P89 · 装箱与覆盖 · 经典问题 · 难

等腰直角三角形装入正方形 · n = 23

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 = 23

目标最小化容器正方形边长

已知最好（未证明）3.464来源精度区间：3.464–3.465（区间内视为追平）本站展示证书 3.46400001Erich Friedman 编录；Found by Maurizio Morandi in December 2007. 原式：s = 3.464+。

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

3.46400001

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": [
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.204134726"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.781514924",
      "y": "0.000000003"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.855654945",
      "y": "0.552614563"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345053",
      "y": "0.433035151"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.433035150"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345051",
      "y": "0.144345053"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.144345053"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.855654950",
      "y": "0.841304660"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.204134728",
      "y": "0.999999997"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690100"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.204134726",
      "y": "0.577380201"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.433035150"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.144345050"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035148",
      "y": "0.144345050"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.795865274",
      "y": "0.408269471"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.577380199",
      "y": "0.204134728"
    },
    {
      "turn": "-0.414213562",
      "x": "0.795865271",
      "y": "0.393919107"
    },
    {
      "quarter": 3,
      "turn": "0.012428988",
      "x": "0.859287111",
      "y": "0.852111969"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.795865272"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690097"
    },
    {
      "turn": "-0.414213562",
      "x": "0.552614501",
      "y": "0.517590526"
    },
    {
      "quarter": 1,
      "turn": "0.335237318",
      "x": "0.580791168",
      "y": "0.613337157"
    },
    {
      "quarter": 2,
      "turn": "0.012428983",
      "x": "0.570507806",
      "y": "0.852111970"
    }
  ],
  "radius": "0.288683602"
}

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": 23,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.999999997",
      "y": "0.204134726"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.781514924",
      "y": "0.000000003"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.855654945",
      "y": "0.552614563"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345053",
      "y": "0.433035151"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.433035150"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.144345051",
      "y": "0.144345053"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.433035149",
      "y": "0.144345053"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.855654950",
      "y": "0.841304660"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.204134728",
      "y": "0.999999997"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690100"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.204134726",
      "y": "0.577380201"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.433035150"
    },
    {
      "turn": "0.000000000",
      "x": "0.144345050",
      "y": "0.144345050"
    },
    {
      "turn": "0.000000000",
      "x": "0.433035148",
      "y": "0.144345050"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.795865274",
      "y": "0.408269471"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.577380199",
      "y": "0.204134728"
    },
    {
      "turn": "-0.414213562",
      "x": "0.795865271",
      "y": "0.393919107"
    },
    {
      "quarter": 3,
      "turn": "0.012428988",
      "x": "0.859287111",
      "y": "0.852111969"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.000000003",
      "y": "0.795865272"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.401094281",
      "y": "0.788690097"
    },
    {
      "turn": "-0.414213562",
      "x": "0.552614501",
      "y": "0.517590526"
    },
    {
      "quarter": 1,
      "turn": "0.335237318",
      "x": "0.580791168",
      "y": "0.613337157"
    },
    {
      "quarter": 2,
      "turn": "0.012428983",
      "x": "0.570507806",
      "y": "0.852111970"
    }
  ],
  "radius": "0.288683602"
}

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