P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 40

将 n 个边长为 1 的正方形装入等边三角形容器,允许边界接触但内部不得重叠,最小化容器等边三角形的边长。

子题n = 40
目标最小化 容器三角形边长

纪录对比

容器三角形边长 · 越小越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[10.4641, 10.46411)Bhavithran AnanthanErich Friedman ↗
本站纪录暂无选手纪录——

严格定义

  • 容器编辑器与证书将容器统一归一化为顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形;公开得分按文献的单位边长图形规范折算。允许接触边界。
  • 提交提交 {radius, placements:[{x,y,turn},…]},每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标;圆的 turn 固定为 0,多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4],且 0 < radius ≤ 4。
  • 约束恰好 n 个全等图形完全位于容器内,任意两图形内部互不重叠。多边形允许任意角度独立旋转。
  • 精度输入的十进制小数按精确有理数解析,正多边形顶点由精确代数数定义,可行性检验不设浮点容差。
  • 目标容器三角形边长 = 1/(2 sin(π/4) r),越小越好。页面、排行榜与文献纪录统一使用该单位并自动换算。
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文献来源与坐标证书

文献目标值按原始发表精度换算,九位小数坐标证书单独存储并与文献目标对照展示。

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正方形装入等边三角形 n = 40 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路,经审核采纳后可获证明分。
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},…]},每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标;圆的 turn 固定为 0,多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4],且 0 < radius ≤ 4。

起步布局答案

{
  "placements": [
    {
      "turn": "0.139014541",
      "x": "0.490110712",
      "y": "0.749757764"
    },
    {
      "turn": "-0.123429023",
      "x": "0.561519369",
      "y": "0.660620124"
    },
    {
      "turn": "0.131652498",
      "x": "0.432368715",
      "y": "0.653319760"
    },
    {
      "turn": "-0.131652498",
      "x": "0.611127628",
      "y": "0.577981885"
    },
    {
      "turn": "0.131652498",
      "x": "0.381454130",
      "y": "0.565133113"
    },
    {
      "turn": "0.131652498",
      "x": "0.489902389",
      "y": "0.561841364"
    },
    {
      "turn": "-0.134829526",
      "x": "0.652094646",
      "y": "0.490844535"
    },
    {
      "turn": "0.123879053",
      "x": "0.334120273",
      "y": "0.481238862"
    },
    {
      "turn": "0.131652498",
      "x": "0.443875654",
      "y": "0.478066161"
    },
    {
      "turn": "0.152448479",
      "x": "0.527449966",
      "y": "0.427875972"
    },
    {
      "turn": "-0.133299825",
      "x": "0.706836569",
      "y": "0.411900215"
    },
    {
      "turn": "0.127536167",
      "x": "0.285711072",
      "y": "0.398444519"
    },
    {
      "turn": "0.131652498",
      "x": "0.396333211",
      "y": "0.395166053"
    },
    {
      "turn": "0.197249610",
      "x": "0.604267467",
      "y": "0.360036818"
    },
    {
      "turn": "0.156096206",
      "x": "0.479998137",
      "y": "0.344345038"
    },
    {
      "turn": "-0.130619735",
      "x": "0.748321517",
      "y": "0.325239844"
    },
    {
      "turn": "0.132014560",
      "x": "0.238075541",
      "y": "0.315114613"
    },
    {
      "turn": "0.131652498",
      "x": "0.347534848",
      "y": "0.312991050"
    },
    {
      "turn": "0.245026390",
      "x": "0.552631750",
      "y": "0.260343476"
    },
    {
      "turn": "0.282386149",
      "x": "0.434417420",
      "y": "0.249024471"
    },
    {
      "turn": "-0.134136147",
      "x": "0.801121920",
      "y": "0.244986703"
    },
    {
      "turn": "0.298251692",
      "x": "0.651528258",
      "y": "0.244202218"
    },
    {
      "turn": "0.131652498",
      "x": "0.189548887",
      "y": "0.232743482"
    },
    {
      "turn": "0.131652498",
      "x": "0.312202526",
      "y": "0.223041426"
    },
    {
      "turn": "-0.105171451",
      "x": "0.737367815",
      "y": "0.162395751"
    },
    {
      "turn": "-0.127593111",
      "x": "0.850573838",
      "y": "0.162488850"
    },
    {
      "turn": "0.371675391",
      "x": "0.611963484",
      "y": "0.147812075"
    },
    {
      "turn": "0.131652498",
      "x": "0.141293211",
      "y": "0.149162199"
    },
    {
      "turn": "0.363879433",
      "x": "0.515428020",
      "y": "0.149004586"
    },
    {
      "turn": "0.131652498",
      "x": "0.250990335",
      "y": "0.148033550"
    },
    {
      "turn": "0.347033730",
      "x": "0.416851128",
      "y": "0.149481748"
    },
    {
      "turn": "0.131652498",
      "x": "0.093323166",
      "y": "0.066075643"
    },
    {
      "turn": "0.131652498",
      "x": "0.203207925",
      "y": "0.065271987"
    },
    {
      "turn": "0.414213562",
      "x": "0.419219108",
      "y": "0.047870714"
    },
    {
      "turn": "0.414213562",
      "x": "0.705913572",
      "y": "0.048517341"
    },
    {
      "turn": "0.414213562",
      "x": "0.801478393",
      "y": "0.048996581"
    },
    {
      "turn": "0.414213562",
      "x": "0.897043214",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.323654287",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.514783929",
      "y": "0.047784526"
    },
    {
      "turn": "0.414213562",
      "x": "0.610348751",
      "y": "0.047784542"
    }
  ],
  "radius": "0.067574532"
}
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": 40,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.139014541",
      "x": "0.490110712",
      "y": "0.749757764"
    },
    {
      "turn": "-0.123429023",
      "x": "0.561519369",
      "y": "0.660620124"
    },
    {
      "turn": "0.131652498",
      "x": "0.432368715",
      "y": "0.653319760"
    },
    {
      "turn": "-0.131652498",
      "x": "0.611127628",
      "y": "0.577981885"
    },
    {
      "turn": "0.131652498",
      "x": "0.381454130",
      "y": "0.565133113"
    },
    {
      "turn": "0.131652498",
      "x": "0.489902389",
      "y": "0.561841364"
    },
    {
      "turn": "-0.134829526",
      "x": "0.652094646",
      "y": "0.490844535"
    },
    {
      "turn": "0.123879053",
      "x": "0.334120273",
      "y": "0.481238862"
    },
    {
      "turn": "0.131652498",
      "x": "0.443875654",
      "y": "0.478066161"
    },
    {
      "turn": "0.152448479",
      "x": "0.527449966",
      "y": "0.427875972"
    },
    {
      "turn": "-0.133299825",
      "x": "0.706836569",
      "y": "0.411900215"
    },
    {
      "turn": "0.127536167",
      "x": "0.285711072",
      "y": "0.398444519"
    },
    {
      "turn": "0.131652498",
      "x": "0.396333211",
      "y": "0.395166053"
    },
    {
      "turn": "0.197249610",
      "x": "0.604267467",
      "y": "0.360036818"
    },
    {
      "turn": "0.156096206",
      "x": "0.479998137",
      "y": "0.344345038"
    },
    {
      "turn": "-0.130619735",
      "x": "0.748321517",
      "y": "0.325239844"
    },
    {
      "turn": "0.132014560",
      "x": "0.238075541",
      "y": "0.315114613"
    },
    {
      "turn": "0.131652498",
      "x": "0.347534848",
      "y": "0.312991050"
    },
    {
      "turn": "0.245026390",
      "x": "0.552631750",
      "y": "0.260343476"
    },
    {
      "turn": "0.282386149",
      "x": "0.434417420",
      "y": "0.249024471"
    },
    {
      "turn": "-0.134136147",
      "x": "0.801121920",
      "y": "0.244986703"
    },
    {
      "turn": "0.298251692",
      "x": "0.651528258",
      "y": "0.244202218"
    },
    {
      "turn": "0.131652498",
      "x": "0.189548887",
      "y": "0.232743482"
    },
    {
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      "x": "0.312202526",
      "y": "0.223041426"
    },
    {
      "turn": "-0.105171451",
      "x": "0.737367815",
      "y": "0.162395751"
    },
    {
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      "x": "0.850573838",
      "y": "0.162488850"
    },
    {
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      "x": "0.611963484",
      "y": "0.147812075"
    },
    {
      "turn": "0.131652498",
      "x": "0.141293211",
      "y": "0.149162199"
    },
    {
      "turn": "0.363879433",
      "x": "0.515428020",
      "y": "0.149004586"
    },
    {
      "turn": "0.131652498",
      "x": "0.250990335",
      "y": "0.148033550"
    },
    {
      "turn": "0.347033730",
      "x": "0.416851128",
      "y": "0.149481748"
    },
    {
      "turn": "0.131652498",
      "x": "0.093323166",
      "y": "0.066075643"
    },
    {
      "turn": "0.131652498",
      "x": "0.203207925",
      "y": "0.065271987"
    },
    {
      "turn": "0.414213562",
      "x": "0.419219108",
      "y": "0.047870714"
    },
    {
      "turn": "0.414213562",
      "x": "0.705913572",
      "y": "0.048517341"
    },
    {
      "turn": "0.414213562",
      "x": "0.801478393",
      "y": "0.048996581"
    },
    {
      "turn": "0.414213562",
      "x": "0.897043214",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.323654287",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.514783929",
      "y": "0.047784526"
    },
    {
      "turn": "0.414213562",
      "x": "0.610348751",
      "y": "0.047784542"
    }
  ],
  "radius": "0.067574532"
}

提交 {radius, placements:[{x,y,turn},…]},每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标;圆的 turn 固定为 0,多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4],且 0 < radius ≤ 4。 · 验证器 v1.0.0

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