P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 18

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 = 18

目标最大化最小边长

已知最好（未证明）8√481/481文献换算值 0.364768602本站展示证书 0.364768497论文原值（单位正方形边长下的容器半径）：√481 / 8 = 2.741+r = √481 / 8，连续分数为 64/481；David W. Cantrell 于 2002 年 3 月给出。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 0.133056056790360548；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18 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

0.364768497

最小边长

答案来源David W. Cantrell

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": "0.999916138",
      "cy": "0.994899030",
      "ux": "0.182363657",
      "uy": "-0.002740639"
    },
    {
      "cx": "0.828579020",
      "cy": "1.729880122",
      "ux": "0.182363657",
      "uy": "-0.002740625"
    },
    {
      "cx": "0.823089563",
      "cy": "1.364606524",
      "ux": "0.182363657",
      "uy": "-0.002740624"
    },
    {
      "cx": "0.635119879",
      "cy": "0.995801389",
      "ux": "0.182363657",
      "uy": "-0.002740633"
    },
    {
      "cx": "1.364669424",
      "cy": "0.991137761",
      "ux": "0.182363657",
      "uy": "-0.002740644"
    },
    {
      "cx": "0.267112400",
      "cy": "0.783024050",
      "ux": "0.182363657",
      "uy": "-0.002740633"
    },
    {
      "cx": "1.726021601",
      "cy": "0.761066935",
      "ux": "0.182363657",
      "uy": "-0.002740641"
    },
    {
      "cx": "0.994419463",
      "cy": "0.629147472",
      "ux": "0.182363657",
      "uy": "-0.002740635"
    },
    {
      "cx": "0.824572802",
      "cy": "0.260952726",
      "ux": "0.182384249",
      "uy": "-0.000010458"
    },
    {
      "cx": "1.193363259",
      "cy": "1.728180202",
      "ux": "0.182363657",
      "uy": "-0.002740624"
    },
    {
      "cx": "0.460460826",
      "cy": "1.509737837",
      "ux": "0.182363656",
      "uy": "-0.002740654"
    },
    {
      "cx": "1.554642619",
      "cy": "1.493262076",
      "ux": "0.182363657",
      "uy": "-0.002740622"
    },
    {
      "cx": "1.187861178",
      "cy": "1.362066317",
      "ux": "0.182363657",
      "uy": "-0.002740623"
    },
    {
      "cx": "0.272428588",
      "cy": "1.147753944",
      "ux": "0.182363656",
      "uy": "-0.002740651"
    },
    {
      "cx": "1.743828606",
      "cy": "1.125609124",
      "ux": "0.182363657",
      "uy": "-0.002740639"
    },
    {
      "cx": "0.629621256",
      "cy": "0.629918956",
      "ux": "0.182363657",
      "uy": "-0.002740632"
    },
    {
      "cx": "1.359159519",
      "cy": "0.624506847",
      "ux": "0.182363657",
      "uy": "-0.002740636"
    },
    {
      "cx": "1.191334196",
      "cy": "0.256373691",
      "ux": "0.182383887",
      "uy": "-0.000363542"
    }
  ]
}

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": 18
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.999916138",
      "cy": "0.994899030",
      "ux": "0.182363657",
      "uy": "-0.002740639"
    },
    {
      "cx": "0.828579020",
      "cy": "1.729880122",
      "ux": "0.182363657",
      "uy": "-0.002740625"
    },
    {
      "cx": "0.823089563",
      "cy": "1.364606524",
      "ux": "0.182363657",
      "uy": "-0.002740624"
    },
    {
      "cx": "0.635119879",
      "cy": "0.995801389",
      "ux": "0.182363657",
      "uy": "-0.002740633"
    },
    {
      "cx": "1.364669424",
      "cy": "0.991137761",
      "ux": "0.182363657",
      "uy": "-0.002740644"
    },
    {
      "cx": "0.267112400",
      "cy": "0.783024050",
      "ux": "0.182363657",
      "uy": "-0.002740633"
    },
    {
      "cx": "1.726021601",
      "cy": "0.761066935",
      "ux": "0.182363657",
      "uy": "-0.002740641"
    },
    {
      "cx": "0.994419463",
      "cy": "0.629147472",
      "ux": "0.182363657",
      "uy": "-0.002740635"
    },
    {
      "cx": "0.824572802",
      "cy": "0.260952726",
      "ux": "0.182384249",
      "uy": "-0.000010458"
    },
    {
      "cx": "1.193363259",
      "cy": "1.728180202",
      "ux": "0.182363657",
      "uy": "-0.002740624"
    },
    {
      "cx": "0.460460826",
      "cy": "1.509737837",
      "ux": "0.182363656",
      "uy": "-0.002740654"
    },
    {
      "cx": "1.554642619",
      "cy": "1.493262076",
      "ux": "0.182363657",
      "uy": "-0.002740622"
    },
    {
      "cx": "1.187861178",
      "cy": "1.362066317",
      "ux": "0.182363657",
      "uy": "-0.002740623"
    },
    {
      "cx": "0.272428588",
      "cy": "1.147753944",
      "ux": "0.182363656",
      "uy": "-0.002740651"
    },
    {
      "cx": "1.743828606",
      "cy": "1.125609124",
      "ux": "0.182363657",
      "uy": "-0.002740639"
    },
    {
      "cx": "0.629621256",
      "cy": "0.629918956",
      "ux": "0.182363657",
      "uy": "-0.002740632"
    },
    {
      "cx": "1.359159519",
      "cy": "0.624506847",
      "ux": "0.182363657",
      "uy": "-0.002740636"
    },
    {
      "cx": "1.191334196",
      "cy": "0.256373691",
      "ux": "0.182383887",
      "uy": "-0.000363542"
    }
  ]
}

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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