P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 17

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 = 17
目标最小化 容器三角形边长
已知最好(未证明)7.301来源精度区间:7.301–7.302(区间内视为追平)本站展示证书 7.301221441Erich Friedman 编录;Found by David W. Cantrell in July 2002. 原式:s = 7.301+。

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
正方形装入等边三角形 n = 17 的当前纪录构型,7.301221441
当前第一名

7.301221441

容器三角形边长

已追平已知最好
答案来源David W. Cantrell
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": "0.131652498",
      "x": "0.485528115",
      "y": "0.703995978"
    },
    {
      "turn": "0.131652498",
      "x": "0.417046422",
      "y": "0.585382206"
    },
    {
      "turn": "0.131652498",
      "x": "0.564603964",
      "y": "0.567032591"
    },
    {
      "turn": "0.131652498",
      "x": "0.348564729",
      "y": "0.466768433"
    },
    {
      "turn": "0.131652498",
      "x": "0.497116220",
      "y": "0.447844962"
    },
    {
      "turn": "0.131652498",
      "x": "0.280083035",
      "y": "0.348154661"
    },
    {
      "turn": "0.414213562",
      "x": "0.645985401",
      "y": "0.342944326"
    },
    {
      "turn": "0.131652498",
      "x": "0.427640577",
      "y": "0.329805047"
    },
    {
      "turn": "0.131652498",
      "x": "0.211601342",
      "y": "0.229540889"
    },
    {
      "turn": "0.414213562",
      "x": "0.578515685",
      "y": "0.205518674"
    },
    {
      "turn": "0.414213562",
      "x": "0.715479072",
      "y": "0.205518674"
    },
    {
      "turn": "0.414213562",
      "x": "0.441552299",
      "y": "0.167775621"
    },
    {
      "turn": "-0.231047659",
      "x": "0.304868993",
      "y": "0.087031700"
    },
    {
      "turn": "0.414223413",
      "x": "0.149356608",
      "y": "0.069343064"
    },
    {
      "turn": "0.414213562",
      "x": "0.578515685",
      "y": "0.068548295"
    },
    {
      "turn": "0.414213562",
      "x": "0.715479072",
      "y": "0.068548295"
    },
    {
      "turn": "0.414213562",
      "x": "0.852442459",
      "y": "0.068481693"
    }
  ],
  "radius": "0.096847738"
}
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": 17,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.485528115",
      "y": "0.703995978"
    },
    {
      "turn": "0.131652498",
      "x": "0.417046422",
      "y": "0.585382206"
    },
    {
      "turn": "0.131652498",
      "x": "0.564603964",
      "y": "0.567032591"
    },
    {
      "turn": "0.131652498",
      "x": "0.348564729",
      "y": "0.466768433"
    },
    {
      "turn": "0.131652498",
      "x": "0.497116220",
      "y": "0.447844962"
    },
    {
      "turn": "0.131652498",
      "x": "0.280083035",
      "y": "0.348154661"
    },
    {
      "turn": "0.414213562",
      "x": "0.645985401",
      "y": "0.342944326"
    },
    {
      "turn": "0.131652498",
      "x": "0.427640577",
      "y": "0.329805047"
    },
    {
      "turn": "0.131652498",
      "x": "0.211601342",
      "y": "0.229540889"
    },
    {
      "turn": "0.414213562",
      "x": "0.578515685",
      "y": "0.205518674"
    },
    {
      "turn": "0.414213562",
      "x": "0.715479072",
      "y": "0.205518674"
    },
    {
      "turn": "0.414213562",
      "x": "0.441552299",
      "y": "0.167775621"
    },
    {
      "turn": "-0.231047659",
      "x": "0.304868993",
      "y": "0.087031700"
    },
    {
      "turn": "0.414223413",
      "x": "0.149356608",
      "y": "0.069343064"
    },
    {
      "turn": "0.414213562",
      "x": "0.578515685",
      "y": "0.068548295"
    },
    {
      "turn": "0.414213562",
      "x": "0.715479072",
      "y": "0.068548295"
    },
    {
      "turn": "0.414213562",
      "x": "0.852442459",
      "y": "0.068481693"
    }
  ],
  "radius": "0.096847738"
}

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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