P91 · 装箱与覆盖 · 经典问题 · 难

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

纪录对比

容器正方形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[8.914, 8.915)	Maurizio Morandi	Erich Friedman ↗
本站纪录	8.914213735	匿名	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器编辑器与证书将容器统一归一化为单位正方形 [0,1]²；公开得分按文献的单位方格三格骨牌规范折算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内骨牌的单个方格边长，对应容器正方形边长 = 1/radius。x、y 为骨牌 2 × 2 外接正方形的中心；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
约束恰好 n 个全等 L 形三格骨牌完全位于容器内，任意两骨牌内部互不重叠。每个骨牌允许任意角度独立旋转。
精度输入的十进制小数按精确有理数解析，可行性检验不设浮点容差。
目标容器正方形边长 = 1/r（其中 r 为骨牌单格边长在单位正方形容器内的长度），越小越好。页面、排行榜与文献纪录统一使用该单位并自动换算。

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文献来源与坐标证书

文献目标值按原始发表精度换算，九位小数坐标证书单独存储并与文献目标对照展示。

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构型与历史

纪录详情

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纪录保持者匿名

解题方式人工

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历史纪录（1 次易主）

匿名人工
8.914213814→8.914213735
2026-09-26

ANSWER FORMAT

答案怎么写

单位正方形 [0,1]²

提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内骨牌的单个方格边长，对应容器正方形边长 = 1/radius。x、y 为骨牌 2 × 2 外接正方形的中心；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。坐标与 turn 的范围限于 [−4, 4]，且 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": "0.112180393"
}

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": "0.112180393"
}

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