极值竞技场
中/EN
P18 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

可倾斜等正方形装入单位正方形 · n = 29

文献中也称squares in squarespacking unit squares in a squaretilted square packing

在单位正方形里放 n 个大小完全一样的小正方形，每个都可以任意旋转且互不重叠，让它们的共同边长尽可能大。

子题n = 29
目标最大化 最小边长

纪录对比

最小边长 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.168525115, 0.168525125]见来源署名kingbird.myphotos.cc ↗
本站纪录0.168525119zzzcy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

  • 容器单位正方形 [0, 1] × [0, 1]，左下角为原点 (0, 0)，右上角为 (1, 1)
  • 提交每个正方形表示为 {cx, cy, ux, uy}：中心坐标 (cx, cy) 与一条半边向量 (ux, uy)，另一条正交半边向量固定为 (−uy, ux)
  • 约束每个正方形可任意旋转，且完全位于单位正方形内；任意两个正方形内部不重叠，允许边界接触。公开指标取所有正方形中的最小边长
  • 目标最大化共同边长（验证器在内部通过精确比较边长平方来判定高低）
放大编辑并提交 ↗

当前展示：本站纪录

1y0
0x1
已验证构造29 个正方形

帮助理解

倾斜排布如何节省空间

当 n 不是完全平方数时，与坐标轴平行的网格排布往往会留下无法利用的条状余隙。允许正方形旋转后，部分正方形可以斜嵌在余隙中以腾出空间；例如 n = 5 的最优构型就是在四个正放的正方形中央嵌入一个旋转 45° 的正方形。

已知证明与开放前沿

这是经典的正方形装箱问题（文献常记最小容器边长为 s(n)）：n = 3, 4, 6..9, 14..16, 25 的最优解已获得数学证明（Göbel、Kearney–Shiu 等），其余规模仍处于开放状态。Erich Friedman 的 squares 综述持续汇总该问题族的最新构造。

查看来源
可倾斜等正方形装入单位正方形 n = 29 的当前纪录构型，0.168525119
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zzzcy
解题方式AI · OpenAI: GPT-6 Astra
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（9 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.16852511981→0.16852511987
  2. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.1685251194→0.1685251198
  3. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.1685251191→0.1685251194
  4. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.1685251190→0.1685251191
  5. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.168525118→0.168525119
  6. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.168524983→0.168525118
  7. Thomas Schadt; optimized by David Ellsworth人工
    0.166666666→0.168524983
  8. lird人工
    0.157134838→0.166666666
  9. lird人工
    0.034482758→0.157134838
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和向量用同一个单位，直接写成小数，例如 "0.1"，最多九位小数。

每个正方形提交 {cx, cy, ux, uy}，其中 (cx, cy) 为中心坐标，(ux, uy) 为一条半边向量，另一条正交半边向量固定取 (−uy, ux)。所有数值均以十进制字符串表示（如 "0.1"）。由于倾斜正方形的最优边长通常是无理数，无法直接写成精确的有限小数，因此通过提交有理半边向量来唯一确定正方形及其边长。系统以所有正方形中的最小边长作为公开指标，故应使各正方形尺寸尽量一致。

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.084262560",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.084262560",
      "cy": "0.252787680",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.252787680",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.252787681",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.421312801",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212319",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.578687199",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.084262560",
      "cy": "0.915637440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.252787680",
      "cy": "0.915737440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.915637440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.747012320",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212320",
      "cy": "0.915737440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212319",
      "cy": "0.747212320",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.578387200",
      "cy": "0.915637440",
      "ux": "0.084262561",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.405955562",
      "cy": "0.219954580",
      "ux": "0.076206272",
      "uy": "0.035955294"
    },
    {
      "cx": "0.564150879",
      "cy": "0.280477196",
      "ux": "0.076391049",
      "uy": "0.035561027"
    },
    {
      "cx": "0.719522804",
      "cy": "0.346035868",
      "ux": "0.076391049",
      "uy": "0.035561027"
    },
    {
      "cx": "0.424205293",
      "cy": "0.774444934",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.266516197",
      "cy": "0.714544070",
      "ux": "0.078770739",
      "uy": "0.029922397"
    },
    {
      "cx": "0.108693136",
      "cy": "0.654592307",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.759417004",
      "cy": "0.557242853",
      "ux": "0.080359852",
      "uy": "-0.025347055"
    },
    {
      "cx": "0.576905087",
      "cy": "0.652176000",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.419363611",
      "cy": "0.592331206",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.261822134",
      "cy": "0.532486412",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.108693135",
      "cy": "0.461025765",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.260099272",
      "cy": "0.339010841",
      "ux": "0.076792115",
      "uy": "0.034686455"
    },
    {
      "cx": "0.417780607",
      "cy": "0.411455342",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.575322083",
      "cy": "0.471300136",
      "ux": "0.078770738",
      "uy": "0.029922397"
    }
  ]
}
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": 29
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.084262560",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.084262560",
      "cy": "0.252787680",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.252787680",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.252787681",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.421312801",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212319",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.578687199",
      "cy": "0.084262560",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.084262560",
      "cy": "0.915637440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.252787680",
      "cy": "0.915737440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.915637440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.915737440",
      "cy": "0.747012320",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212320",
      "cy": "0.915737440",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.747212319",
      "cy": "0.747212320",
      "ux": "0.084262560",
      "uy": "0.000000000"
    },
    {
      "cx": "0.578387200",
      "cy": "0.915637440",
      "ux": "0.084262561",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.405955562",
      "cy": "0.219954580",
      "ux": "0.076206272",
      "uy": "0.035955294"
    },
    {
      "cx": "0.564150879",
      "cy": "0.280477196",
      "ux": "0.076391049",
      "uy": "0.035561027"
    },
    {
      "cx": "0.719522804",
      "cy": "0.346035868",
      "ux": "0.076391049",
      "uy": "0.035561027"
    },
    {
      "cx": "0.424205293",
      "cy": "0.774444934",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.266516197",
      "cy": "0.714544070",
      "ux": "0.078770739",
      "uy": "0.029922397"
    },
    {
      "cx": "0.108693136",
      "cy": "0.654592307",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.759417004",
      "cy": "0.557242853",
      "ux": "0.080359852",
      "uy": "-0.025347055"
    },
    {
      "cx": "0.576905087",
      "cy": "0.652176000",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.419363611",
      "cy": "0.592331206",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.261822134",
      "cy": "0.532486412",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.108693135",
      "cy": "0.461025765",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.260099272",
      "cy": "0.339010841",
      "ux": "0.076792115",
      "uy": "0.034686455"
    },
    {
      "cx": "0.417780607",
      "cy": "0.411455342",
      "ux": "0.078770738",
      "uy": "0.029922397"
    },
    {
      "cx": "0.575322083",
      "cy": "0.471300136",
      "ux": "0.078770738",
      "uy": "0.029922397"
    }
  ]
}

每个正方形提交 {cx, cy, ux, uy}，其中 (cx, cy) 为中心坐标，(ux, uy) 为一条半边向量，另一条正交半边向量固定取 (−uy, ux)。所有数值均以十进制字符串表示（如 "0.1"）。由于倾斜正方形的最优边长通常是无理数，无法直接写成精确的有限小数，因此通过提交有理半边向量来唯一确定正方形及其边长。系统以所有正方形中的最小边长作为公开指标，故应使各正方形尺寸尽量一致。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

n=11疑似已证明最优NUE_13#311 条回复共同边长输入有些buglird#281 条回复我需要共同边长选项!!!lird#282 条回复