P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 20

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 = 20

目标最大化最小边长

已知最好（未证明）0.345661942来源精度区间：0.3455425017277125–0.3456619426201176（区间内视为追平）本站展示证书 0.345594818论文原值（单位正方形边长下的容器半径）：2.893+David W. Cantrell 于 2002 年 6 月给出 r = 2.893+ 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 0.119435778792790036；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20 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

0.345594818

最小边长

已追平已知最好

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": "0.653263044",
      "cy": "1.337136240",
      "ux": "0.172792498",
      "uy": "-0.001302903"
    },
    {
      "cx": "0.998901095",
      "cy": "1.341553184",
      "ux": "0.172792498",
      "uy": "-0.001302896"
    },
    {
      "cx": "1.690036863",
      "cy": "1.331775032",
      "ux": "0.172792498",
      "uy": "-0.001302909"
    },
    {
      "cx": "0.650657224",
      "cy": "0.991551128",
      "ux": "0.172792498",
      "uy": "-0.001302916"
    },
    {
      "cx": "1.002527785",
      "cy": "0.993123018",
      "ux": "0.172789811",
      "uy": "-0.001620535"
    },
    {
      "cx": "1.695123619",
      "cy": "0.986131952",
      "ux": "0.172792497",
      "uy": "-0.001302950"
    },
    {
      "cx": "1.388133100",
      "cy": "0.298614186",
      "ux": "0.157226871",
      "uy": "0.071684419"
    },
    {
      "cx": "0.624103755",
      "cy": "0.301175930",
      "ux": "0.059253715",
      "uy": "0.162320492"
    },
    {
      "cx": "0.963763111",
      "cy": "0.223809891",
      "ux": "0.059253715",
      "uy": "0.162320492"
    },
    {
      "cx": "1.001511318",
      "cy": "1.687723888",
      "ux": "0.172792498",
      "uy": "-0.001302904"
    },
    {
      "cx": "1.344466364",
      "cy": "1.336316490",
      "ux": "0.172792498",
      "uy": "-0.001302906"
    },
    {
      "cx": "0.307601381",
      "cy": "1.329588046",
      "ux": "0.172792498",
      "uy": "-0.001302905"
    },
    {
      "cx": "1.348720146",
      "cy": "0.988743903",
      "ux": "0.172792498",
      "uy": "-0.001302896"
    },
    {
      "cx": "0.302373489",
      "cy": "0.984022730",
      "ux": "0.172792498",
      "uy": "-0.001302909"
    },
    {
      "cx": "0.831156770",
      "cy": "0.644585322",
      "ux": "0.172792498",
      "uy": "-0.001302885"
    },
    {
      "cx": "1.176760403",
      "cy": "0.644435731",
      "ux": "0.172792498",
      "uy": "-0.001302850"
    },
    {
      "cx": "1.674366085",
      "cy": "0.640683718",
      "ux": "0.172792498",
      "uy": "-0.001302897"
    },
    {
      "cx": "0.655868740",
      "cy": "1.682721426",
      "ux": "0.172792499",
      "uy": "-0.001302777"
    },
    {
      "cx": "1.347072527",
      "cy": "1.681949642",
      "ux": "0.172792498",
      "uy": "-0.001302903"
    },
    {
      "cx": "0.328147839",
      "cy": "0.638223644",
      "ux": "0.172792498",
      "uy": "-0.001302911"
    }
  ]
}

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": 20
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.653263044",
      "cy": "1.337136240",
      "ux": "0.172792498",
      "uy": "-0.001302903"
    },
    {
      "cx": "0.998901095",
      "cy": "1.341553184",
      "ux": "0.172792498",
      "uy": "-0.001302896"
    },
    {
      "cx": "1.690036863",
      "cy": "1.331775032",
      "ux": "0.172792498",
      "uy": "-0.001302909"
    },
    {
      "cx": "0.650657224",
      "cy": "0.991551128",
      "ux": "0.172792498",
      "uy": "-0.001302916"
    },
    {
      "cx": "1.002527785",
      "cy": "0.993123018",
      "ux": "0.172789811",
      "uy": "-0.001620535"
    },
    {
      "cx": "1.695123619",
      "cy": "0.986131952",
      "ux": "0.172792497",
      "uy": "-0.001302950"
    },
    {
      "cx": "1.388133100",
      "cy": "0.298614186",
      "ux": "0.157226871",
      "uy": "0.071684419"
    },
    {
      "cx": "0.624103755",
      "cy": "0.301175930",
      "ux": "0.059253715",
      "uy": "0.162320492"
    },
    {
      "cx": "0.963763111",
      "cy": "0.223809891",
      "ux": "0.059253715",
      "uy": "0.162320492"
    },
    {
      "cx": "1.001511318",
      "cy": "1.687723888",
      "ux": "0.172792498",
      "uy": "-0.001302904"
    },
    {
      "cx": "1.344466364",
      "cy": "1.336316490",
      "ux": "0.172792498",
      "uy": "-0.001302906"
    },
    {
      "cx": "0.307601381",
      "cy": "1.329588046",
      "ux": "0.172792498",
      "uy": "-0.001302905"
    },
    {
      "cx": "1.348720146",
      "cy": "0.988743903",
      "ux": "0.172792498",
      "uy": "-0.001302896"
    },
    {
      "cx": "0.302373489",
      "cy": "0.984022730",
      "ux": "0.172792498",
      "uy": "-0.001302909"
    },
    {
      "cx": "0.831156770",
      "cy": "0.644585322",
      "ux": "0.172792498",
      "uy": "-0.001302885"
    },
    {
      "cx": "1.176760403",
      "cy": "0.644435731",
      "ux": "0.172792498",
      "uy": "-0.001302850"
    },
    {
      "cx": "1.674366085",
      "cy": "0.640683718",
      "ux": "0.172792498",
      "uy": "-0.001302897"
    },
    {
      "cx": "0.655868740",
      "cy": "1.682721426",
      "ux": "0.172792499",
      "uy": "-0.001302777"
    },
    {
      "cx": "1.347072527",
      "cy": "1.681949642",
      "ux": "0.172792498",
      "uy": "-0.001302903"
    },
    {
      "cx": "0.328147839",
      "cy": "0.638223644",
      "ux": "0.172792498",
      "uy": "-0.001302911"
    }
  ]
}

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？