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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 = 26

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 = 26
目标最小化 正方形边长
已知最好（未证明）150.722283637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 ↗
150.722283637y0
0x150.722283637
已验证构造side = 150.722283637

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

150.722283637

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": [
    [
      "116.568371164",
      "6.16149546"
    ],
    [
      "57.700266496",
      "131.392030078"
    ],
    [
      "4.670525447",
      "4.599767046"
    ],
    [
      "48.32778094",
      "4.065196952"
    ],
    [
      "5.056563733",
      "97.292061145"
    ],
    [
      "6.006003172",
      "51.610919016"
    ],
    [
      "24.341982804",
      "143.643094411"
    ],
    [
      "8.078366553",
      "142.66039854"
    ],
    [
      "141.722283637",
      "41.527178404"
    ],
    [
      "122.735275582",
      "40.824664969"
    ],
    [
      "67.878085621",
      "139.711821888"
    ],
    [
      "104.982612368",
      "54.180124812"
    ],
    [
      "43.763966431",
      "137.722283637"
    ],
    [
      "64.093219883",
      "14"
    ],
    [
      "135.696182709",
      "135.696182709"
    ],
    [
      "134.722283637",
      "16"
    ],
    [
      "133.641066062",
      "103.572446312"
    ],
    [
      "132.722283637",
      "66.983022527"
    ],
    [
      "19",
      "117.455414943"
    ],
    [
      "100.261439078",
      "86.738871549"
    ],
    [
      "98.193367059",
      "21.885427231"
    ],
    [
      "99.081138483",
      "128.722283637"
    ],
    [
      "23.008436836",
      "75.360144055"
    ],
    [
      "59.992861462",
      "104.471374384"
    ],
    [
      "67.368517066",
      "56.029659876"
    ],
    [
      "26",
      "26.202729161"
    ]
  ],
  "side": "150.722283637"
}
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": 26
}

当前第一名的答案

{
  "centers": [
    [
      "116.568371164",
      "6.16149546"
    ],
    [
      "57.700266496",
      "131.392030078"
    ],
    [
      "4.670525447",
      "4.599767046"
    ],
    [
      "48.32778094",
      "4.065196952"
    ],
    [
      "5.056563733",
      "97.292061145"
    ],
    [
      "6.006003172",
      "51.610919016"
    ],
    [
      "24.341982804",
      "143.643094411"
    ],
    [
      "8.078366553",
      "142.66039854"
    ],
    [
      "141.722283637",
      "41.527178404"
    ],
    [
      "122.735275582",
      "40.824664969"
    ],
    [
      "67.878085621",
      "139.711821888"
    ],
    [
      "104.982612368",
      "54.180124812"
    ],
    [
      "43.763966431",
      "137.722283637"
    ],
    [
      "64.093219883",
      "14"
    ],
    [
      "135.696182709",
      "135.696182709"
    ],
    [
      "134.722283637",
      "16"
    ],
    [
      "133.641066062",
      "103.572446312"
    ],
    [
      "132.722283637",
      "66.983022527"
    ],
    [
      "19",
      "117.455414943"
    ],
    [
      "100.261439078",
      "86.738871549"
    ],
    [
      "98.193367059",
      "21.885427231"
    ],
    [
      "99.081138483",
      "128.722283637"
    ],
    [
      "23.008436836",
      "75.360144055"
    ],
    [
      "59.992861462",
      "104.471374384"
    ],
    [
      "67.368517066",
      "56.029659876"
    ],
    [
      "26",
      "26.202729161"
    ]
  ],
  "side": "150.722283637"
}

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