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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 = 19

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 = 19
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[0.2046823915, 0.2046823925]见来源署名kingbird.myphotos.cc ↗
本站纪录0.204682392zzzcy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19 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 = 19 的当前纪录构型，0.204682392
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（4 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.204682392000000000→0.204682392000000009
  2. Robert Wainwright, based on Charles F. Cottingham's packing人工
    0.204599997→0.204682392
  3. lird人工
    0.2→0.204599997
  4. lird人工
    0.052631578→0.2
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": "0.102341197",
      "cy": "0.102530304",
      "ux": "0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.102543310",
      "cy": "0.307322003",
      "ux": "0.102350095",
      "uy": "-0.000058104"
    },
    {
      "cx": "0.307023589",
      "cy": "0.102341197",
      "ux": "0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.896307675",
      "cy": "0.897546555",
      "ux": "-0.102343650",
      "uy": "0.000067264"
    },
    {
      "cx": "0.896131601",
      "cy": "0.687140768",
      "ux": "-0.102349945",
      "uy": "0.000434671"
    },
    {
      "cx": "0.895717786",
      "cy": "0.476008975",
      "ux": "-0.102533036",
      "uy": "0.000815018"
    },
    {
      "cx": "0.690371969",
      "cy": "0.897658803",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.690713943",
      "cy": "0.692976411",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.482789700",
      "cy": "0.895247668",
      "ux": "-0.102410782",
      "uy": "0.001960256"
    },
    {
      "cx": "0.897658803",
      "cy": "0.102341197",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.733582527",
      "cy": "0.287679621",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.583139985",
      "cy": "0.145109769",
      "ux": "-0.072369664",
      "uy": "0.072554253"
    },
    {
      "cx": "0.602341196",
      "cy": "0.445902906",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.452862156",
      "cy": "0.305917330",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.102484655",
      "cy": "0.897573875",
      "ux": "-0.000006579",
      "uy": "-0.102350603"
    },
    {
      "cx": "0.294196064",
      "cy": "0.735845417",
      "ux": "0.064825944",
      "uy": "-0.079274263"
    },
    {
      "cx": "0.144175551",
      "cy": "0.595422269",
      "ux": "0.064779360",
      "uy": "-0.079350727"
    },
    {
      "cx": "0.450096227",
      "cy": "0.596420617",
      "ux": "0.065512561",
      "uy": "-0.078631019"
    },
    {
      "cx": "0.299554733",
      "cy": "0.456048656",
      "ux": "0.065593104",
      "uy": "-0.078610344"
    }
  ]
}
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": 19
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.102341197",
      "cy": "0.102530304",
      "ux": "0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.102543310",
      "cy": "0.307322003",
      "ux": "0.102350095",
      "uy": "-0.000058104"
    },
    {
      "cx": "0.307023589",
      "cy": "0.102341197",
      "ux": "0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.896307675",
      "cy": "0.897546555",
      "ux": "-0.102343650",
      "uy": "0.000067264"
    },
    {
      "cx": "0.896131601",
      "cy": "0.687140768",
      "ux": "-0.102349945",
      "uy": "0.000434671"
    },
    {
      "cx": "0.895717786",
      "cy": "0.476008975",
      "ux": "-0.102533036",
      "uy": "0.000815018"
    },
    {
      "cx": "0.690371969",
      "cy": "0.897658803",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.690713943",
      "cy": "0.692976411",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.482789700",
      "cy": "0.895247668",
      "ux": "-0.102410782",
      "uy": "0.001960256"
    },
    {
      "cx": "0.897658803",
      "cy": "0.102341197",
      "ux": "-0.102341196",
      "uy": "-0.000000001"
    },
    {
      "cx": "0.733582527",
      "cy": "0.287679621",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.583139985",
      "cy": "0.145109769",
      "ux": "-0.072369664",
      "uy": "0.072554253"
    },
    {
      "cx": "0.602341196",
      "cy": "0.445902906",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.452862156",
      "cy": "0.305917330",
      "ux": "-0.072366154",
      "uy": "0.072366154"
    },
    {
      "cx": "0.102484655",
      "cy": "0.897573875",
      "ux": "-0.000006579",
      "uy": "-0.102350603"
    },
    {
      "cx": "0.294196064",
      "cy": "0.735845417",
      "ux": "0.064825944",
      "uy": "-0.079274263"
    },
    {
      "cx": "0.144175551",
      "cy": "0.595422269",
      "ux": "0.064779360",
      "uy": "-0.079350727"
    },
    {
      "cx": "0.450096227",
      "cy": "0.596420617",
      "ux": "0.065512561",
      "uy": "-0.078631019"
    },
    {
      "cx": "0.299554733",
      "cy": "0.456048656",
      "ux": "0.065593104",
      "uy": "-0.078610344"
    }
  ]
}

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

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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