P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 34

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 = 34

目标最大化最小边长

已知最好（未证明）0.275549652来源精度区间：0.2755488933956441–0.2755496526696629（区间内视为追平）本站展示证书 0.275549211论文原值（单位正方形边长下的容器半径）：3.62911+Jonathan Viquerat 于 2026 年 8 月给出 r = 3.62911+ 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 0.075927367926578312；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34 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

0.275549211

最小边长

已追平已知最好

答案来源Jonathan Viquerat

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": "1.788987030",
      "cy": "0.721583403",
      "ux": "0.026187863",
      "uy": "0.135262848"
    },
    {
      "cx": "0.977536701",
      "cy": "1.745578752",
      "ux": "0.025612183",
      "uy": "0.135373033"
    },
    {
      "cx": "1.605240862",
      "cy": "0.462051189",
      "ux": "0.025301326",
      "uy": "0.135431477"
    },
    {
      "cx": "0.818133069",
      "cy": "0.757162525",
      "ux": "0.025612431",
      "uy": "0.135372987"
    },
    {
      "cx": "0.935351712",
      "cy": "1.467144202",
      "ux": "0.025800121",
      "uy": "0.135337341"
    },
    {
      "cx": "0.211121689",
      "cy": "1.278079856",
      "ux": "0.025857141",
      "uy": "0.135326459"
    },
    {
      "cx": "0.697992938",
      "cy": "1.750303466",
      "ux": "0.025612197",
      "uy": "0.135373031"
    },
    {
      "cx": "1.524634429",
      "cy": "1.613720034",
      "ux": "0.025113739",
      "uy": "0.135466388"
    },
    {
      "cx": "1.474640118",
      "cy": "1.342172859",
      "ux": "0.024927064",
      "uy": "0.135500862"
    },
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      "cy": "1.477689553",
      "ux": "0.025612199",
      "uy": "0.135373031"
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      "ux": "0.025852758",
      "uy": "0.135327296"
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      "cy": "1.003063998",
      "ux": "0.025612430",
      "uy": "0.135372987"
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      "ux": "0.025612432",
      "uy": "0.135372986"
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      "cy": "0.487201270",
      "ux": "0.025612422",
      "uy": "0.135372988"
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      "cy": "1.528915553",
      "ux": "0.025612211",
      "uy": "0.135373028"
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    {
      "cx": "0.753506840",
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      "uy": "0.135373031"
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    {
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      "cy": "0.954440611",
      "ux": "0.025612198",
      "uy": "0.135373031"
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    {
      "cx": "0.244076279",
      "cy": "0.710665888",
      "ux": "0.025612863",
      "uy": "0.135372905"
    },
    {
      "cx": "1.749388530",
      "cy": "1.304809278",
      "ux": "0.023648792",
      "uy": "0.135729793"
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    {
      "cx": "0.711535292",
      "cy": "0.216455200",
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      "uy": "0.135372990"
    }
  ]
}

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": 34
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.788987030",
      "cy": "0.721583403",
      "ux": "0.026187863",
      "uy": "0.135262848"
    },
    {
      "cx": "0.977536701",
      "cy": "1.745578752",
      "ux": "0.025612183",
      "uy": "0.135373033"
    },
    {
      "cx": "1.605240862",
      "cy": "0.462051189",
      "ux": "0.025301326",
      "uy": "0.135431477"
    },
    {
      "cx": "0.818133069",
      "cy": "0.757162525",
      "ux": "0.025612431",
      "uy": "0.135372987"
    },
    {
      "cx": "0.935351712",
      "cy": "1.467144202",
      "ux": "0.025800121",
      "uy": "0.135337341"
    },
    {
      "cx": "0.211121689",
      "cy": "1.278079856",
      "ux": "0.025857141",
      "uy": "0.135326459"
    },
    {
      "cx": "0.697992938",
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      "ux": "0.025612197",
      "uy": "0.135373031"
    },
    {
      "cx": "1.524634429",
      "cy": "1.613720034",
      "ux": "0.025113739",
      "uy": "0.135466388"
    },
    {
      "cx": "1.474640118",
      "cy": "1.342172859",
      "ux": "0.024927064",
      "uy": "0.135500862"
    },
    {
      "cx": "1.569232919",
      "cy": "1.044425136",
      "ux": "0.011294463",
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      "uy": "0.135729793"
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    {
      "cx": "0.711535292",
      "cy": "0.216455200",
      "ux": "0.025612414",
      "uy": "0.135372990"
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  ]
}

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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