P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 22

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 = 22

目标最大化最小边长

已知最好（未证明）0.329772885来源精度区间：0.3297717979158422–0.3297728854138156（区间内视为追平）本站展示证书 0.329771769论文原值（单位正方形边长下的容器半径）：3.03239+Haowei Lin 于 2026 年 7 月给出 r = 3.03239+ 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 0.108749419743005248；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22 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

0.329771769

最小边长

答案来源Haowei Lin

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": "0.449828027",
      "cy": "1.146977502",
      "ux": "0.143726263",
      "uy": "0.080809136"
    },
    {
      "cx": "0.639626582",
      "cy": "1.729570335",
      "ux": "0.143726265",
      "uy": "0.080809133"
    },
    {
      "cx": "0.393850824",
      "cy": "1.493826083",
      "ux": "0.143726264",
      "uy": "0.080809133"
    },
    {
      "cx": "0.829163489",
      "cy": "1.381608871",
      "ux": "0.064233708",
      "uy": "0.151859757"
    },
    {
      "cx": "1.452868455",
      "cy": "1.153230043",
      "ux": "0.060206586",
      "uy": "0.153500886"
    },
    {
      "cx": "1.329428855",
      "cy": "1.561195070",
      "ux": "0.060206357",
      "uy": "0.153500976"
    },
    {
      "cx": "1.063310079",
      "cy": "1.785842965",
      "ux": "0.060206357",
      "uy": "0.153500976"
    },
    {
      "cx": "1.634622899",
      "cy": "1.436172735",
      "ux": "0.060206356",
      "uy": "0.153500976"
    },
    {
      "cx": "1.138899299",
      "cy": "0.912671622",
      "ux": "0.063871537",
      "uy": "0.152012440"
    },
    {
      "cx": "0.757238757",
      "cy": "0.329246626",
      "ux": "0.054654715",
      "uy": "0.155564190"
    },
    {
      "cx": "0.827773901",
      "cy": "1.023515817",
      "ux": "0.063871539",
      "uy": "0.152012439"
    },
    {
      "cx": "1.757293921",
      "cy": "0.681805493",
      "ux": "0.063871527",
      "uy": "0.152012444"
    },
    {
      "cx": "1.143378210",
      "cy": "1.270606264",
      "ux": "0.061328039",
      "uy": "0.153056286"
    },
    {
      "cx": "1.447892641",
      "cy": "0.796753156",
      "ux": "0.063871538",
      "uy": "0.152012440"
    },
    {
      "cx": "0.886950203",
      "cy": "0.643023199",
      "ux": "0.052560133",
      "uy": "0.156284316"
    },
    {
      "cx": "1.072352560",
      "cy": "0.224562683",
      "ux": "0.052534370",
      "uy": "0.156292978"
    },
    {
      "cx": "0.450083600",
      "cy": "0.451402172",
      "ux": "0.054220886",
      "uy": "0.155715929"
    },
    {
      "cx": "1.506703214",
      "cy": "0.416772312",
      "ux": "0.052560161",
      "uy": "0.156284307"
    },
    {
      "cx": "0.194168620",
      "cy": "0.840696445",
      "ux": "0.161755418",
      "uy": "0.031977178"
    },
    {
      "cx": "0.577772367",
      "cy": "0.758226094",
      "ux": "0.052560131",
      "uy": "0.156284317"
    },
    {
      "cx": "1.197298891",
      "cy": "0.531301684",
      "ux": "0.052560160",
      "uy": "0.156284307"
    },
    {
      "cx": "1.789452037",
      "cy": "1.039745411",
      "ux": "0.035985123",
      "uy": "0.160911236"
    }
  ]
}

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": 22
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.449828027",
      "cy": "1.146977502",
      "ux": "0.143726263",
      "uy": "0.080809136"
    },
    {
      "cx": "0.639626582",
      "cy": "1.729570335",
      "ux": "0.143726265",
      "uy": "0.080809133"
    },
    {
      "cx": "0.393850824",
      "cy": "1.493826083",
      "ux": "0.143726264",
      "uy": "0.080809133"
    },
    {
      "cx": "0.829163489",
      "cy": "1.381608871",
      "ux": "0.064233708",
      "uy": "0.151859757"
    },
    {
      "cx": "1.452868455",
      "cy": "1.153230043",
      "ux": "0.060206586",
      "uy": "0.153500886"
    },
    {
      "cx": "1.329428855",
      "cy": "1.561195070",
      "ux": "0.060206357",
      "uy": "0.153500976"
    },
    {
      "cx": "1.063310079",
      "cy": "1.785842965",
      "ux": "0.060206357",
      "uy": "0.153500976"
    },
    {
      "cx": "1.634622899",
      "cy": "1.436172735",
      "ux": "0.060206356",
      "uy": "0.153500976"
    },
    {
      "cx": "1.138899299",
      "cy": "0.912671622",
      "ux": "0.063871537",
      "uy": "0.152012440"
    },
    {
      "cx": "0.757238757",
      "cy": "0.329246626",
      "ux": "0.054654715",
      "uy": "0.155564190"
    },
    {
      "cx": "0.827773901",
      "cy": "1.023515817",
      "ux": "0.063871539",
      "uy": "0.152012439"
    },
    {
      "cx": "1.757293921",
      "cy": "0.681805493",
      "ux": "0.063871527",
      "uy": "0.152012444"
    },
    {
      "cx": "1.143378210",
      "cy": "1.270606264",
      "ux": "0.061328039",
      "uy": "0.153056286"
    },
    {
      "cx": "1.447892641",
      "cy": "0.796753156",
      "ux": "0.063871538",
      "uy": "0.152012440"
    },
    {
      "cx": "0.886950203",
      "cy": "0.643023199",
      "ux": "0.052560133",
      "uy": "0.156284316"
    },
    {
      "cx": "1.072352560",
      "cy": "0.224562683",
      "ux": "0.052534370",
      "uy": "0.156292978"
    },
    {
      "cx": "0.450083600",
      "cy": "0.451402172",
      "ux": "0.054220886",
      "uy": "0.155715929"
    },
    {
      "cx": "1.506703214",
      "cy": "0.416772312",
      "ux": "0.052560161",
      "uy": "0.156284307"
    },
    {
      "cx": "0.194168620",
      "cy": "0.840696445",
      "ux": "0.161755418",
      "uy": "0.031977178"
    },
    {
      "cx": "0.577772367",
      "cy": "0.758226094",
      "ux": "0.052560131",
      "uy": "0.156284317"
    },
    {
      "cx": "1.197298891",
      "cy": "0.531301684",
      "ux": "0.052560160",
      "uy": "0.156284307"
    },
    {
      "cx": "1.789452037",
      "cy": "1.039745411",
      "ux": "0.035985123",
      "uy": "0.160911236"
    }
  ]
}

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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