P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 30

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 = 30

目标最大化最小边长

已知最好（未证明）0.289049086来源精度区间：0.2890482508245101–0.2890490863158382（区间内视为追平）本站展示证书 0.289048409论文原值（单位正方形边长下的容器半径）：3.45962+Milan Kovacic 于 2026 年 5 月给出 r = 3.45962+ 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 0.083548982816364776；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30 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

0.289048409

最小边长

已追平已知最好

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": "1.533113666",
      "cy": "1.126265015",
      "ux": "0.144522481",
      "uy": "-0.000705939"
    },
    {
      "cx": "1.275665988",
      "cy": "1.416574503",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "1.243548663",
      "cy": "1.127679465",
      "ux": "0.144522481",
      "uy": "-0.000705960"
    },
    {
      "cx": "1.019090745",
      "cy": "0.155238693",
      "ux": "0.144487313",
      "uy": "0.003265285"
    },
    {
      "cx": "0.727143940",
      "cy": "0.236345910",
      "ux": "0.144520556",
      "uy": "0.001027030"
    },
    {
      "cx": "1.195499108",
      "cy": "1.790205531",
      "ux": "0.075629959",
      "uy": "0.123155816"
    },
    {
      "cx": "1.670586915",
      "cy": "1.454199526",
      "ux": "0.082855954",
      "uy": "0.118415103"
    },
    {
      "cx": "1.484426532",
      "cy": "1.692327296",
      "ux": "0.082855934",
      "uy": "0.118415117"
    },
    {
      "cx": "0.253416827",
      "cy": "1.373528066",
      "ux": "0.075629964",
      "uy": "0.123155813"
    },
    {
      "cx": "0.736055267",
      "cy": "1.054748739",
      "ux": "0.075629969",
      "uy": "0.123155810"
    },
    {
      "cx": "1.223427755",
      "cy": "0.838725827",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "0.740388833",
      "cy": "1.752879463",
      "ux": "0.075629955",
      "uy": "0.123155819"
    },
    {
      "cx": "0.446370324",
      "cy": "1.594235760",
      "ux": "0.075629959",
      "uy": "0.123155816"
    },
    {
      "cx": "0.488643436",
      "cy": "1.204217214",
      "ux": "0.075629964",
      "uy": "0.123155813"
    },
    {
      "cx": "0.479526853",
      "cy": "0.804678742",
      "ux": "0.135317718",
      "uy": "0.050757866"
    },
    {
      "cx": "0.930389587",
      "cy": "1.274608488",
      "ux": "0.075629967",
      "uy": "0.123155812"
    },
    {
      "cx": "1.512472783",
      "cy": "0.837313869",
      "ux": "0.144522481",
      "uy": "-0.000705956"
    },
    {
      "cx": "0.393186393",
      "cy": "0.463578094",
      "ux": "0.135317719",
      "uy": "0.050757865"
    },
    {
      "cx": "0.190078061",
      "cy": "1.004820334",
      "ux": "0.135317718",
      "uy": "0.050757867"
    },
    {
      "cx": "0.904339485",
      "cy": "0.746902404",
      "ux": "0.135317729",
      "uy": "0.050757838"
    },
    {
      "cx": "0.663832170",
      "cy": "0.565066488",
      "ux": "0.135317719",
      "uy": "0.050757863"
    },
    {
      "cx": "1.026843692",
      "cy": "0.484139739",
      "ux": "0.135317729",
      "uy": "0.050757838"
    },
    {
      "cx": "0.976715614",
      "cy": "1.585360067",
      "ux": "0.075629958",
      "uy": "0.123155817"
    },
    {
      "cx": "0.683272095",
      "cy": "1.426363281",
      "ux": "0.075629983",
      "uy": "0.123155802"
    },
    {
      "cx": "1.310706226",
      "cy": "0.246766942",
      "ux": "0.143900059",
      "uy": "0.013417106"
    },
    {
      "cx": "0.208891339",
      "cy": "0.703163007",
      "ux": "0.135317718",
      "uy": "0.050757866"
    },
    {
      "cx": "1.356036093",
      "cy": "0.549026079",
      "ux": "0.144522481",
      "uy": "-0.000705958"
    },
    {
      "cx": "1.822387569",
      "cy": "1.107834627",
      "ux": "0.144522481",
      "uy": "-0.000705953"
    },
    {
      "cx": "1.644998021",
      "cy": "0.530597254",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "1.801434704",
      "cy": "0.818885050",
      "ux": "0.144522481",
      "uy": "-0.000705956"
    }
  ]
}

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": 30
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.533113666",
      "cy": "1.126265015",
      "ux": "0.144522481",
      "uy": "-0.000705939"
    },
    {
      "cx": "1.275665988",
      "cy": "1.416574503",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "1.243548663",
      "cy": "1.127679465",
      "ux": "0.144522481",
      "uy": "-0.000705960"
    },
    {
      "cx": "1.019090745",
      "cy": "0.155238693",
      "ux": "0.144487313",
      "uy": "0.003265285"
    },
    {
      "cx": "0.727143940",
      "cy": "0.236345910",
      "ux": "0.144520556",
      "uy": "0.001027030"
    },
    {
      "cx": "1.195499108",
      "cy": "1.790205531",
      "ux": "0.075629959",
      "uy": "0.123155816"
    },
    {
      "cx": "1.670586915",
      "cy": "1.454199526",
      "ux": "0.082855954",
      "uy": "0.118415103"
    },
    {
      "cx": "1.484426532",
      "cy": "1.692327296",
      "ux": "0.082855934",
      "uy": "0.118415117"
    },
    {
      "cx": "0.253416827",
      "cy": "1.373528066",
      "ux": "0.075629964",
      "uy": "0.123155813"
    },
    {
      "cx": "0.736055267",
      "cy": "1.054748739",
      "ux": "0.075629969",
      "uy": "0.123155810"
    },
    {
      "cx": "1.223427755",
      "cy": "0.838725827",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "0.740388833",
      "cy": "1.752879463",
      "ux": "0.075629955",
      "uy": "0.123155819"
    },
    {
      "cx": "0.446370324",
      "cy": "1.594235760",
      "ux": "0.075629959",
      "uy": "0.123155816"
    },
    {
      "cx": "0.488643436",
      "cy": "1.204217214",
      "ux": "0.075629964",
      "uy": "0.123155813"
    },
    {
      "cx": "0.479526853",
      "cy": "0.804678742",
      "ux": "0.135317718",
      "uy": "0.050757866"
    },
    {
      "cx": "0.930389587",
      "cy": "1.274608488",
      "ux": "0.075629967",
      "uy": "0.123155812"
    },
    {
      "cx": "1.512472783",
      "cy": "0.837313869",
      "ux": "0.144522481",
      "uy": "-0.000705956"
    },
    {
      "cx": "0.393186393",
      "cy": "0.463578094",
      "ux": "0.135317719",
      "uy": "0.050757865"
    },
    {
      "cx": "0.190078061",
      "cy": "1.004820334",
      "ux": "0.135317718",
      "uy": "0.050757867"
    },
    {
      "cx": "0.904339485",
      "cy": "0.746902404",
      "ux": "0.135317729",
      "uy": "0.050757838"
    },
    {
      "cx": "0.663832170",
      "cy": "0.565066488",
      "ux": "0.135317719",
      "uy": "0.050757863"
    },
    {
      "cx": "1.026843692",
      "cy": "0.484139739",
      "ux": "0.135317729",
      "uy": "0.050757838"
    },
    {
      "cx": "0.976715614",
      "cy": "1.585360067",
      "ux": "0.075629958",
      "uy": "0.123155817"
    },
    {
      "cx": "0.683272095",
      "cy": "1.426363281",
      "ux": "0.075629983",
      "uy": "0.123155802"
    },
    {
      "cx": "1.310706226",
      "cy": "0.246766942",
      "ux": "0.143900059",
      "uy": "0.013417106"
    },
    {
      "cx": "0.208891339",
      "cy": "0.703163007",
      "ux": "0.135317718",
      "uy": "0.050757866"
    },
    {
      "cx": "1.356036093",
      "cy": "0.549026079",
      "ux": "0.144522481",
      "uy": "-0.000705958"
    },
    {
      "cx": "1.822387569",
      "cy": "1.107834627",
      "ux": "0.144522481",
      "uy": "-0.000705953"
    },
    {
      "cx": "1.644998021",
      "cy": "0.530597254",
      "ux": "0.144522481",
      "uy": "-0.000705957"
    },
    {
      "cx": "1.801434704",
      "cy": "0.818885050",
      "ux": "0.144522481",
      "uy": "-0.000705956"
    }
  ]
}

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？