P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 27

文献中也称squares in circlespacking unit squares in a circle

在半径 1 的圆内放 n 个可任意旋转的正方形，边长完全相同、互不重叠，使共同边长尽可能大。

子题n = 27

目标最大化最小边长

已知最好（未证明）0.306701426来源精度区间：0.3067004855068685–0.3067014261616317（区间内视为追平）本站展示证书 0.306672835论文原值（单位正方形边长下的容器半径）：3.26050+Milan Kovacic 于 2026 年 5 月给出 r = 3.26050+ 的排法（单位正方形的最小外接圆半径）。本站从公开图数字化各正方形的位置与方向，按图中的分离轴解出该固定接触结构并向内修复到九位定点网格，证书分数为 0.094048227817777936；来源截断区间与证书分开比较，最优性未知

严格定义

容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
提交每个正方形写 {cx, cy, ux, uy}：中心加一条半边向量，另一条固定取 (−uy, ux)
约束每个正方形的角度完全自由；全部落在容器内；两两内部不重叠，贴边接触允许。计分只看最小的那个正方形，所以边长不一致占不到便宜
目标让共同边长尽可能大。内部以其平方精确比较

放大来摆，然后提交 ↗

2y0

0x2

已验证构造27 个正方形

帮助理解

哪里有优化空间

圆形容器没有角：正方形的直边贴不住弧形边界，最优解几乎总是倾斜的，正方形彼此以角相抵。

前沿在哪里

n = 4 已证明；n ≤ 35 的其余已知最好值取自 Friedman 的 squares-in-circles 汇总（1997 年起多人贡献），全部未证明；n = 36–40 没有任何公开记录，从零开始。

查看来源

当前第一名

0.306672835

最小边长

答案来源Milan Kovacic

解题方式公开参考构造

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以坐标范围是 0 到 2。坐标和向量用同一个单位，直接写成小数，例如 "0.4"，最多九位小数。

每个正方形提交 {cx,cy,ux,uy}：中心加一条半边向量，另一条半边向量固定取 (-uy,ux)。每个数写成十进制字符串，例如 "0.4"；角度完全自由，计分取最小正方形的边长，所以把它们写得一样大最划算。边长本身几乎总是无理数，写不成有限小数，所以你写的是那条半边向量，边长由它精确定出。

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.263844100",
      "cy": "1.305424515",
      "ux": "0.153324672",
      "uy": "-0.001897870"
    },
    {
      "cx": "1.798024971",
      "cy": "1.156127990",
      "ux": "0.153334971",
      "uy": "0.000666079"
    },
    {
      "cx": "1.798920939",
      "cy": "0.849455235",
      "ux": "0.153334971",
      "uy": "0.000666080"
    },
    {
      "cx": "0.822679109",
      "cy": "1.779295159",
      "ux": "0.153027135",
      "uy": "0.009734115"
    },
    {
      "cx": "1.186305181",
      "cy": "1.496845730",
      "ux": "0.153324673",
      "uy": "-0.001897866"
    },
    {
      "cx": "1.127054333",
      "cy": "0.192978016",
      "ux": "0.153334972",
      "uy": "0.000665967"
    },
    {
      "cx": "1.139610486",
      "cy": "1.804122011",
      "ux": "0.153324661",
      "uy": "-0.001898781"
    },
    {
      "cx": "1.473475140",
      "cy": "0.322614812",
      "ux": "0.131309443",
      "uy": "0.079182621"
    },
    {
      "cx": "0.512843729",
      "cy": "0.385481406",
      "ux": "0.153334984",
      "uy": "0.000663224"
    },
    {
      "cx": "0.572896389",
      "cy": "0.692446970",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "1.492953012",
      "cy": "0.686304005",
      "ux": "0.153334971",
      "uy": "0.000666081"
    },
    {
      "cx": "0.820299283",
      "cy": "0.210851894",
      "ux": "0.153334975",
      "uy": "0.000665196"
    },
    {
      "cx": "1.186762824",
      "cy": "0.574316354",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.514922437",
      "cy": "1.616109910",
      "ux": "0.153329064",
      "uy": "-0.001501792"
    },
    {
      "cx": "0.878999572",
      "cy": "0.824461061",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "0.880331725",
      "cy": "0.517790203",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "1.494388157",
      "cy": "1.608839335",
      "ux": "0.153324673",
      "uy": "-0.001897802"
    },
    {
      "cx": "1.490593041",
      "cy": "1.302189524",
      "ux": "0.153324673",
      "uy": "-0.001897868"
    },
    {
      "cx": "1.185430666",
      "cy": "0.880987333",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "0.266225310",
      "cy": "0.691165734",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.879144922",
      "cy": "1.459432661",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    },
    {
      "cx": "1.182509506",
      "cy": "1.190193759",
      "ux": "0.153324673",
      "uy": "-0.001897868"
    },
    {
      "cx": "1.491620850",
      "cy": "0.992974864",
      "ux": "0.153334971",
      "uy": "0.000666079"
    },
    {
      "cx": "0.165326655",
      "cy": "0.997404086",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.570577101",
      "cy": "1.308312914",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    },
    {
      "cx": "0.566781304",
      "cy": "1.001658761",
      "ux": "0.153324672",
      "uy": "-0.001897870"
    },
    {
      "cx": "0.875113581",
      "cy": "1.133749049",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    }
  ]
}

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": 27
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.263844100",
      "cy": "1.305424515",
      "ux": "0.153324672",
      "uy": "-0.001897870"
    },
    {
      "cx": "1.798024971",
      "cy": "1.156127990",
      "ux": "0.153334971",
      "uy": "0.000666079"
    },
    {
      "cx": "1.798920939",
      "cy": "0.849455235",
      "ux": "0.153334971",
      "uy": "0.000666080"
    },
    {
      "cx": "0.822679109",
      "cy": "1.779295159",
      "ux": "0.153027135",
      "uy": "0.009734115"
    },
    {
      "cx": "1.186305181",
      "cy": "1.496845730",
      "ux": "0.153324673",
      "uy": "-0.001897866"
    },
    {
      "cx": "1.127054333",
      "cy": "0.192978016",
      "ux": "0.153334972",
      "uy": "0.000665967"
    },
    {
      "cx": "1.139610486",
      "cy": "1.804122011",
      "ux": "0.153324661",
      "uy": "-0.001898781"
    },
    {
      "cx": "1.473475140",
      "cy": "0.322614812",
      "ux": "0.131309443",
      "uy": "0.079182621"
    },
    {
      "cx": "0.512843729",
      "cy": "0.385481406",
      "ux": "0.153334984",
      "uy": "0.000663224"
    },
    {
      "cx": "0.572896389",
      "cy": "0.692446970",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "1.492953012",
      "cy": "0.686304005",
      "ux": "0.153334971",
      "uy": "0.000666081"
    },
    {
      "cx": "0.820299283",
      "cy": "0.210851894",
      "ux": "0.153334975",
      "uy": "0.000665196"
    },
    {
      "cx": "1.186762824",
      "cy": "0.574316354",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.514922437",
      "cy": "1.616109910",
      "ux": "0.153329064",
      "uy": "-0.001501792"
    },
    {
      "cx": "0.878999572",
      "cy": "0.824461061",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "0.880331725",
      "cy": "0.517790203",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "1.494388157",
      "cy": "1.608839335",
      "ux": "0.153324673",
      "uy": "-0.001897802"
    },
    {
      "cx": "1.490593041",
      "cy": "1.302189524",
      "ux": "0.153324673",
      "uy": "-0.001897868"
    },
    {
      "cx": "1.185430666",
      "cy": "0.880987333",
      "ux": "0.153334971",
      "uy": "0.000666070"
    },
    {
      "cx": "0.266225310",
      "cy": "0.691165734",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.879144922",
      "cy": "1.459432661",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    },
    {
      "cx": "1.182509506",
      "cy": "1.190193759",
      "ux": "0.153324673",
      "uy": "-0.001897868"
    },
    {
      "cx": "1.491620850",
      "cy": "0.992974864",
      "ux": "0.153334971",
      "uy": "0.000666079"
    },
    {
      "cx": "0.165326655",
      "cy": "0.997404086",
      "ux": "0.153334971",
      "uy": "0.000666072"
    },
    {
      "cx": "0.570577101",
      "cy": "1.308312914",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    },
    {
      "cx": "0.566781304",
      "cy": "1.001658761",
      "ux": "0.153324672",
      "uy": "-0.001897870"
    },
    {
      "cx": "0.875113581",
      "cy": "1.133749049",
      "ux": "0.153324673",
      "uy": "-0.001897867"
    }
  ]
}

每个正方形提交 {cx,cy,ux,uy}：中心加一条半边向量，另一条半边向量固定取 (-uy,ux)。每个数写成十进制字符串，例如 "0.4"；角度完全自由，计分取最小正方形的边长，所以把它们写得一样大最划算。边长本身几乎总是无理数，写不成有限小数，所以你写的是那条半边向量，边长由它精确定出。 · 验证器 v1.0.0

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