P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 27

把 n 个边长为 1 的等边三角形装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 27

目标最小化容器正方形边长

已知最好（未证明）3.6726来源精度区间：3.6726–3.6727（区间内视为追平）本站展示证书 3.672637086Erich Friedman 编录；27. Found by Ian Watson in April 2026. 原式：s = 3.6726+。

严格定义

容器编辑器及证书统一缩放到单位正方形 [0,1]²；仅是坐标约定，分数按文献的单位图形计算。边界接触允许。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。
约束恰好 n 个同样大的图形，完整位于容器内，任意两图形内部不重叠。多边形可独立旋转，不限直角。
精度输入小数精确表示有理数；正多边形的顶点用代数数定义，不截断成小数。可行性验算没有浮点容差。有限小数证书不等于连续最优性证明。
目标容器正方形边长 = 1/(2 sin(π/3) r)，越小越好。页面、排行榜和外部纪录使用相同单位，自动计算，无需手工换算。

放大来摆，然后提交 ↗

已验证构造已验证构造

帮助理解

外部来源与证书

外部目标按原始印刷精度换算，本站坐标证书另存。复现损失不降低文献目标；源表的构造不自动视为最优性证明。

查看来源

当前第一名

3.672637086

容器正方形边长

已追平已知最好

答案来源27. Found by Ian Watson

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

单位正方形 [0,1]²

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.276183091",
      "y": "0.921398404"
    },
    {
      "turn": "0.267949192",
      "x": "0.548467008",
      "y": "0.921398404"
    },
    {
      "turn": "0.267949192",
      "x": "0.820750925",
      "y": "0.921398404"
    },
    {
      "turn": "-0.267949192",
      "x": "0.412325049",
      "y": "0.842796807"
    },
    {
      "turn": "-0.267949192",
      "x": "0.684608967",
      "y": "0.842796807"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.834132480"
    },
    {
      "turn": "-0.411941066",
      "x": "0.888409818",
      "y": "0.720787979"
    },
    {
      "turn": "-0.577350269",
      "x": "0.157203193",
      "y": "0.697990521"
    },
    {
      "turn": "0.000000000",
      "x": "0.314406386",
      "y": "0.651365675"
    },
    {
      "turn": "0.267949192",
      "x": "0.542609375",
      "y": "0.685593614"
    },
    {
      "turn": "-0.267949192",
      "x": "0.653570700",
      "y": "0.563377880"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.561848562"
    },
    {
      "turn": "0.133627905",
      "x": "0.848312488",
      "y": "0.568784531"
    },
    {
      "turn": "-0.577350269",
      "x": "0.393007982",
      "y": "0.515223716"
    },
    {
      "turn": "-0.577350269",
      "x": "0.157203193",
      "y": "0.425706604"
    },
    {
      "turn": "-0.529774025",
      "x": "0.911699891",
      "y": "0.392033450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.314406386",
      "y": "0.379081758"
    },
    {
      "turn": "0.267949192",
      "x": "0.653570700",
      "y": "0.406174687"
    },
    {
      "turn": "-0.267949192",
      "x": "0.525030541",
      "y": "0.314406386"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.289564645"
    },
    {
      "turn": "0.036432739",
      "x": "0.823399782",
      "y": "0.260122545"
    },
    {
      "turn": "-0.529774025",
      "x": "0.911699891",
      "y": "0.130061272"
    },
    {
      "turn": "0.267949192",
      "x": "0.278831947",
      "y": "0.157203193"
    },
    {
      "turn": "0.267949192",
      "x": "0.551115864",
      "y": "0.157203193"
    },
    {
      "turn": "-0.267949192",
      "x": "0.414973906",
      "y": "0.078601596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.687257823",
      "y": "0.078601596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.142689988",
      "y": "0.078601596"
    }
  ],
  "radius": "0.157203191"
}

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": 27,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.276183091",
      "y": "0.921398404"
    },
    {
      "turn": "0.267949192",
      "x": "0.548467008",
      "y": "0.921398404"
    },
    {
      "turn": "0.267949192",
      "x": "0.820750925",
      "y": "0.921398404"
    },
    {
      "turn": "-0.267949192",
      "x": "0.412325049",
      "y": "0.842796807"
    },
    {
      "turn": "-0.267949192",
      "x": "0.684608967",
      "y": "0.842796807"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.834132480"
    },
    {
      "turn": "-0.411941066",
      "x": "0.888409818",
      "y": "0.720787979"
    },
    {
      "turn": "-0.577350269",
      "x": "0.157203193",
      "y": "0.697990521"
    },
    {
      "turn": "0.000000000",
      "x": "0.314406386",
      "y": "0.651365675"
    },
    {
      "turn": "0.267949192",
      "x": "0.542609375",
      "y": "0.685593614"
    },
    {
      "turn": "-0.267949192",
      "x": "0.653570700",
      "y": "0.563377880"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.561848562"
    },
    {
      "turn": "0.133627905",
      "x": "0.848312488",
      "y": "0.568784531"
    },
    {
      "turn": "-0.577350269",
      "x": "0.393007982",
      "y": "0.515223716"
    },
    {
      "turn": "-0.577350269",
      "x": "0.157203193",
      "y": "0.425706604"
    },
    {
      "turn": "-0.529774025",
      "x": "0.911699891",
      "y": "0.392033450"
    },
    {
      "turn": "-0.000000000",
      "x": "0.314406386",
      "y": "0.379081758"
    },
    {
      "turn": "0.267949192",
      "x": "0.653570700",
      "y": "0.406174687"
    },
    {
      "turn": "-0.267949192",
      "x": "0.525030541",
      "y": "0.314406386"
    },
    {
      "turn": "0.000000000",
      "x": "0.078601596",
      "y": "0.289564645"
    },
    {
      "turn": "0.036432739",
      "x": "0.823399782",
      "y": "0.260122545"
    },
    {
      "turn": "-0.529774025",
      "x": "0.911699891",
      "y": "0.130061272"
    },
    {
      "turn": "0.267949192",
      "x": "0.278831947",
      "y": "0.157203193"
    },
    {
      "turn": "0.267949192",
      "x": "0.551115864",
      "y": "0.157203193"
    },
    {
      "turn": "-0.267949192",
      "x": "0.414973906",
      "y": "0.078601596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.687257823",
      "y": "0.078601596"
    },
    {
      "turn": "-0.267949192",
      "x": "0.142689988",
      "y": "0.078601596"
    }
  ],
  "radius": "0.157203191"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。 · 验证器 v1.0.0

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