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P91 · 装箱与覆盖 · 经典问题 · 难

L 形三格骨牌装入正方形 · n = 28

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 = 28
目标最小化 容器正方形边长
已知最好（未证明）9.828来源精度区间：9.828–9.829（区间内视为追平）Erich Friedman 编录；Found by Maurizio Morandi in May 2026. 原式：s = 7 + 2√2 = 9.828+。

上方保留外部已知最好数值，本站暂未复现其构型。起步布局是普通本站基线，不是外部纪录。

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
L 形三格骨牌装入正方形 n = 28 的当前纪录构型，10
当前第一名

10

容器正方形边长

纪录保持者创始基准
解题方式人工
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": "0.000000000",
      "x": "0.100000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.100000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.200000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.400000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.500000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.700000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.800000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.800000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.800000000"
    }
  ],
  "radius": "0.100000000"
}
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": 28,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.100000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.100000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.200000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.200000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.400000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.400000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.500000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.500000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.700000000",
      "y": "0.700000000"
    },
    {
      "turn": "0.000000000",
      "x": "0.900000000",
      "y": "0.700000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.100000000",
      "y": "0.800000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.300000000",
      "y": "0.800000000"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.500000000",
      "y": "0.800000000"
    }
  ],
  "radius": "0.100000000"
}

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。 · 验证器 v1.0.0

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讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？