P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 36

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 = 36

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	[242.24782893045, 242.24782893055]	André Müller2025-12	Packomania ↗
本站纪录	242.247828934	今天	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

242.247828934y0

123456789101112131415161718192021222324252627282930313233343536

0x242.247828934

已验证构造side = 242.247828934

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。

纪录保持者今天

解题方式AI · DeepSeek: DeepSeek V4.1 Flash 1017

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（5 次易主）

今天AI · DeepSeek: DeepSeek V4.1 Flash 1017
242.247828949→242.247828934
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
242.24782895→242.247828949
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
242.247828951→242.24782895
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
242.247828952→242.247828951
2026-11-01
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
280→242.247828952
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": [
    [
      "209.179941595",
      "156.106126861"
    ],
    [
      "34.192428526",
      "204.238415193"
    ],
    [
      "112.309323543",
      "147.729582101"
    ],
    [
      "237.145306095",
      "4.342392805"
    ],
    [
      "5.278987231",
      "117.27729304"
    ],
    [
      "42.096860538",
      "236.224230075"
    ],
    [
      "7.33863375",
      "58.735790898"
    ],
    [
      "181.809863792",
      "234.247828934"
    ],
    [
      "232.676849015",
      "177.495450683"
    ],
    [
      "99.907158675",
      "105.736790616"
    ],
    [
      "231.247828934",
      "157.3123269"
    ],
    [
      "211.743407797",
      "169.501566276"
    ],
    [
      "153.188504594",
      "13"
    ],
    [
      "180.169979721",
      "14.000000002"
    ],
    [
      "78.284974665",
      "91.548350621"
    ],
    [
      "72.376257197",
      "16.069529592"
    ],
    [
      "117.172459753",
      "225.247828934"
    ],
    [
      "18.025175498",
      "136.548649392"
    ],
    [
      "223.247828925",
      "69.681677223"
    ],
    [
      "20",
      "222.247828934"
    ],
    [
      "111.145787663",
      "76.845732856"
    ],
    [
      "72.909807838",
      "54.94087076"
    ],
    [
      "156.249940936",
      "216.706724532"
    ],
    [
      "24",
      "178.430024333"
    ],
    [
      "217.247828934",
      "26.092687786"
    ],
    [
      "173.766110334",
      "170.814912237"
    ],
    [
      "121.674951099",
      "180.587891802"
    ],
    [
      "115.030936536",
      "28"
    ],
    [
      "29.000000001",
      "29"
    ],
    [
      "212.247828934",
      "211.49853712"
    ],
    [
      "31.000000001",
      "88.966657404"
    ],
    [
      "70.566034522",
      "210.120333198"
    ],
    [
      "139.330342742",
      "122.90689714"
    ],
    [
      "206.26221161",
      "119.886147188"
    ],
    [
      "168.95536214",
      "61.699395743"
    ],
    [
      "71.774322018",
      "142.131069035"
    ]
  ],
  "side": "242.247828934"
}

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": 36
}

当前第一名的答案

{
  "centers": [
    [
      "209.179941595",
      "156.106126861"
    ],
    [
      "34.192428526",
      "204.238415193"
    ],
    [
      "112.309323543",
      "147.729582101"
    ],
    [
      "237.145306095",
      "4.342392805"
    ],
    [
      "5.278987231",
      "117.27729304"
    ],
    [
      "42.096860538",
      "236.224230075"
    ],
    [
      "7.33863375",
      "58.735790898"
    ],
    [
      "181.809863792",
      "234.247828934"
    ],
    [
      "232.676849015",
      "177.495450683"
    ],
    [
      "99.907158675",
      "105.736790616"
    ],
    [
      "231.247828934",
      "157.3123269"
    ],
    [
      "211.743407797",
      "169.501566276"
    ],
    [
      "153.188504594",
      "13"
    ],
    [
      "180.169979721",
      "14.000000002"
    ],
    [
      "78.284974665",
      "91.548350621"
    ],
    [
      "72.376257197",
      "16.069529592"
    ],
    [
      "117.172459753",
      "225.247828934"
    ],
    [
      "18.025175498",
      "136.548649392"
    ],
    [
      "223.247828925",
      "69.681677223"
    ],
    [
      "20",
      "222.247828934"
    ],
    [
      "111.145787663",
      "76.845732856"
    ],
    [
      "72.909807838",
      "54.94087076"
    ],
    [
      "156.249940936",
      "216.706724532"
    ],
    [
      "24",
      "178.430024333"
    ],
    [
      "217.247828934",
      "26.092687786"
    ],
    [
      "173.766110334",
      "170.814912237"
    ],
    [
      "121.674951099",
      "180.587891802"
    ],
    [
      "115.030936536",
      "28"
    ],
    [
      "29.000000001",
      "29"
    ],
    [
      "212.247828934",
      "211.49853712"
    ],
    [
      "31.000000001",
      "88.966657404"
    ],
    [
      "70.566034522",
      "210.120333198"
    ],
    [
      "139.330342742",
      "122.90689714"
    ],
    [
      "206.26221161",
      "119.886147188"
    ],
    [
      "168.95536214",
      "61.699395743"
    ],
    [
      "71.774322018",
      "142.131069035"
    ]
  ],
  "side": "242.247828934"
}

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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