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正方形装入等边三角形 · n = 29

把 n 个边长为 1 的正方形装入等边三角形，允许接触但内部不能重叠，最小化容器三角形边长。

子题n = 29

目标最小化容器三角形边长

已知最好（未证明）9.154来源精度区间：9.154–9.155（区间内视为追平）本站展示证书 9.154700661Erich Friedman 编录；Found by Erich Friedman in 1997. 原式：s = 8 + 2 / √3 = 9.154+。

严格定义

容器编辑器及证书统一缩放到顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形；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。
目标容器三角形边长 = 1/(2 sin(π/4) 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

9.154700661

容器三角形边长

已追平已知最好

答案来源Erich Friedman

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

顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形

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": "-0.414213562",
      "x": "0.117682743",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.226916245",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.773083755",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.882317257",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.226916245",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.773083755",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.281532996",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.390766498",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.609233502",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.718467004",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.390766498",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.609233502",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.600784261"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.600784261"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.710017763"
    }
  ],
  "radius": "0.077239749"
}

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": 29,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.117682743",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.226916245",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.773083755",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.882317257",
      "y": "0.054616751"
    },
    {
      "turn": "-0.414213562",
      "x": "0.226916245",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.773083755",
      "y": "0.163850253"
    },
    {
      "turn": "-0.414213562",
      "x": "0.281532996",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.390766498",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.609233502",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.718467004",
      "y": "0.273083755"
    },
    {
      "turn": "-0.414213562",
      "x": "0.336149747",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.663850253",
      "y": "0.382317257"
    },
    {
      "turn": "-0.414213562",
      "x": "0.390766498",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.609233502",
      "y": "0.491550759"
    },
    {
      "turn": "-0.414213562",
      "x": "0.445383249",
      "y": "0.600784261"
    },
    {
      "turn": "-0.414213562",
      "x": "0.554616751",
      "y": "0.600784261"
    },
    {
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.710017763"
    }
  ],
  "radius": "0.077239749"
}

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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