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 = 29

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 = 29
目标最小化 正方形边长
已知最好（未证明）176.505704217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 ↗
176.505704217y0
0x176.505704217
已验证构造side = 176.505704217

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

176.505704217

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": [
    [
      "171.937030569",
      "44.312992584"
    ],
    [
      "130.83178046",
      "174.217017055"
    ],
    [
      "137.744478898",
      "117.957528417"
    ],
    [
      "39.784668056",
      "172.458049246"
    ],
    [
      "123.126426035",
      "171.339113897"
    ],
    [
      "170.439260427",
      "121.751468894"
    ],
    [
      "86.511490439",
      "136.744460439"
    ],
    [
      "77.324722954",
      "168.438054186"
    ],
    [
      "75.161183084",
      "150.919091477"
    ],
    [
      "84.577312725",
      "10"
    ],
    [
      "61.372567974",
      "11"
    ],
    [
      "12.148877251",
      "64.420950836"
    ],
    [
      "13.004035759",
      "122.439268833"
    ],
    [
      "55.2928564",
      "162.505704217"
    ],
    [
      "161.505704217",
      "62.390600439"
    ],
    [
      "16.011470777",
      "93.562465699"
    ],
    [
      "81.690201275",
      "81.361435757"
    ],
    [
      "110.745556414",
      "19.961075356"
    ],
    [
      "102.375453837",
      "157.505704217"
    ],
    [
      "44.16110675",
      "70.957630471"
    ],
    [
      "21",
      "155.505704217"
    ],
    [
      "154.025412542",
      "98.626637994"
    ],
    [
      "75.767573295",
      "41.802334682"
    ],
    [
      "152.505704217",
      "24.443268516"
    ],
    [
      "108.660473432",
      "113.557777472"
    ],
    [
      "120.508440706",
      "62.864293072"
    ],
    [
      "27",
      "27.202671722"
    ],
    [
      "148.505704217",
      "148.505704217"
    ],
    [
      "54.91199826",
      "118.763679584"
    ]
  ],
  "side": "176.505704217"
}
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": 29
}

当前第一名的答案

{
  "centers": [
    [
      "171.937030569",
      "44.312992584"
    ],
    [
      "130.83178046",
      "174.217017055"
    ],
    [
      "137.744478898",
      "117.957528417"
    ],
    [
      "39.784668056",
      "172.458049246"
    ],
    [
      "123.126426035",
      "171.339113897"
    ],
    [
      "170.439260427",
      "121.751468894"
    ],
    [
      "86.511490439",
      "136.744460439"
    ],
    [
      "77.324722954",
      "168.438054186"
    ],
    [
      "75.161183084",
      "150.919091477"
    ],
    [
      "84.577312725",
      "10"
    ],
    [
      "61.372567974",
      "11"
    ],
    [
      "12.148877251",
      "64.420950836"
    ],
    [
      "13.004035759",
      "122.439268833"
    ],
    [
      "55.2928564",
      "162.505704217"
    ],
    [
      "161.505704217",
      "62.390600439"
    ],
    [
      "16.011470777",
      "93.562465699"
    ],
    [
      "81.690201275",
      "81.361435757"
    ],
    [
      "110.745556414",
      "19.961075356"
    ],
    [
      "102.375453837",
      "157.505704217"
    ],
    [
      "44.16110675",
      "70.957630471"
    ],
    [
      "21",
      "155.505704217"
    ],
    [
      "154.025412542",
      "98.626637994"
    ],
    [
      "75.767573295",
      "41.802334682"
    ],
    [
      "152.505704217",
      "24.443268516"
    ],
    [
      "108.660473432",
      "113.557777472"
    ],
    [
      "120.508440706",
      "62.864293072"
    ],
    [
      "27",
      "27.202671722"
    ],
    [
      "148.505704217",
      "148.505704217"
    ],
    [
      "54.91199826",
      "118.763679584"
    ]
  ],
  "side": "176.505704217"
}

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