P89 · 装箱与覆盖 · 经典问题 · 难

等腰直角三角形装入正方形 · n = 19

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 = 19

目标最小化容器正方形边长

已知最好（未证明）3.201来源精度区间：3.201–3.202（区间内视为追平）本站展示证书 3.201496211Erich Friedman 编录；Found by Maurizio Morandi in February 2008. 原式：s = 3.201+。

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

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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

3.201496211

容器正方形边长

已追平已知最好

答案来源Maurizio Morandi

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": 2,
      "turn": "0.000000000",
      "x": "0.219115100",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.531469060"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.531469058"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.532960235",
      "y": "0.185175259"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.249604552",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.561958510",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.156176980",
      "y": "0.465025926"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.219115100"
    },
    {
      "quarter": 3,
      "turn": "-0.075206786",
      "x": "0.177779241",
      "y": "0.131069708"
    },
    {
      "turn": "0.000000000",
      "x": "0.531469058",
      "y": "0.843823017"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.843823017"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.531469060"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.279850674",
      "y": "0.562219841"
    },
    {
      "turn": "-0.414213562",
      "x": "0.279850671",
      "y": "0.562219839"
    },
    {
      "turn": "-0.048915272",
      "x": "0.859811014",
      "y": "0.170674536"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500718276",
      "y": "0.341352240"
    },
    {
      "quarter": 2,
      "turn": "-0.337772885",
      "x": "0.029493573",
      "y": "0.781109919"
    }
  ],
  "radius": "0.312353954"
}

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": 19,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.219115100",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.843823020"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.531469060",
      "y": "0.531469060"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.531469058"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.532960235",
      "y": "0.185175259"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.249604552",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.561958510",
      "y": "0.156176980"
    },
    {
      "turn": "0.000000000",
      "x": "0.156176980",
      "y": "0.465025926"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.219115100"
    },
    {
      "quarter": 3,
      "turn": "-0.075206786",
      "x": "0.177779241",
      "y": "0.131069708"
    },
    {
      "turn": "0.000000000",
      "x": "0.531469058",
      "y": "0.843823017"
    },
    {
      "turn": "0.000000000",
      "x": "0.843823017",
      "y": "0.843823017"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.843823020",
      "y": "0.531469060"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.279850674",
      "y": "0.562219841"
    },
    {
      "turn": "-0.414213562",
      "x": "0.279850671",
      "y": "0.562219839"
    },
    {
      "turn": "-0.048915272",
      "x": "0.859811014",
      "y": "0.170674536"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.500718276",
      "y": "0.341352240"
    },
    {
      "quarter": 2,
      "turn": "-0.337772885",
      "x": "0.029493573",
      "y": "0.781109919"
    }
  ],
  "radius": "0.312353954"
}

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