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L 形三格骨牌装入正方形 · n = 23

把 n 个由三个单位方格拼成的 L 形三格骨牌装入正方形，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）8.914来源精度区间：8.914–8.915（区间内视为追平）本站展示证书 8.914213814Erich Friedman 编录；Found by Maurizio Morandi in June 2012. 原式：s = 15/2 + √2 = 8.914+。

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

8.914213814

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": 1,
      "turn": "0.414213551",
      "x": "0.388810090",
      "y": "0.667280109"
    },
    {
      "quarter": 2,
      "turn": "-0.414213557",
      "x": "0.634713372",
      "y": "0.421376822"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.570797819",
      "y": "0.112180395"
    },
    {
      "quarter": 3,
      "turn": "0.414213552",
      "x": "0.308758507",
      "y": "0.747331691"
    },
    {
      "quarter": 1,
      "turn": "-0.410087471",
      "x": "0.158804200",
      "y": "0.916322717"
    },
    {
      "quarter": 3,
      "turn": "0.414208431",
      "x": "0.555257537",
      "y": "0.500832654"
    },
    {
      "turn": "-0.413819943",
      "x": "0.802537265",
      "y": "0.271721892"
    },
    {
      "quarter": 2,
      "turn": "-0.000000008",
      "x": "0.775639208",
      "y": "0.775639207"
    },
    {
      "turn": "-0.000000011",
      "x": "0.112180398",
      "y": "0.673082374"
    },
    {
      "turn": "-0.000000005",
      "x": "0.112180397",
      "y": "0.448721582"
    },
    {
      "quarter": 3,
      "turn": "-0.000000003",
      "x": "0.663458813",
      "y": "0.887819604"
    },
    {
      "quarter": 1,
      "turn": "-0.000000012",
      "x": "0.887819602",
      "y": "0.663458809"
    },
    {
      "quarter": 2,
      "turn": "-0.000000007",
      "x": "0.224360792",
      "y": "0.448721584"
    },
    {
      "quarter": 2,
      "turn": "-0.000000017",
      "x": "0.887819601",
      "y": "0.439098020"
    },
    {
      "turn": "0.000000000",
      "x": "0.112180396",
      "y": "0.112180396"
    },
    {
      "turn": "-0.000000001",
      "x": "0.336541186",
      "y": "0.112180396"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.448721582",
      "y": "0.224360791"
    },
    {
      "quarter": 2,
      "turn": "-0.000000001",
      "x": "0.439098025",
      "y": "0.887819604"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.887819605",
      "y": "0.887819602"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.663458811",
      "y": "0.663458813"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.336541187",
      "y": "0.336541187"
    },
    {
      "quarter": 2,
      "turn": "0.000000001",
      "x": "0.112180398",
      "y": "0.224360792"
    },
    {
      "quarter": 1,
      "turn": "0.015538692",
      "x": "0.884309660",
      "y": "0.115612152"
    }
  ],
  "radius": "0.112180392"
}

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": [
    {
      "quarter": 1,
      "turn": "0.414213551",
      "x": "0.388810090",
      "y": "0.667280109"
    },
    {
      "quarter": 2,
      "turn": "-0.414213557",
      "x": "0.634713372",
      "y": "0.421376822"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.570797819",
      "y": "0.112180395"
    },
    {
      "quarter": 3,
      "turn": "0.414213552",
      "x": "0.308758507",
      "y": "0.747331691"
    },
    {
      "quarter": 1,
      "turn": "-0.410087471",
      "x": "0.158804200",
      "y": "0.916322717"
    },
    {
      "quarter": 3,
      "turn": "0.414208431",
      "x": "0.555257537",
      "y": "0.500832654"
    },
    {
      "turn": "-0.413819943",
      "x": "0.802537265",
      "y": "0.271721892"
    },
    {
      "quarter": 2,
      "turn": "-0.000000008",
      "x": "0.775639208",
      "y": "0.775639207"
    },
    {
      "turn": "-0.000000011",
      "x": "0.112180398",
      "y": "0.673082374"
    },
    {
      "turn": "-0.000000005",
      "x": "0.112180397",
      "y": "0.448721582"
    },
    {
      "quarter": 3,
      "turn": "-0.000000003",
      "x": "0.663458813",
      "y": "0.887819604"
    },
    {
      "quarter": 1,
      "turn": "-0.000000012",
      "x": "0.887819602",
      "y": "0.663458809"
    },
    {
      "quarter": 2,
      "turn": "-0.000000007",
      "x": "0.224360792",
      "y": "0.448721584"
    },
    {
      "quarter": 2,
      "turn": "-0.000000017",
      "x": "0.887819601",
      "y": "0.439098020"
    },
    {
      "turn": "0.000000000",
      "x": "0.112180396",
      "y": "0.112180396"
    },
    {
      "turn": "-0.000000001",
      "x": "0.336541186",
      "y": "0.112180396"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.448721582",
      "y": "0.224360791"
    },
    {
      "quarter": 2,
      "turn": "-0.000000001",
      "x": "0.439098025",
      "y": "0.887819604"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.887819605",
      "y": "0.887819602"
    },
    {
      "quarter": 2,
      "turn": "-0.000000006",
      "x": "0.663458811",
      "y": "0.663458813"
    },
    {
      "quarter": 1,
      "turn": "-0.000000003",
      "x": "0.336541187",
      "y": "0.336541187"
    },
    {
      "quarter": 2,
      "turn": "0.000000001",
      "x": "0.112180398",
      "y": "0.224360792"
    },
    {
      "quarter": 1,
      "turn": "0.015538692",
      "x": "0.884309660",
      "y": "0.115612152"
    }
  ],
  "radius": "0.112180392"
}

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