P89 · 装箱与覆盖 · 经典问题 · 难

等腰直角三角形装入正方形 · n = 21

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 = 21

目标最小化容器正方形边长

已知最好（未证明）3.340来源精度区间：3.34–3.341（区间内视为追平）本站展示证书 3.340065737Erich Friedman 编录；Found by Maurizio Morandi in March 2008. 原式：s = 3.340+。

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

当前第一名

3.340065737

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）＋

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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": [
    {
      "turn": "-0.000000068",
      "x": "0.850302354",
      "y": "0.550906961"
    },
    {
      "quarter": 3,
      "turn": "0.414213555",
      "x": "0.233350328",
      "y": "0.577144755"
    },
    {
      "turn": "0.414213531",
      "x": "0.445054780",
      "y": "0.445054803"
    },
    {
      "quarter": 2,
      "turn": "-0.000000018",
      "x": "0.251511727",
      "y": "0.850302339"
    },
    {
      "quarter": 2,
      "turn": "-0.000000029",
      "x": "0.550907039",
      "y": "0.850302336"
    },
    {
      "quarter": 2,
      "turn": "0.414213472",
      "x": "0.656759220",
      "y": "0.233350315"
    },
    {
      "quarter": 2,
      "turn": "-0.000000077",
      "x": "0.850302323",
      "y": "0.251511641"
    },
    {
      "turn": "-0.000000019",
      "x": "0.449092972",
      "y": "0.149697661"
    },
    {
      "quarter": 2,
      "turn": "-0.000000020",
      "x": "0.550907025",
      "y": "0.550907022"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.149697659",
      "y": "0.149697659"
    },
    {
      "turn": "0.414213471",
      "x": "0.577144787",
      "y": "0.233350324"
    },
    {
      "quarter": 2,
      "turn": "-0.000000033",
      "x": "0.850302335",
      "y": "0.850302335"
    },
    {
      "turn": "-0.000000037",
      "x": "0.850302350",
      "y": "0.850302313"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.149697656"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.449092969"
    },
    {
      "turn": "-0.000000026",
      "x": "0.550907038",
      "y": "0.850302332"
    },
    {
      "quarter": 2,
      "turn": "-0.000000042",
      "x": "0.850302332",
      "y": "0.550907003"
    },
    {
      "quarter": 2,
      "turn": "0.414213529",
      "x": "0.445054784",
      "y": "0.445054806"
    },
    {
      "quarter": 2,
      "turn": "0.384561065",
      "x": "0.788572377",
      "y": "0.010822938"
    },
    {
      "quarter": 2,
      "turn": "-0.384561063",
      "x": "0.010822939",
      "y": "0.788572377"
    },
    {
      "quarter": 1,
      "turn": "0.414213548",
      "x": "0.233350335",
      "y": "0.656759269"
    }
  ],
  "radius": "0.299395305"
}

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": 21,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "-0.000000068",
      "x": "0.850302354",
      "y": "0.550906961"
    },
    {
      "quarter": 3,
      "turn": "0.414213555",
      "x": "0.233350328",
      "y": "0.577144755"
    },
    {
      "turn": "0.414213531",
      "x": "0.445054780",
      "y": "0.445054803"
    },
    {
      "quarter": 2,
      "turn": "-0.000000018",
      "x": "0.251511727",
      "y": "0.850302339"
    },
    {
      "quarter": 2,
      "turn": "-0.000000029",
      "x": "0.550907039",
      "y": "0.850302336"
    },
    {
      "quarter": 2,
      "turn": "0.414213472",
      "x": "0.656759220",
      "y": "0.233350315"
    },
    {
      "quarter": 2,
      "turn": "-0.000000077",
      "x": "0.850302323",
      "y": "0.251511641"
    },
    {
      "turn": "-0.000000019",
      "x": "0.449092972",
      "y": "0.149697661"
    },
    {
      "quarter": 2,
      "turn": "-0.000000020",
      "x": "0.550907025",
      "y": "0.550907022"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.149697659",
      "y": "0.149697659"
    },
    {
      "turn": "0.414213471",
      "x": "0.577144787",
      "y": "0.233350324"
    },
    {
      "quarter": 2,
      "turn": "-0.000000033",
      "x": "0.850302335",
      "y": "0.850302335"
    },
    {
      "turn": "-0.000000037",
      "x": "0.850302350",
      "y": "0.850302313"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.149697656"
    },
    {
      "turn": "0.000000000",
      "x": "0.149697656",
      "y": "0.449092969"
    },
    {
      "turn": "-0.000000026",
      "x": "0.550907038",
      "y": "0.850302332"
    },
    {
      "quarter": 2,
      "turn": "-0.000000042",
      "x": "0.850302332",
      "y": "0.550907003"
    },
    {
      "quarter": 2,
      "turn": "0.414213529",
      "x": "0.445054784",
      "y": "0.445054806"
    },
    {
      "quarter": 2,
      "turn": "0.384561065",
      "x": "0.788572377",
      "y": "0.010822938"
    },
    {
      "quarter": 2,
      "turn": "-0.384561063",
      "x": "0.010822939",
      "y": "0.788572377"
    },
    {
      "quarter": 1,
      "turn": "0.414213548",
      "x": "0.233350335",
      "y": "0.656759269"
    }
  ],
  "radius": "0.299395305"
}

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