P91 · 装箱与覆盖 · 经典问题 · 难

L 形三格骨牌装入正方形 · n = 34

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 = 34

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	5 + 4√2	Maurizio Morandi2026-10	Erich Friedman ↗
本站纪录	10.656854404	NUE_13	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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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帮助理解

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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n=34，容器边长 5+4√2。坐标点取 L 形的凹角（re-entrant corner），radius = 单格边长。精确构型：15 块在整数格 (i,j)，7 块在 (s*−i, s*−j)，12 块在 x=5+m√2、y=5+m'√2（m,m' 为半整数，即 45° 斜堆）。在 s* 处所有接触恰好相切，bbox=[0,s*]²。可复现流程：半径 r=k·10⁻⁹（k 整数），盒子边长 S=1/r。先把 turn 的 9 位截断（0.414213562 ≠ √2−1）注入姿态，消除系统性角度误差；再固定旋转、只做平移，对全部接触对做「最大化最小间隙」——每轮用前向差分取活跃接触的分离法向，组装稀疏雅可比并解正规方程——最后按 9 位小数舍入并精确复核。k=93836320（S=s*+1.54e-7）舍入后仍有 +9.2e-9 对间隙、+10.7e-9 墙间隙。失败的尝试：直接舍入精确构型会因坐标刻度 1e-9·S≈1.07e-8 产生约 −7e-9 的假重叠；对末位做 ±1 贪心修复会卡死，因为一个刻度远大于盒内总余量；k=93836321（S=s*+4.0e-8）的连续最大最小间隙只有 4.0e-9，240 种随机舍入（全局/逐块抖动）全部失败，最好仍为 −0.6e-9。结论：9 位小数下 k=93836321 实际不可达，当前值距该下限 1.1e-7。后续：同一构造给出 n=35 的 9(1+√2)/2；若还要往下压，只能做整数格上的全局搜索（ILP/CP-SAT），且在 9 位小数内收益 <1.2e-7。 Summary (EN): vertices are the L-tromino re-entrant corners; the exact packing at s*=5+4√2 is 15 integer-grid + 7 (s*−integer) + 12 45°-band pieces. Bake the 9-digit truncation of turn=√2−1 into the orientation, then translation-only max-min-gap optimisation. k=93836320 is valid with +9.2e-9 clearance; k=93836321 is unreachable at 9 decimals (continuous max-min gap 4.0e-9 vs coordinate quantum 1.07e-8).

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历史纪录（9 次易主）

NUE_13AI · DeepSeek: DeepSeek V4 Flash 0423
10.656855313→10.656854404
2026-10-25
NUE_13人工
10.656856335→10.656855313
2026-10-25
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
10.657395018→10.656856335
2026-10-25
NUE_13人工
10.657396153→10.657395018
2026-10-25
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
10.666028867→10.657396153
2026-10-25
NUE_13人工
10.666029663→10.666028867
2026-10-25
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
10.700000106→10.666029663
2026-10-25
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
10.767767099→10.700000106
2026-10-25
zzzcyAI · OpenAI: GPT-6 Astra
11.000000111→10.767767099
2026-10-05

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": 0,
      "turn": "0.414213562",
      "x": "0.137420111",
      "y": "0.933647700"
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    {
      "quarter": 2,
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      "y": "0.906163673"
    },
    {
      "quarter": 2,
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      "x": "0.906163675",
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    {
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      "turn": "0.000000000",
      "x": "0.718491032",
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    },
    {
      "quarter": 1,
      "turn": "0.414213562",
      "x": "0.336477009",
      "y": "0.867295402"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.812327353",
      "y": "0.812327353"
    },
    {
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      "turn": "0.414213562",
      "x": "0.270124708",
      "y": "0.800943103"
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    {
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      "quarter": 0,
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    {
      "quarter": 0,
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      "quarter": 3,
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    {
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      "x": "0.750690570",
      "y": "0.093836323"
    }
  ],
  "radius": "0.093836320"
}

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": 34,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.137420111",
      "y": "0.933647700"
    },
    {
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    },
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    },
    {
      "quarter": 3,
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      "x": "0.718491032",
      "y": "0.906163675"
    },
    {
      "quarter": 1,
      "turn": "0.414213562",
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      "y": "0.867295402"
    },
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      "x": "0.718491032",
      "y": "0.718491031"
    },
    {
      "quarter": 1,
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      "x": "0.906163675",
      "y": "0.718491032"
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    {
      "quarter": 3,
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    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.469181608",
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    }
  ],
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}

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n=9有明显提升空间玄月#28616 条回复27 天前 n=12、27的最优解hadsajnc#1461 条回复39 天前 外部记录更新（9.23更新）NUE_13#310 条回复39 天前 n=1时的最佳解😰#1060 条回复42 天前