P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 15

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 = 15

目标最大化最小边长

已知最好（未证明）0.394788787来源精度区间：0.3946329913180741–0.3947887879984209（区间内视为追平）本站展示证书 0.394765481论文原值（单位正方形边长下的容器半径）：2.533+David W. Cantrell 于 2002 年 6 月给出 r = 2.533+ 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 0.15583978539397736；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15 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

0.394765481

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": "1.314034654",
      "cy": "1.141370769",
      "ux": "0.147365504",
      "uy": "0.131313955"
    },
    {
      "cx": "0.693477165",
      "cy": "1.157911098",
      "ux": "0.147365507",
      "uy": "0.131313952"
    },
    {
      "cx": "1.292386316",
      "cy": "1.672565023",
      "ux": "0.147365507",
      "uy": "0.131313951"
    },
    {
      "cx": "0.217057921",
      "cy": "1.001620045",
      "ux": "0.197382319",
      "uy": "-0.000408419"
    },
    {
      "cx": "0.783456362",
      "cy": "0.285597020",
      "ux": "0.197364734",
      "uy": "-0.002666152"
    },
    {
      "cx": "1.178075666",
      "cy": "0.272108831",
      "ux": "0.197364734",
      "uy": "-0.002666152"
    },
    {
      "cx": "1.008449420",
      "cy": "1.397823531",
      "ux": "0.147365506",
      "uy": "0.131313952"
    },
    {
      "cx": "0.408860250",
      "cy": "1.477319022",
      "ux": "0.147365508",
      "uy": "0.131313950"
    },
    {
      "cx": "1.626350261",
      "cy": "1.441399875",
      "ux": "0.147365508",
      "uy": "0.131313951"
    },
    {
      "cx": "0.745821510",
      "cy": "1.692554555",
      "ux": "0.147365507",
      "uy": "0.131313951"
    },
    {
      "cx": "1.584692234",
      "cy": "0.571093627",
      "ux": "0.010598269",
      "uy": "0.197098004"
    },
    {
      "cx": "0.790035285",
      "cy": "0.680309658",
      "ux": "0.197364734",
      "uy": "-0.002666153"
    },
    {
      "cx": "1.185743062",
      "cy": "0.666806770",
      "ux": "0.197364734",
      "uy": "-0.002666154"
    },
    {
      "cx": "0.392959385",
      "cy": "0.604244879",
      "ux": "0.197364734",
      "uy": "-0.002666151"
    },
    {
      "cx": "1.781269883",
      "cy": "0.955859125",
      "ux": "0.010598267",
      "uy": "0.197098004"
    }
  ]
}

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": 15
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.314034654",
      "cy": "1.141370769",
      "ux": "0.147365504",
      "uy": "0.131313955"
    },
    {
      "cx": "0.693477165",
      "cy": "1.157911098",
      "ux": "0.147365507",
      "uy": "0.131313952"
    },
    {
      "cx": "1.292386316",
      "cy": "1.672565023",
      "ux": "0.147365507",
      "uy": "0.131313951"
    },
    {
      "cx": "0.217057921",
      "cy": "1.001620045",
      "ux": "0.197382319",
      "uy": "-0.000408419"
    },
    {
      "cx": "0.783456362",
      "cy": "0.285597020",
      "ux": "0.197364734",
      "uy": "-0.002666152"
    },
    {
      "cx": "1.178075666",
      "cy": "0.272108831",
      "ux": "0.197364734",
      "uy": "-0.002666152"
    },
    {
      "cx": "1.008449420",
      "cy": "1.397823531",
      "ux": "0.147365506",
      "uy": "0.131313952"
    },
    {
      "cx": "0.408860250",
      "cy": "1.477319022",
      "ux": "0.147365508",
      "uy": "0.131313950"
    },
    {
      "cx": "1.626350261",
      "cy": "1.441399875",
      "ux": "0.147365508",
      "uy": "0.131313951"
    },
    {
      "cx": "0.745821510",
      "cy": "1.692554555",
      "ux": "0.147365507",
      "uy": "0.131313951"
    },
    {
      "cx": "1.584692234",
      "cy": "0.571093627",
      "ux": "0.010598269",
      "uy": "0.197098004"
    },
    {
      "cx": "0.790035285",
      "cy": "0.680309658",
      "ux": "0.197364734",
      "uy": "-0.002666153"
    },
    {
      "cx": "1.185743062",
      "cy": "0.666806770",
      "ux": "0.197364734",
      "uy": "-0.002666154"
    },
    {
      "cx": "0.392959385",
      "cy": "0.604244879",
      "ux": "0.197364734",
      "uy": "-0.002666151"
    },
    {
      "cx": "1.781269883",
      "cy": "0.955859125",
      "ux": "0.010598267",
      "uy": "0.197098004"
    }
  ]
}

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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