P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 26

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 = 26

目标最大化最小边长

已知最好（未证明）2√41/41文献换算值 0.312347523本站展示证书 0.312347439论文原值（单位正方形边长下的容器半径）：√41 / 2 = 3.201+r = √41 / 2，连续分数为 4/41；Erich Friedman 于 1997 年给出。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 0.097560922905179944；最优性未知提交只收小数，所以纪录追的是这个可写值；如果你认为自己求出了理论最优的闭式，欢迎邮件投稿。 minmaxarena.com@gmail.com

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26 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

0.312347439

最小边长

答案来源Erich Friedman

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": "0.851755280",
      "cy": "1.784305082",
      "ux": "0.156172569",
      "uy": "-0.000599515"
    },
    {
      "cx": "0.379988960",
      "cy": "1.474702568",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.692321175",
      "cy": "1.471610223",
      "ux": "0.156166012",
      "uy": "-0.001551631"
    },
    {
      "cx": "1.317045114",
      "cy": "1.471416763",
      "ux": "0.156166012",
      "uy": "-0.001551632"
    },
    {
      "cx": "1.629317404",
      "cy": "1.462293798",
      "ux": "0.156166012",
      "uy": "-0.001551634"
    },
    {
      "cx": "0.224784325",
      "cy": "1.163881696",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.537802051",
      "cy": "1.160771639",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.851412193",
      "cy": "1.153228829",
      "ux": "0.156167594",
      "uy": "-0.001383300"
    },
    {
      "cx": "1.479667605",
      "cy": "1.151417745",
      "ux": "0.156166012",
      "uy": "-0.001551633"
    },
    {
      "cx": "1.793400833",
      "cy": "1.148300550",
      "ux": "0.156166012",
      "uy": "-0.001551625"
    },
    {
      "cx": "0.370682594",
      "cy": "0.537706200",
      "ux": "0.156166012",
      "uy": "-0.001551620"
    },
    {
      "cx": "0.682942868",
      "cy": "0.527374081",
      "ux": "0.156166012",
      "uy": "-0.001551620"
    },
    {
      "cx": "1.307616647",
      "cy": "0.522131311",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "1.620011047",
      "cy": "0.525297440",
      "ux": "0.156166012",
      "uy": "-0.001551623"
    },
    {
      "cx": "0.849245651",
      "cy": "0.208938190",
      "ux": "0.156173412",
      "uy": "-0.000310221"
    },
    {
      "cx": "1.165780716",
      "cy": "1.787601980",
      "ux": "0.156172865",
      "uy": "-0.000516779"
    },
    {
      "cx": "1.004636812",
      "cy": "1.466850256",
      "ux": "0.156166012",
      "uy": "-0.001551633"
    },
    {
      "cx": "1.164603633",
      "cy": "1.150845323",
      "ux": "0.156168358",
      "uy": "-0.001294207"
    },
    {
      "cx": "0.220693155",
      "cy": "0.851559407",
      "ux": "0.156166012",
      "uy": "-0.001551612"
    },
    {
      "cx": "0.535123379",
      "cy": "0.848435302",
      "ux": "0.156166012",
      "uy": "-0.001551624"
    },
    {
      "cx": "0.848728692",
      "cy": "0.840564199",
      "ux": "0.156169133",
      "uy": "-0.001196931"
    },
    {
      "cx": "1.475781668",
      "cy": "0.839093405",
      "ux": "0.156166012",
      "uy": "-0.001551625"
    },
    {
      "cx": "1.790029356",
      "cy": "0.835971138",
      "ux": "0.156166012",
      "uy": "-0.001551623"
    },
    {
      "cx": "0.995291997",
      "cy": "0.525985441",
      "ux": "0.156166012",
      "uy": "-0.001551618"
    },
    {
      "cx": "1.162124512",
      "cy": "0.208882754",
      "ux": "0.156173156",
      "uy": "-0.000419783"
    },
    {
      "cx": "1.161683445",
      "cy": "0.838015889",
      "ux": "0.156164601",
      "uy": "-0.001687727"
    }
  ]
}

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": 26
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.851755280",
      "cy": "1.784305082",
      "ux": "0.156172569",
      "uy": "-0.000599515"
    },
    {
      "cx": "0.379988960",
      "cy": "1.474702568",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.692321175",
      "cy": "1.471610223",
      "ux": "0.156166012",
      "uy": "-0.001551631"
    },
    {
      "cx": "1.317045114",
      "cy": "1.471416763",
      "ux": "0.156166012",
      "uy": "-0.001551632"
    },
    {
      "cx": "1.629317404",
      "cy": "1.462293798",
      "ux": "0.156166012",
      "uy": "-0.001551634"
    },
    {
      "cx": "0.224784325",
      "cy": "1.163881696",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.537802051",
      "cy": "1.160771639",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "0.851412193",
      "cy": "1.153228829",
      "ux": "0.156167594",
      "uy": "-0.001383300"
    },
    {
      "cx": "1.479667605",
      "cy": "1.151417745",
      "ux": "0.156166012",
      "uy": "-0.001551633"
    },
    {
      "cx": "1.793400833",
      "cy": "1.148300550",
      "ux": "0.156166012",
      "uy": "-0.001551625"
    },
    {
      "cx": "0.370682594",
      "cy": "0.537706200",
      "ux": "0.156166012",
      "uy": "-0.001551620"
    },
    {
      "cx": "0.682942868",
      "cy": "0.527374081",
      "ux": "0.156166012",
      "uy": "-0.001551620"
    },
    {
      "cx": "1.307616647",
      "cy": "0.522131311",
      "ux": "0.156166012",
      "uy": "-0.001551622"
    },
    {
      "cx": "1.620011047",
      "cy": "0.525297440",
      "ux": "0.156166012",
      "uy": "-0.001551623"
    },
    {
      "cx": "0.849245651",
      "cy": "0.208938190",
      "ux": "0.156173412",
      "uy": "-0.000310221"
    },
    {
      "cx": "1.165780716",
      "cy": "1.787601980",
      "ux": "0.156172865",
      "uy": "-0.000516779"
    },
    {
      "cx": "1.004636812",
      "cy": "1.466850256",
      "ux": "0.156166012",
      "uy": "-0.001551633"
    },
    {
      "cx": "1.164603633",
      "cy": "1.150845323",
      "ux": "0.156168358",
      "uy": "-0.001294207"
    },
    {
      "cx": "0.220693155",
      "cy": "0.851559407",
      "ux": "0.156166012",
      "uy": "-0.001551612"
    },
    {
      "cx": "0.535123379",
      "cy": "0.848435302",
      "ux": "0.156166012",
      "uy": "-0.001551624"
    },
    {
      "cx": "0.848728692",
      "cy": "0.840564199",
      "ux": "0.156169133",
      "uy": "-0.001196931"
    },
    {
      "cx": "1.475781668",
      "cy": "0.839093405",
      "ux": "0.156166012",
      "uy": "-0.001551625"
    },
    {
      "cx": "1.790029356",
      "cy": "0.835971138",
      "ux": "0.156166012",
      "uy": "-0.001551623"
    },
    {
      "cx": "0.995291997",
      "cy": "0.525985441",
      "ux": "0.156166012",
      "uy": "-0.001551618"
    },
    {
      "cx": "1.162124512",
      "cy": "0.208882754",
      "ux": "0.156173156",
      "uy": "-0.000419783"
    },
    {
      "cx": "1.161683445",
      "cy": "0.838015889",
      "ux": "0.156164601",
      "uy": "-0.001687727"
    }
  ]
}

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？