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P89 · 装箱与覆盖 · 经典问题 · 难

等腰直角三角形装入正方形 · n = 16

把 n 个直角边为 1 的等腰直角三角形（七巧板的小三角形）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 = 16
目标最小化 容器正方形边长
公开图形的本站重建值2.828来源精度区间：2.828–2.829（区间内视为追平）本站展示证书 2.828427242Erich Friedman 编录；Trivial. 原式：s = 2√2 = 2.828+。

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,quarter},…]}，x、y、turn 用最多九位小数的十进制字符串。radius 是零件的短边长（直角边、多米诺的短边、方格边长），容器正方形边长 = 1/radius。x、y 是零件外接矩形的中心；quarter 是 0–3 的整数，先绕中心精确转 quarter 个直角，再转 turn = tan(θ/2) 给出的有理角。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/r（r 是零件短边在单位正方形里的长度），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
等腰直角三角形装入正方形 n = 16 的当前纪录构型，2.828427242
当前第一名

2.828427242

容器正方形边长

答案来源Trivial.
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,quarter},…]}，x、y、turn 用最多九位小数的十进制字符串。radius 是零件的短边长（直角边、多米诺的短边、方格边长），容器正方形边长 = 1/radius。x、y 是零件外接矩形的中心；quarter 是 0–3 的整数，先绕中心精确转 quarter 个直角，再转 turn = tan(θ/2) 给出的有理角。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.250000010"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.000000020"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.000000020"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.500000000"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999980",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999980",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.500000000"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.749999990"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.999999980"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.999999980"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000020",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000020",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.500000000"
    }
  ],
  "radius": "0.353553376"
}
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": 16,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.250000010"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.000000020"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.000000020"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.500000000"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999980",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.999999980",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.500000000"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.749999990"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.749999990",
      "y": "0.999999980"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.999999980"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500000000",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000020",
      "y": "0.749999990"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000020",
      "y": "0.250000010"
    },
    {
      "turn": "-0.414213562",
      "x": "0.250000010",
      "y": "0.500000000"
    }
  ],
  "radius": "0.353553376"
}

提交 {radius, placements:[{x,y,turn,quarter},…]}，x、y、turn 用最多九位小数的十进制字符串。radius 是零件的短边长（直角边、多米诺的短边、方格边长），容器正方形边长 = 1/radius。x、y 是零件外接矩形的中心；quarter 是 0–3 的整数，先绕中心精确转 quarter 个直角，再转 turn = tan(θ/2) 给出的有理角。坐标和 turn 限于 ±4，0 < radius ≤ 4。 · 验证器 v1.0.0

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