P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 63

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 = 63

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	[551.96174627795, 551.96174627805]	André Müller2025-12	Packomania ↗
本站纪录	551.961451435	NUE_13	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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

551.961451435y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263

0x551.961451435

已验证构造side = 551.961451435

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（3 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
551.961451436→551.961451435
2026-11-01
condorAI · OpenAI: GPT-6.1 Sol
551.96174642→551.961451436
2026-10-27
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
640→551.96174642
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

{
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    ],
    [
      "335.505216045",
      "404.234269243"
    ],
    [
      "263.407498845",
      "517.706011791"
    ],
    [
      "492.031223388",
      "96.333651837"
    ],
    [
      "535.905451436",
      "88.790774875"
    ],
    [
      "410.968458752",
      "172.667090655"
    ],
    [
      "94.239190747",
      "7.000000000"
    ],
    [
      "8.000000000",
      "218.739444677"
    ],
    [
      "536.219129722",
      "447.607648228"
    ],
    [
      "541.961451429",
      "222.398104491"
    ],
    [
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      "535.326848001"
    ],
    [
      "449.192647131",
      "538.974217715"
    ],
    [
      "13.000000000",
      "328.750320154"
    ],
    [
      "146.931201272",
      "536.657794757"
    ],
    [
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    [
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    ],
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      "344.318097337"
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    ]
  ],
  "side": "551.961451435"
}

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": 63
}

当前第一名的答案

{
  "centers": [
    [
      "362.661644561",
      "226.469352666"
    ],
    [
      "335.505216045",
      "404.234269243"
    ],
    [
      "263.407498845",
      "517.706011791"
    ],
    [
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      "63.569782630"
    ],
    [
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    ],
    [
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      "309.047817806"
    ],
    [
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      "532.961451433"
    ],
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      "344.318097337"
    ],
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      "118.635556469"
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      "312.161485229"
    ],
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      "63.045206625",
      "173.894988736"
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  ],
  "side": "551.961451435"
}

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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