P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 28

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 = 28

目标最大化最小边长

已知最好（未证明）0.300167793来源精度区间：0.3001668927923925–0.3001677937967324（区间内视为追平）本站展示证书 0.300166913论文原值（单位正方形边长下的容器半径）：3.33147+Milan Kovacic 于 2026 年 5 月给出 r = 3.33147+ 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 0.09010017601131044；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28 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

0.300166913

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": "0.885793076",
      "cy": "1.813270299",
      "ux": "0.001512134",
      "uy": "0.150075840"
    },
    {
      "cx": "0.303687952",
      "cy": "0.618407947",
      "ux": "0.150075835",
      "uy": "-0.001512630"
    },
    {
      "cx": "0.598785356",
      "cy": "0.315251376",
      "ux": "0.150075834",
      "uy": "-0.001512639"
    },
    {
      "cx": "0.220012037",
      "cy": "1.219615865",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "0.778609026",
      "cy": "0.913803325",
      "ux": "0.150075834",
      "uy": "-0.001512638"
    },
    {
      "cx": "1.510317218",
      "cy": "1.645153777",
      "ux": "0.076171792",
      "uy": "0.129317061"
    },
    {
      "cx": "1.347797797",
      "cy": "1.392513484",
      "ux": "0.076171790",
      "uy": "0.129317062"
    },
    {
      "cx": "1.699594268",
      "cy": "1.370697165",
      "ux": "0.074873273",
      "uy": "0.130073199"
    },
    {
      "cx": "1.555167203",
      "cy": "0.411791320",
      "ux": "0.076171757",
      "uy": "0.129317081"
    },
    {
      "cx": "1.794294298",
      "cy": "0.967676586",
      "ux": "0.076171758",
      "uy": "0.129317081"
    },
    {
      "cx": "1.441373053",
      "cy": "0.809690524",
      "ux": "0.083803211",
      "uy": "0.124507293"
    },
    {
      "cx": "0.478347827",
      "cy": "0.916829785",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "1.744926074",
      "cy": "0.648386743",
      "ux": "0.076171758",
      "uy": "0.129317081"
    },
    {
      "cx": "1.218828960",
      "cy": "1.809534908",
      "ux": "0.039164089",
      "uy": "0.144883465"
    },
    {
      "cx": "1.532756331",
      "cy": "1.115090547",
      "ux": "0.076171787",
      "uy": "0.129317064"
    },
    {
      "cx": "0.903027967",
      "cy": "0.516764658",
      "ux": "0.150075834",
      "uy": "-0.001512634"
    },
    {
      "cx": "0.405764138",
      "cy": "1.517925856",
      "ux": "0.150075834",
      "uy": "-0.001512668"
    },
    {
      "cx": "1.006067569",
      "cy": "1.511875657",
      "ux": "0.150075834",
      "uy": "-0.001512672"
    },
    {
      "cx": "1.119503701",
      "cy": "1.114947559",
      "ux": "0.150075834",
      "uy": "-0.001512667"
    },
    {
      "cx": "0.897619681",
      "cy": "0.181499954",
      "ux": "0.150075836",
      "uy": "-0.001512463"
    },
    {
      "cx": "1.203179707",
      "cy": "0.513739656",
      "ux": "0.150075834",
      "uy": "-0.001512634"
    },
    {
      "cx": "1.198125489",
      "cy": "0.213608175",
      "ux": "0.150075836",
      "uy": "-0.001512463"
    },
    {
      "cx": "0.705915851",
      "cy": "1.514900559",
      "ux": "0.150075834",
      "uy": "-0.001512667"
    },
    {
      "cx": "0.520163758",
      "cy": "1.216590535",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "1.077894632",
      "cy": "0.815184698",
      "ux": "0.150075834",
      "uy": "-0.001512659"
    },
    {
      "cx": "0.820315482",
      "cy": "1.213565235",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "0.178150263",
      "cy": "0.919855542",
      "ux": "0.150075834",
      "uy": "-0.001512654"
    },
    {
      "cx": "0.603839708",
      "cy": "0.615382638",
      "ux": "0.150075834",
      "uy": "-0.001512640"
    }
  ]
}

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": 28
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.885793076",
      "cy": "1.813270299",
      "ux": "0.001512134",
      "uy": "0.150075840"
    },
    {
      "cx": "0.303687952",
      "cy": "0.618407947",
      "ux": "0.150075835",
      "uy": "-0.001512630"
    },
    {
      "cx": "0.598785356",
      "cy": "0.315251376",
      "ux": "0.150075834",
      "uy": "-0.001512639"
    },
    {
      "cx": "0.220012037",
      "cy": "1.219615865",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "0.778609026",
      "cy": "0.913803325",
      "ux": "0.150075834",
      "uy": "-0.001512638"
    },
    {
      "cx": "1.510317218",
      "cy": "1.645153777",
      "ux": "0.076171792",
      "uy": "0.129317061"
    },
    {
      "cx": "1.347797797",
      "cy": "1.392513484",
      "ux": "0.076171790",
      "uy": "0.129317062"
    },
    {
      "cx": "1.699594268",
      "cy": "1.370697165",
      "ux": "0.074873273",
      "uy": "0.130073199"
    },
    {
      "cx": "1.555167203",
      "cy": "0.411791320",
      "ux": "0.076171757",
      "uy": "0.129317081"
    },
    {
      "cx": "1.794294298",
      "cy": "0.967676586",
      "ux": "0.076171758",
      "uy": "0.129317081"
    },
    {
      "cx": "1.441373053",
      "cy": "0.809690524",
      "ux": "0.083803211",
      "uy": "0.124507293"
    },
    {
      "cx": "0.478347827",
      "cy": "0.916829785",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "1.744926074",
      "cy": "0.648386743",
      "ux": "0.076171758",
      "uy": "0.129317081"
    },
    {
      "cx": "1.218828960",
      "cy": "1.809534908",
      "ux": "0.039164089",
      "uy": "0.144883465"
    },
    {
      "cx": "1.532756331",
      "cy": "1.115090547",
      "ux": "0.076171787",
      "uy": "0.129317064"
    },
    {
      "cx": "0.903027967",
      "cy": "0.516764658",
      "ux": "0.150075834",
      "uy": "-0.001512634"
    },
    {
      "cx": "0.405764138",
      "cy": "1.517925856",
      "ux": "0.150075834",
      "uy": "-0.001512668"
    },
    {
      "cx": "1.006067569",
      "cy": "1.511875657",
      "ux": "0.150075834",
      "uy": "-0.001512672"
    },
    {
      "cx": "1.119503701",
      "cy": "1.114947559",
      "ux": "0.150075834",
      "uy": "-0.001512667"
    },
    {
      "cx": "0.897619681",
      "cy": "0.181499954",
      "ux": "0.150075836",
      "uy": "-0.001512463"
    },
    {
      "cx": "1.203179707",
      "cy": "0.513739656",
      "ux": "0.150075834",
      "uy": "-0.001512634"
    },
    {
      "cx": "1.198125489",
      "cy": "0.213608175",
      "ux": "0.150075836",
      "uy": "-0.001512463"
    },
    {
      "cx": "0.705915851",
      "cy": "1.514900559",
      "ux": "0.150075834",
      "uy": "-0.001512667"
    },
    {
      "cx": "0.520163758",
      "cy": "1.216590535",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "1.077894632",
      "cy": "0.815184698",
      "ux": "0.150075834",
      "uy": "-0.001512659"
    },
    {
      "cx": "0.820315482",
      "cy": "1.213565235",
      "ux": "0.150075834",
      "uy": "-0.001512666"
    },
    {
      "cx": "0.178150263",
      "cy": "0.919855542",
      "ux": "0.150075834",
      "uy": "-0.001512654"
    },
    {
      "cx": "0.603839708",
      "cy": "0.615382638",
      "ux": "0.150075834",
      "uy": "-0.001512640"
    }
  ]
}

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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