P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 23

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 = 23

目标最大化最小边长

已知最好（未证明）0.326011123来源精度区间：0.3260100606704722–0.3260111234995339（区间内视为追平）本站展示证书 0.326009547论文原值（单位正方形边长下的容器半径）：3.06738+Milan Kovacic 于 2026 年 5 月给出 r = 3.06738+ 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 0.10628222499918314；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23 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

0.326009547

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": "1.333021036",
      "cy": "1.706006008",
      "ux": "0.162996773",
      "uy": "-0.001615027"
    },
    {
      "cx": "1.072427782",
      "cy": "0.844866531",
      "ux": "0.162996773",
      "uy": "-0.001615052"
    },
    {
      "cx": "0.681033881",
      "cy": "1.712466228",
      "ux": "0.162996773",
      "uy": "-0.001615049"
    },
    {
      "cx": "1.004930337",
      "cy": "1.497586477",
      "ux": "0.162996773",
      "uy": "-0.001615052"
    },
    {
      "cx": "1.023208928",
      "cy": "1.171379795",
      "ux": "0.162996773",
      "uy": "-0.001615050"
    },
    {
      "cx": "0.243584896",
      "cy": "0.771885730",
      "ux": "0.162996752",
      "uy": "-0.001617199"
    },
    {
      "cx": "1.814139140",
      "cy": "1.058380487",
      "ux": "0.162985507",
      "uy": "0.002506144"
    },
    {
      "cx": "0.696082374",
      "cy": "1.060265967",
      "ux": "0.162996773",
      "uy": "-0.001615047"
    },
    {
      "cx": "1.413250369",
      "cy": "1.051074187",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "0.886860900",
      "cy": "0.521086230",
      "ux": "0.036119824",
      "uy": "0.158952555"
    },
    {
      "cx": "1.469458172",
      "cy": "0.376336247",
      "ux": "0.159272256",
      "uy": "0.034682919"
    },
    {
      "cx": "0.788947737",
      "cy": "0.209015059",
      "ux": "0.036119825",
      "uy": "0.158952555"
    },
    {
      "cx": "0.499106943",
      "cy": "0.404757479",
      "ux": "0.036119825",
      "uy": "0.158952555"
    },
    {
      "cx": "0.597020105",
      "cy": "0.716828655",
      "ux": "0.036119824",
      "uy": "0.158952555"
    },
    {
      "cx": "1.741521641",
      "cy": "0.723234384",
      "ux": "0.011546091",
      "uy": "0.162595339"
    },
    {
      "cx": "1.175244097",
      "cy": "0.195239101",
      "ux": "0.159272255",
      "uy": "0.034682923"
    },
    {
      "cx": "1.350177628",
      "cy": "1.376152462",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "1.401333090",
      "cy": "0.724842833",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "1.676131126",
      "cy": "1.382306527",
      "ux": "0.162985507",
      "uy": "0.002506144"
    },
    {
      "cx": "0.352136504",
      "cy": "1.422860031",
      "ux": "0.162996751",
      "uy": "-0.001617275"
    },
    {
      "cx": "1.008160427",
      "cy": "1.823580057",
      "ux": "0.162996773",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.677803778",
      "cy": "1.386472653",
      "ux": "0.162996773",
      "uy": "-0.001615047"
    },
    {
      "cx": "0.223928928",
      "cy": "1.098106445",
      "ux": "0.162996751",
      "uy": "-0.001617270"
    }
  ]
}

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": 23
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.333021036",
      "cy": "1.706006008",
      "ux": "0.162996773",
      "uy": "-0.001615027"
    },
    {
      "cx": "1.072427782",
      "cy": "0.844866531",
      "ux": "0.162996773",
      "uy": "-0.001615052"
    },
    {
      "cx": "0.681033881",
      "cy": "1.712466228",
      "ux": "0.162996773",
      "uy": "-0.001615049"
    },
    {
      "cx": "1.004930337",
      "cy": "1.497586477",
      "ux": "0.162996773",
      "uy": "-0.001615052"
    },
    {
      "cx": "1.023208928",
      "cy": "1.171379795",
      "ux": "0.162996773",
      "uy": "-0.001615050"
    },
    {
      "cx": "0.243584896",
      "cy": "0.771885730",
      "ux": "0.162996752",
      "uy": "-0.001617199"
    },
    {
      "cx": "1.814139140",
      "cy": "1.058380487",
      "ux": "0.162985507",
      "uy": "0.002506144"
    },
    {
      "cx": "0.696082374",
      "cy": "1.060265967",
      "ux": "0.162996773",
      "uy": "-0.001615047"
    },
    {
      "cx": "1.413250369",
      "cy": "1.051074187",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "0.886860900",
      "cy": "0.521086230",
      "ux": "0.036119824",
      "uy": "0.158952555"
    },
    {
      "cx": "1.469458172",
      "cy": "0.376336247",
      "ux": "0.159272256",
      "uy": "0.034682919"
    },
    {
      "cx": "0.788947737",
      "cy": "0.209015059",
      "ux": "0.036119825",
      "uy": "0.158952555"
    },
    {
      "cx": "0.499106943",
      "cy": "0.404757479",
      "ux": "0.036119825",
      "uy": "0.158952555"
    },
    {
      "cx": "0.597020105",
      "cy": "0.716828655",
      "ux": "0.036119824",
      "uy": "0.158952555"
    },
    {
      "cx": "1.741521641",
      "cy": "0.723234384",
      "ux": "0.011546091",
      "uy": "0.162595339"
    },
    {
      "cx": "1.175244097",
      "cy": "0.195239101",
      "ux": "0.159272255",
      "uy": "0.034682923"
    },
    {
      "cx": "1.350177628",
      "cy": "1.376152462",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "1.401333090",
      "cy": "0.724842833",
      "ux": "0.162985508",
      "uy": "0.002506137"
    },
    {
      "cx": "1.676131126",
      "cy": "1.382306527",
      "ux": "0.162985507",
      "uy": "0.002506144"
    },
    {
      "cx": "0.352136504",
      "cy": "1.422860031",
      "ux": "0.162996751",
      "uy": "-0.001617275"
    },
    {
      "cx": "1.008160427",
      "cy": "1.823580057",
      "ux": "0.162996773",
      "uy": "-0.001615048"
    },
    {
      "cx": "0.677803778",
      "cy": "1.386472653",
      "ux": "0.162996773",
      "uy": "-0.001615047"
    },
    {
      "cx": "0.223928928",
      "cy": "1.098106445",
      "ux": "0.162996751",
      "uy": "-0.001617270"
    }
  ]
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？