.009261812"
    ],
    [
      "49.685667095",
      "160.820337893"
    ],
    [
      "98.267177254",
      "62.877839783"
    ],
    [
      "9.024512801",
      "93.080448401"
    ],
    [
      "10.694503901",
      "112.083422918"
    ],
    [
      "11",
      "45.631435515"
    ],
    [
      "118.106851324",
      "67.285794376"
    ],
    [
      "111.952030151",
      "154.806922971"
    ],
    [
      "14",
      "70.450782808"
    ],
    [
      "114.223717844",
      "126.567971764"
    ],
    [
      "37.333276994",
      "51.594532259"
    ],
    [
      "150.820337893",
      "17"
    ],
    [
      "19.80362263",
      "18"
    ],
    [
      "71.560536651",
      "58.908539124"
    ],
    [
      "147.820337893",
      "53.87817783"
    ],
    [
      "58.688066821",
      "21"
    ],
    [
      "145.820337893",
      "145.820337893"
    ],
    [
      "95.069768117",
      "93.712504932"
    ],
    [
      "24",
      "143.463883592"
    ],
    [
      "142.820337893",
      "98.599537381"
    ],
    [
      "75.643027663",
      "140.443502212"
    ],
    [
      "45.182695033",
      "97.071067001"
    ],
    [
      "107.185489434",
      "28"
    ]
  ],
  "side": "167.820337893"
}

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