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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 = 24

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 = 24
目标最小化 正方形边长
已知最好（未证明）134.072386878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 ↗
134.072386878y0
0x134.072386878
已验证构造side = 134.072386878

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

134.072386878

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": [
    [
      "2.197573667",
      "86.259698922"
    ],
    [
      "49.104672627",
      "25.092961216"
    ],
    [
      "76.813805267",
      "130.85006306"
    ],
    [
      "40.253747155",
      "129.984531236"
    ],
    [
      "5.063000356",
      "93.371943709"
    ],
    [
      "6",
      "46.386707506"
    ],
    [
      "40.560975364",
      "96.572492816"
    ],
    [
      "125.994505183",
      "40.709402085"
    ],
    [
      "125.072386878",
      "59.341418384"
    ],
    [
      "80.66811287",
      "95.189862671"
    ],
    [
      "82.933955713",
      "71.123361092"
    ],
    [
      "56.37472895",
      "12"
    ],
    [
      "121.072386878",
      "121.072386878"
    ],
    [
      "60.335541196",
      "81.897554307"
    ],
    [
      "92.735454276",
      "119.072386878"
    ],
    [
      "84.087541872",
      "16"
    ],
    [
      "117.072386878",
      "17"
    ],
    [
      "100.620117537",
      "47.892116043"
    ],
    [
      "58.99096227",
      "114.911947079"
    ],
    [
      "20",
      "114.072386878"
    ],
    [
      "61.696302317",
      "45.455600338"
    ],
    [
      "111.755335543",
      "87.335277421"
    ],
    [
      "23.148233498",
      "23.000000001"
    ],
    [
      "24.15577577",
      "70.26908189"
    ]
  ],
  "side": "134.072386878"
}
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": 24
}

当前第一名的答案

{
  "centers": [
    [
      "2.197573667",
      "86.259698922"
    ],
    [
      "49.104672627",
      "25.092961216"
    ],
    [
      "76.813805267",
      "130.85006306"
    ],
    [
      "40.253747155",
      "129.984531236"
    ],
    [
      "5.063000356",
      "93.371943709"
    ],
    [
      "6",
      "46.386707506"
    ],
    [
      "40.560975364",
      "96.572492816"
    ],
    [
      "125.994505183",
      "40.709402085"
    ],
    [
      "125.072386878",
      "59.341418384"
    ],
    [
      "80.66811287",
      "95.189862671"
    ],
    [
      "82.933955713",
      "71.123361092"
    ],
    [
      "56.37472895",
      "12"
    ],
    [
      "121.072386878",
      "121.072386878"
    ],
    [
      "60.335541196",
      "81.897554307"
    ],
    [
      "92.735454276",
      "119.072386878"
    ],
    [
      "84.087541872",
      "16"
    ],
    [
      "117.072386878",
      "17"
    ],
    [
      "100.620117537",
      "47.892116043"
    ],
    [
      "58.99096227",
      "114.911947079"
    ],
    [
      "20",
      "114.072386878"
    ],
    [
      "61.696302317",
      "45.455600338"
    ],
    [
      "111.755335543",
      "87.335277421"
    ],
    [
      "23.148233498",
      "23.000000001"
    ],
    [
      "24.15577577",
      "70.26908189"
    ]
  ],
  "side": "134.072386878"
}

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