P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 17

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 = 17

目标最大化最小边长

已知最好（未证明）0.373570625来源精度区间：0.3735692298496757–0.3735706253945840（区间内视为追平）本站展示证书 0.373569949论文原值（单位正方形边长下的容器半径）：2.67687+Milan Kovacic 于 2026 年 5 月给出 r = 2.67687+ 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 0.139554506980699732；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17 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

0.373569949

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": "1.226958937",
      "cy": "0.270468569",
      "ux": "0.186504578",
      "uy": "0.010230797"
    },
    {
      "cx": "0.957503999",
      "cy": "1.779972864",
      "ux": "0.186600182",
      "uy": "-0.008306553"
    },
    {
      "cx": "0.616803397",
      "cy": "0.853425531",
      "ux": "0.186600182",
      "uy": "-0.008306561"
    },
    {
      "cx": "1.034382299",
      "cy": "1.385735795",
      "ux": "0.186728624",
      "uy": "0.004587808"
    },
    {
      "cx": "1.590830298",
      "cy": "0.559046560",
      "ux": "0.186728625",
      "uy": "0.004587763"
    },
    {
      "cx": "0.836620534",
      "cy": "0.244681856",
      "ux": "0.186600183",
      "uy": "-0.008306548"
    },
    {
      "cx": "1.683876164",
      "cy": "1.308698067",
      "ux": "0.186728625",
      "uy": "0.004587758"
    },
    {
      "cx": "1.029057434",
      "cy": "1.011922241",
      "ux": "0.186728624",
      "uy": "0.004587810"
    },
    {
      "cx": "0.993580001",
      "cy": "0.637367857",
      "ux": "0.186728624",
      "uy": "0.004587809"
    },
    {
      "cx": "1.778255497",
      "cy": "0.937334163",
      "ux": "0.186728625",
      "uy": "0.004587760"
    },
    {
      "cx": "1.404798201",
      "cy": "0.928158636",
      "ux": "0.186728625",
      "uy": "0.004587767"
    },
    {
      "cx": "0.243603013",
      "cy": "0.870038641",
      "ux": "0.186600182",
      "uy": "-0.008306560"
    },
    {
      "cx": "0.575692109",
      "cy": "1.603135466",
      "ux": "0.186600183",
      "uy": "-0.008306552"
    },
    {
      "cx": "0.473417074",
      "cy": "0.485868460",
      "ux": "0.186600183",
      "uy": "-0.008306547"
    },
    {
      "cx": "1.392003827",
      "cy": "1.692342065",
      "ux": "0.092029877",
      "uy": "0.162539622"
    },
    {
      "cx": "0.278982995",
      "cy": "1.242403614",
      "ux": "0.186600182",
      "uy": "-0.008306560"
    },
    {
      "cx": "0.652183367",
      "cy": "1.225790490",
      "ux": "0.186600182",
      "uy": "-0.008306561"
    }
  ]
}

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": 17
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.226958937",
      "cy": "0.270468569",
      "ux": "0.186504578",
      "uy": "0.010230797"
    },
    {
      "cx": "0.957503999",
      "cy": "1.779972864",
      "ux": "0.186600182",
      "uy": "-0.008306553"
    },
    {
      "cx": "0.616803397",
      "cy": "0.853425531",
      "ux": "0.186600182",
      "uy": "-0.008306561"
    },
    {
      "cx": "1.034382299",
      "cy": "1.385735795",
      "ux": "0.186728624",
      "uy": "0.004587808"
    },
    {
      "cx": "1.590830298",
      "cy": "0.559046560",
      "ux": "0.186728625",
      "uy": "0.004587763"
    },
    {
      "cx": "0.836620534",
      "cy": "0.244681856",
      "ux": "0.186600183",
      "uy": "-0.008306548"
    },
    {
      "cx": "1.683876164",
      "cy": "1.308698067",
      "ux": "0.186728625",
      "uy": "0.004587758"
    },
    {
      "cx": "1.029057434",
      "cy": "1.011922241",
      "ux": "0.186728624",
      "uy": "0.004587810"
    },
    {
      "cx": "0.993580001",
      "cy": "0.637367857",
      "ux": "0.186728624",
      "uy": "0.004587809"
    },
    {
      "cx": "1.778255497",
      "cy": "0.937334163",
      "ux": "0.186728625",
      "uy": "0.004587760"
    },
    {
      "cx": "1.404798201",
      "cy": "0.928158636",
      "ux": "0.186728625",
      "uy": "0.004587767"
    },
    {
      "cx": "0.243603013",
      "cy": "0.870038641",
      "ux": "0.186600182",
      "uy": "-0.008306560"
    },
    {
      "cx": "0.575692109",
      "cy": "1.603135466",
      "ux": "0.186600183",
      "uy": "-0.008306552"
    },
    {
      "cx": "0.473417074",
      "cy": "0.485868460",
      "ux": "0.186600183",
      "uy": "-0.008306547"
    },
    {
      "cx": "1.392003827",
      "cy": "1.692342065",
      "ux": "0.092029877",
      "uy": "0.162539622"
    },
    {
      "cx": "0.278982995",
      "cy": "1.242403614",
      "ux": "0.186600182",
      "uy": "-0.008306560"
    },
    {
      "cx": "0.652183367",
      "cy": "1.225790490",
      "ux": "0.186600182",
      "uy": "-0.008306561"
    }
  ]
}

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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