P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 24

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 = 24

目标最大化最小边长

已知最好（未证明）0.321605454来源精度区间：0.3216044201311502–0.3216054544285072（区间内视为追平）本站展示证书 0.321577701论文原值（单位正方形边长下的容器半径）：3.10940+Milan Kovacic 于 2026 年 5 月给出 r = 3.10940+ 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 0.103412218420073488；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24 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

0.321577701

最小边长

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": "1.473650212",
      "cy": "1.612334197",
      "ux": "0.160786757",
      "uy": "-0.000820710"
    },
    {
      "cx": "0.416769884",
      "cy": "1.505560020",
      "ux": "0.160188397",
      "uy": "0.013882807"
    },
    {
      "cx": "0.537579157",
      "cy": "1.141194348",
      "ux": "0.160786756",
      "uy": "-0.000820856"
    },
    {
      "cx": "0.854767970",
      "cy": "0.817991518",
      "ux": "0.160786764",
      "uy": "-0.000819271"
    },
    {
      "cx": "1.152968843",
      "cy": "1.788758945",
      "ux": "0.160786757",
      "uy": "-0.000820708"
    },
    {
      "cx": "1.693337639",
      "cy": "0.640723703",
      "ux": "-0.000000249",
      "uy": "0.160788851"
    },
    {
      "cx": "0.425342833",
      "cy": "0.475699178",
      "ux": "0.046162351",
      "uy": "0.154019778"
    },
    {
      "cx": "0.692332353",
      "cy": "0.246411782",
      "ux": "0.046162355",
      "uy": "0.154019777"
    },
    {
      "cx": "0.859152680",
      "cy": "1.139552647",
      "ux": "0.160786756",
      "uy": "-0.000820854"
    },
    {
      "cx": "1.501334634",
      "cy": "0.965796646",
      "ux": "0.003534129",
      "uy": "0.160750007"
    },
    {
      "cx": "1.377211457",
      "cy": "0.315755397",
      "ux": "-0.005425336",
      "uy": "0.160697295"
    },
    {
      "cx": "1.734609780",
      "cy": "1.296963927",
      "ux": "0.160351447",
      "uy": "0.011851932"
    },
    {
      "cx": "1.823002564",
      "cy": "0.966365460",
      "ux": "0.160750007",
      "uy": "-0.003534133"
    },
    {
      "cx": "1.398456574",
      "cy": "1.289809977",
      "ux": "0.160756384",
      "uy": "-0.003231056"
    },
    {
      "cx": "1.054983913",
      "cy": "0.175631660",
      "ux": "0.160432393",
      "uy": "0.010700557"
    },
    {
      "cx": "0.216138343",
      "cy": "1.168830781",
      "ux": "0.160786756",
      "uy": "-0.000820858"
    },
    {
      "cx": "1.369250684",
      "cy": "0.645254413",
      "ux": "0.160768963",
      "uy": "-0.002528878"
    },
    {
      "cx": "1.044127380",
      "cy": "0.497199761",
      "ux": "0.160432393",
      "uy": "0.010700560"
    },
    {
      "cx": "1.179830065",
      "cy": "0.972658187",
      "ux": "0.160750007",
      "uy": "-0.003534128"
    },
    {
      "cx": "0.752284381",
      "cy": "1.461680141",
      "ux": "0.160786756",
      "uy": "-0.000820850"
    },
    {
      "cx": "0.831356826",
      "cy": "1.782858502",
      "ux": "0.160786757",
      "uy": "-0.000820711"
    },
    {
      "cx": "0.211753379",
      "cy": "0.847269672",
      "ux": "0.160786764",
      "uy": "-0.000819265"
    },
    {
      "cx": "0.533194450",
      "cy": "0.819633218",
      "ux": "0.160786764",
      "uy": "-0.000819271"
    },
    {
      "cx": "1.075038659",
      "cy": "1.460032292",
      "ux": "0.160786754",
      "uy": "-0.000821184"
    }
  ]
}

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": 24
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.473650212",
      "cy": "1.612334197",
      "ux": "0.160786757",
      "uy": "-0.000820710"
    },
    {
      "cx": "0.416769884",
      "cy": "1.505560020",
      "ux": "0.160188397",
      "uy": "0.013882807"
    },
    {
      "cx": "0.537579157",
      "cy": "1.141194348",
      "ux": "0.160786756",
      "uy": "-0.000820856"
    },
    {
      "cx": "0.854767970",
      "cy": "0.817991518",
      "ux": "0.160786764",
      "uy": "-0.000819271"
    },
    {
      "cx": "1.152968843",
      "cy": "1.788758945",
      "ux": "0.160786757",
      "uy": "-0.000820708"
    },
    {
      "cx": "1.693337639",
      "cy": "0.640723703",
      "ux": "-0.000000249",
      "uy": "0.160788851"
    },
    {
      "cx": "0.425342833",
      "cy": "0.475699178",
      "ux": "0.046162351",
      "uy": "0.154019778"
    },
    {
      "cx": "0.692332353",
      "cy": "0.246411782",
      "ux": "0.046162355",
      "uy": "0.154019777"
    },
    {
      "cx": "0.859152680",
      "cy": "1.139552647",
      "ux": "0.160786756",
      "uy": "-0.000820854"
    },
    {
      "cx": "1.501334634",
      "cy": "0.965796646",
      "ux": "0.003534129",
      "uy": "0.160750007"
    },
    {
      "cx": "1.377211457",
      "cy": "0.315755397",
      "ux": "-0.005425336",
      "uy": "0.160697295"
    },
    {
      "cx": "1.734609780",
      "cy": "1.296963927",
      "ux": "0.160351447",
      "uy": "0.011851932"
    },
    {
      "cx": "1.823002564",
      "cy": "0.966365460",
      "ux": "0.160750007",
      "uy": "-0.003534133"
    },
    {
      "cx": "1.398456574",
      "cy": "1.289809977",
      "ux": "0.160756384",
      "uy": "-0.003231056"
    },
    {
      "cx": "1.054983913",
      "cy": "0.175631660",
      "ux": "0.160432393",
      "uy": "0.010700557"
    },
    {
      "cx": "0.216138343",
      "cy": "1.168830781",
      "ux": "0.160786756",
      "uy": "-0.000820858"
    },
    {
      "cx": "1.369250684",
      "cy": "0.645254413",
      "ux": "0.160768963",
      "uy": "-0.002528878"
    },
    {
      "cx": "1.044127380",
      "cy": "0.497199761",
      "ux": "0.160432393",
      "uy": "0.010700560"
    },
    {
      "cx": "1.179830065",
      "cy": "0.972658187",
      "ux": "0.160750007",
      "uy": "-0.003534128"
    },
    {
      "cx": "0.752284381",
      "cy": "1.461680141",
      "ux": "0.160786756",
      "uy": "-0.000820850"
    },
    {
      "cx": "0.831356826",
      "cy": "1.782858502",
      "ux": "0.160786757",
      "uy": "-0.000820711"
    },
    {
      "cx": "0.211753379",
      "cy": "0.847269672",
      "ux": "0.160786764",
      "uy": "-0.000819265"
    },
    {
      "cx": "0.533194450",
      "cy": "0.819633218",
      "ux": "0.160786764",
      "uy": "-0.000819271"
    },
    {
      "cx": "1.075038659",
      "cy": "1.460032292",
      "ux": "0.160786754",
      "uy": "-0.000821184"
    }
  ]
}

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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