P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 39

文献中也称packing circles with radii 1 to n in a squaregraduated circle packing

将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个正方形内，使外围正方形的边长尽可能小。

子题n = 39

目标最小化正方形边长

纪录对比

正方形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[272.59903112615, 272.59903112625]	Wilfred Heap2026-09-10	Packomania ↗
本站纪录	272.599031145	NUE_13	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器左下角位于原点 (0, 0)、右上角位于 (side, side) 的正方形（以最小圆的半径为单位 1，side 即为待最小化的目标值）
提交正方形边长 side 与圆心列表 centers（按半径 1, 2, …, n 依次排列）
约束第 i 个圆的半径恰好为 i；任意两圆内部不重叠；每个圆完全位于正方形容器内
目标最小化容器正方形边长 side

挑战这个纪录 ↗

当前展示：本站纪录

272.599031145y0

123456789101112131415161718192021222324252627282930313233343536373839

0x272.599031145

已验证构造side = 272.599031145

帮助理解

角位分配与小圆填隙

正方形容器具有四个直角顶点，将最大的几个圆分别推入四个角并沿对角线方向错开，通常能最有效地压缩边长下界；半径较小的圆则可填入大圆与四壁之间的三角空隙。每当 n 增加 1 时，最大圆组对四个角位的重新分配往往会改变整体布局。

已知证明与开放前沿

n ≤ 4 的最优边长可由最大圆之间的圆心距投影直接证明（详见各子题说明）。Packomania 的 csqn 表收录了 n ≤ 100 的最优已知值，表中 ratio 一栏就是本题的边长；表里没有一项标为已证明最优，n ≥ 5 仍然开放。圆容器版本曾是 Al Zimmermann 编程竞赛的赛题（也见 Packomania）。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者NUE_13

解题方式AI · DeepSeek: DeepSeek V4.1 Flash

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（4 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
272.599031148→272.599031145
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
272.599031149→272.599031148
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
272.59903115→272.599031149
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
330→272.59903115
2026-10-23

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。

提交容器正方形边长 side 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。所有数值均以十进制字符串表示（如 "3.5"）。

当前第一名的答案

{
  "centers": [
    [
      "31.878781872",
      "164.582315715"
    ],
    [
      "270.598072534",
      "265.395434081"
    ],
    [
      "86.033900028",
      "59.605357673"
    ],
    [
      "43.909753486",
      "41.775839765"
    ],
    [
      "5.002396516",
      "161.593602034"
    ],
    [
      "6.573343467",
      "266.596155330"
    ],
    [
      "7.903533333",
      "106.906773666"
    ],
    [
      "137.524936360",
      "264.595196730"
    ],
    [
      "72.166420576",
      "263.599031142"
    ],
    [
      "213.788275685",
      "238.268072722"
    ],
    [
      "210.220093201",
      "259.165781909"
    ],
    [
      "221.037874802",
      "217.496859161"
    ],
    [
      "74.622352130",
      "70.978555076"
    ],
    [
      "183.745733726",
      "123.357407161"
    ],
    [
      "160.816663207",
      "15.000000000"
    ],
    [
      "155.673169588",
      "45.570320145"
    ],
    [
      "255.590883040",
      "17.008148105"
    ],
    [
      "254.599031144",
      "52.376601915"
    ],
    [
      "253.599031143",
      "159.809797475"
    ],
    [
      "20.000000013",
      "182.983297937"
    ],
    [
      "125.320184505",
      "21.000000001"
    ],
    [
      "250.599031144",
      "200.699894079"
    ],
    [
      "23.011023909",
      "23.011023903"
    ],
    [
      "137.565314510",
      "192.828835351"
    ],
    [
      "247.521541391",
      "247.599031141"
    ],
    [
      "172.139404507",
      "84.207906871"
    ],
    [
      "77.751227004",
      "112.167453304"
    ],
    [
      "28.000000006",
      "135.654659667"
    ],
    [
      "75.964333785",
      "29.000000008"
    ],
    [
      "117.620133419",
      "71.415366841"
    ],
    [
      "31.000000000",
      "76.730980213"
    ],
    [
      "105.811368320",
      "240.167908857"
    ],
    [
      "170.808878877",
      "239.599031133"
    ],
    [
      "205.763160063",
      "34.514415717"
    ],
    [
      "135.871476044",
      "133.800368987"
    ],
    [
      "36.166420574",
      "236.599031138"
    ],
    [
      "78.216023242",
      "176.926332892"
    ],
    [
      "197.050550639",
      "173.626501517"
    ],
    [
      "233.599031145",
      "105.367167115"
    ]
  ],
  "side": "272.599031145"
}

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": 39
}

当前第一名的答案

{
  "centers": [
    [
      "31.878781872",
      "164.582315715"
    ],
    [
      "270.598072534",
      "265.395434081"
    ],
    [
      "86.033900028",
      "59.605357673"
    ],
    [
      "43.909753486",
      "41.775839765"
    ],
    [
      "5.002396516",
      "161.593602034"
    ],
    [
      "6.573343467",
      "266.596155330"
    ],
    [
      "7.903533333",
      "106.906773666"
    ],
    [
      "137.524936360",
      "264.595196730"
    ],
    [
      "72.166420576",
      "263.599031142"
    ],
    [
      "213.788275685",
      "238.268072722"
    ],
    [
      "210.220093201",
      "259.165781909"
    ],
    [
      "221.037874802",
      "217.496859161"
    ],
    [
      "74.622352130",
      "70.978555076"
    ],
    [
      "183.745733726",
      "123.357407161"
    ],
    [
      "160.816663207",
      "15.000000000"
    ],
    [
      "155.673169588",
      "45.570320145"
    ],
    [
      "255.590883040",
      "17.008148105"
    ],
    [
      "254.599031144",
      "52.376601915"
    ],
    [
      "253.599031143",
      "159.809797475"
    ],
    [
      "20.000000013",
      "182.983297937"
    ],
    [
      "125.320184505",
      "21.000000001"
    ],
    [
      "250.599031144",
      "200.699894079"
    ],
    [
      "23.011023909",
      "23.011023903"
    ],
    [
      "137.565314510",
      "192.828835351"
    ],
    [
      "247.521541391",
      "247.599031141"
    ],
    [
      "172.139404507",
      "84.207906871"
    ],
    [
      "77.751227004",
      "112.167453304"
    ],
    [
      "28.000000006",
      "135.654659667"
    ],
    [
      "75.964333785",
      "29.000000008"
    ],
    [
      "117.620133419",
      "71.415366841"
    ],
    [
      "31.000000000",
      "76.730980213"
    ],
    [
      "105.811368320",
      "240.167908857"
    ],
    [
      "170.808878877",
      "239.599031133"
    ],
    [
      "205.763160063",
      "34.514415717"
    ],
    [
      "135.871476044",
      "133.800368987"
    ],
    [
      "36.166420574",
      "236.599031138"
    ],
    [
      "78.216023242",
      "176.926332892"
    ],
    [
      "197.050550639",
      "173.626501517"
    ],
    [
      "233.599031145",
      "105.367167115"
    ]
  ],
  "side": "272.599031145"
}

提交容器正方形边长 side 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。所有数值均以十进制字符串表示（如 "3.5"）。 · 验证器 v1.0.0

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