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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 = 52

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 = 52
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[415.56869321095, 415.56869321105]André Müller2025-12Packomania ↗
本站纪录415.568693213邱仲普纪录详情 ↓

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

415.568693213y0
0x415.568693213
已验证构造side = 415.568693213

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
半径成等差的圆装入正方形 n = 52 的当前纪录构型，415.568693213
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普
解题方式AI · OpenAI: GPT-5.6 Sol
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公开方法与可复现线索（点击展开）＋

p13-n52-v1：从公开边长 415.568693250 的构造出发，固定半径 1,2,…,52，在 1e-9 整数网格上精修圆心和方形边长。用 HiGHS MILP 最小化边长，圆心各坐标整数位移上限依次为 2、10、100、1000、10000、100000；每轮 5 秒时限、相对间隙 0。保留每个圆的四边包含约束，对可能接触的圆对加入平方距离的一阶线性约束。每轮完整整数复核后才接受严格改进；接受的步骤为：上限 2 得分 415.568693248、上限 10 得分 415.568693238、上限 100 得分 415.568693213。其余上限没有获得额外改进。 复现最终证书：按半径顺序，对起始圆心施加以下非零 [dx,dy]（单位 1e-9），边长取 415.568693213：1:(-112,-112), 2:(-112,-112), 3:(-112,-112), 4:(-112,-112), 5:(-112,-112), 6:(-112,-112), 7:(-112,-112), 8:(-112,-112), 9:(-112,-112), 10:(-112,-112), 11:(-112,-112), 12:(-112,-112), 13:(-29,-2), 14:(-112,-112), 15:(-112,-112), 16:(-1,-9), 17:(-112,-112), 18:(-26,-25), 19:(-112,-112), 20:(-21,-18), 21:(-112,-112), 22:(-112,-112), 23:(13,12), 24:(-4,0), 25:(2,7), 26:(-112,-112), 27:(-11,24), 28:(-112,-112), 29:(-27,-27), 30:(-112,-112), 31:(-35,7), 32:(-17,-3), 33:(-23,2), 34:(-2,-19), 35:(-1,-3), 36:(8,6), 37:(-16,-18), 38:(-112,-112), 39:(-112,-112), 40:(-10,-14), 41:(-4,-17), 42:(-4,-7), 43:(-36,19), 44:(-14,-21), 45:(-11,-3), 46:(-19,-10), 47:(-26,20), 48:(-24,-23), 49:(-35,-3), 50:(-35,10), 51:(-29,13), 52:(-7,-3)。 验证方法：所有小数乘 10^9 转为整数，逐圆检查 r≤x,y≤L−r，逐对检查 (xi−xj)^2+(yi−yj)^2≥(ri+rj)^2。两套独立实现均通过全部 208 条边界与 1326 个圆对约束。比起始构造缩短 37 个网格单位；这是站内有限网格构造改进，未证明连续最优性。

在讨论区查看与回复 →
历史纪录（2 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    415.56869325→415.568693213
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    486→415.56869325
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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}
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": 52
}

当前第一名的答案

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}

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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