P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 25

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 = 25

目标最大化最小边长

已知最好（未证明）0.315010237来源精度区间：0.3150092455213560–0.3150102378327296（区间内视为追平）本站展示证书 0.315009887论文原值（单位正方形边长下的容器半径）：3.17450+Haowei Lin 于 2026 年 7 月给出 r = 3.17450+ 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 0.099231229510207204；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25 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

0.315009887

最小边长

已追平已知最好

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": "0.368461615",
      "cy": "0.463124696",
      "ux": "0.119853191",
      "uy": "0.102191096"
    },
    {
      "cx": "1.705016220",
      "cy": "1.397675190",
      "ux": "0.148489432",
      "uy": "0.052523290"
    },
    {
      "cx": "1.008227890",
      "cy": "1.156152531",
      "ux": "0.029064578",
      "uy": "0.154800058"
    },
    {
      "cx": "0.711216166",
      "cy": "0.246901056",
      "ux": "0.025785406",
      "uy": "0.155379923"
    },
    {
      "cx": "0.826530257",
      "cy": "0.866400807",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "1.085238072",
      "cy": "1.810346190",
      "ux": "0.157475824",
      "uy": "0.003028605"
    },
    {
      "cx": "0.231561120",
      "cy": "1.251801976",
      "ux": "0.025785374",
      "uy": "0.155379928"
    },
    {
      "cx": "1.477175229",
      "cy": "0.938764597",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "0.199737048",
      "cy": "0.937765112",
      "ux": "0.025785380",
      "uy": "0.155379927"
    },
    {
      "cx": "1.374033691",
      "cy": "0.317244852",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "1.064359458",
      "cy": "1.466127747",
      "ux": "0.029064568",
      "uy": "0.154800059"
    },
    {
      "cx": "1.827065547",
      "cy": "1.069889571",
      "ux": "0.156945585",
      "uy": "0.013262382"
    },
    {
      "cx": "1.425604461",
      "cy": "0.628004727",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "0.735164650",
      "cy": "1.426736589",
      "ux": "0.025785365",
      "uy": "0.155379929"
    },
    {
      "cx": "1.353032361",
      "cy": "1.278684248",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "1.136845478",
      "cy": "0.812150552",
      "ux": "0.025785382",
      "uy": "0.155379927"
    },
    {
      "cx": "0.774959490",
      "cy": "0.555640924",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "1.085274704",
      "cy": "0.501390671",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "1.021531406",
      "cy": "0.192650820",
      "ux": "0.025785392",
      "uy": "0.155379925"
    },
    {
      "cx": "1.403878464",
      "cy": "1.668869039",
      "ux": "0.157457101",
      "uy": "0.003881861"
    },
    {
      "cx": "1.761364351",
      "cy": "0.727081387",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "0.552210002",
      "cy": "1.137552785",
      "ux": "0.025785395",
      "uy": "0.155379924"
    },
    {
      "cx": "0.500543244",
      "cy": "0.826214551",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "0.434225773",
      "cy": "1.537487753",
      "ux": "0.025785372",
      "uy": "0.155379928"
    },
    {
      "cx": "0.771011570",
      "cy": "1.766582250",
      "ux": "0.157475824",
      "uy": "0.003028614"
    }
  ]
}

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": 25
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.368461615",
      "cy": "0.463124696",
      "ux": "0.119853191",
      "uy": "0.102191096"
    },
    {
      "cx": "1.705016220",
      "cy": "1.397675190",
      "ux": "0.148489432",
      "uy": "0.052523290"
    },
    {
      "cx": "1.008227890",
      "cy": "1.156152531",
      "ux": "0.029064578",
      "uy": "0.154800058"
    },
    {
      "cx": "0.711216166",
      "cy": "0.246901056",
      "ux": "0.025785406",
      "uy": "0.155379923"
    },
    {
      "cx": "0.826530257",
      "cy": "0.866400807",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "1.085238072",
      "cy": "1.810346190",
      "ux": "0.157475824",
      "uy": "0.003028605"
    },
    {
      "cx": "0.231561120",
      "cy": "1.251801976",
      "ux": "0.025785374",
      "uy": "0.155379928"
    },
    {
      "cx": "1.477175229",
      "cy": "0.938764597",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "0.199737048",
      "cy": "0.937765112",
      "ux": "0.025785380",
      "uy": "0.155379927"
    },
    {
      "cx": "1.374033691",
      "cy": "0.317244852",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "1.064359458",
      "cy": "1.466127747",
      "ux": "0.029064568",
      "uy": "0.154800059"
    },
    {
      "cx": "1.827065547",
      "cy": "1.069889571",
      "ux": "0.156945585",
      "uy": "0.013262382"
    },
    {
      "cx": "1.425604461",
      "cy": "0.628004727",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "0.735164650",
      "cy": "1.426736589",
      "ux": "0.025785365",
      "uy": "0.155379929"
    },
    {
      "cx": "1.353032361",
      "cy": "1.278684248",
      "ux": "0.025785384",
      "uy": "0.155379926"
    },
    {
      "cx": "1.136845478",
      "cy": "0.812150552",
      "ux": "0.025785382",
      "uy": "0.155379927"
    },
    {
      "cx": "0.774959490",
      "cy": "0.555640924",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "1.085274704",
      "cy": "0.501390671",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "1.021531406",
      "cy": "0.192650820",
      "ux": "0.025785392",
      "uy": "0.155379925"
    },
    {
      "cx": "1.403878464",
      "cy": "1.668869039",
      "ux": "0.157457101",
      "uy": "0.003881861"
    },
    {
      "cx": "1.761364351",
      "cy": "0.727081387",
      "ux": "0.025785383",
      "uy": "0.155379926"
    },
    {
      "cx": "0.552210002",
      "cy": "1.137552785",
      "ux": "0.025785395",
      "uy": "0.155379924"
    },
    {
      "cx": "0.500543244",
      "cy": "0.826214551",
      "ux": "0.025785379",
      "uy": "0.155379927"
    },
    {
      "cx": "0.434225773",
      "cy": "1.537487753",
      "ux": "0.025785372",
      "uy": "0.155379928"
    },
    {
      "cx": "0.771011570",
      "cy": "1.766582250",
      "ux": "0.157475824",
      "uy": "0.003028614"
    }
  ]
}

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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