论坛 › 趣题妙解 › 有多少个菱形

clion1988 发表于 7 天前

有多少个菱形

如图，现有很多士兵来排队，士兵用红点表达，他们先组合成一个蓝色正六边形，队长在正六边形正中，6名士兵站在六个角点上；然后又有12名士兵加入，他们又形成了品红色正六边形包住蓝色正六边形，品红色图形边长是蓝色边长的2倍，其6个角点分别站一名士兵，每边中心点也站一名士兵；最后18名士兵再加入，他们形成绿色正六边形包住品红色正六边形，其6个角点分别站一名士兵，剩余12名士兵站在其边长1/3均分点上。
明显他们同时形成了若干菱形，例如图中的橙色菱形为其中之一，那么到底总共有多少个菱形呢？

northwolves 发表于 4 天前

遍历了一下，645个。不知道对不对

northwolves 发表于 4 天前

t = ArcSin]; p =
FullSimplify@
Join[{{0, 0}},
   Flatten, Sin}, {2 Cos,
       2 Sin}, {3 Cos,
       3 Sin}, {Sqrt*Cos,
       Sqrt*Sin}, {Sqrt/2*
      Cos,
       Sqrt/2*Sin}, {Sqrt/2*
      Cos,
       Sqrt/2*Sin}}, {k, 0, 5}], 1]]; x =
Select, {y, Subsets}],
Length@# < 4 && Max[#[]] > 3 &]; {Length@p, Length@x,
ReverseSortBy}

{37,645,{{{{1,4},{Sqrt,1},{2,1}},47},{{{1,4},{2,1},{Sqrt,1}},47},{{{1,5},{Sqrt,1}},30},{{{Sqrt,4},{3,1},{2 Sqrt,1}},27},{{{Sqrt,4},{2 Sqrt,1},{1,1}},24},{{{Sqrt,4},{2 Sqrt,1},{3,1}},23},{{{Sqrt,4},{Sqrt,1},{1,1}},20},{{{Sqrt,4},{4,1},{Sqrt,1}},19},{{{1,1},{Sqrt,4},{Sqrt,1}},18},{{{Sqrt,5},{3,1}},17},{{{Sqrt,4},{5,1},{2 Sqrt,1}},13},{{{2,4},{4,1},{2 Sqrt,1}},13},{{{Sqrt/2,4},{Sqrt/2,1},{Sqrt,1}},12},{{{Sqrt/2,4},{Sqrt,1},{Sqrt/2,1}},12},{{{Sqrt/2,4},{1,1},{(7 Sqrt)/2,1}},12},{{{Sqrt/2,4},{1,1},{(5 Sqrt)/2,1}},12},{{{Sqrt/2,5},{Sqrt/2,1}},12},{{{Sqrt,5},{Sqrt,1}},12},{{{Sqrt,4},{1,1},{Sqrt,1}},11},{{{2,4},{2 Sqrt,1},{4,1}},11},{{{Sqrt,1},{Sqrt,4},{1,1}},10},{{{Sqrt,1},{1,4},{2,1}},10},{{{1,1},{Sqrt,4},{2 Sqrt,1}},10},{{{2,5},{2 Sqrt,1}},9},{{{Sqrt,4},{1,1},{2 Sqrt,1}},8},{{{Sqrt,4},{Sqrt,1},{4,1}},7},{{{3,1},{Sqrt,4},{2 Sqrt,1}},7},{{{Sqrt/2,4},{6 Sqrt,1},{19/2,1}},6},{{{Sqrt/2,4},{6 Sqrt,1},{17/2,1}},6},{{{Sqrt/2,4},{(7 Sqrt)/2,1},{19/2,1}},6},{{{Sqrt/2,4},{(7 Sqrt)/2,1},{1,1}},6},{{{Sqrt/2,4},{(5 Sqrt)/2,1},{17/2,1}},6},{{{Sqrt/2,4},{(5 Sqrt)/2,1},{1,1}},6},{{{Sqrt/2,4},{19/2,1},{6 Sqrt,1}},6},{{{Sqrt/2,4},{19/2,1},{(7 Sqrt)/2,1}},6},{{{Sqrt/2,4},{17/2,1},{6 Sqrt,1}},6},{{{Sqrt/2,4},{17/2,1},{(5 Sqrt)/2,1}},6},{{{Sqrt,4},{3 Sqrt,1},{5,1}},6},{{{Sqrt,4},{3 Sqrt,1},{4,1}},6},{{{Sqrt,4},{2 Sqrt,1},{5,1}},6},{{{3,4},{3 Sqrt,1},{6,1}},6},{{{3,4},{6,1},{3 Sqrt,1}},6},{{{1,1},{Sqrt/2,4},{(7 Sqrt)/2,1}},6},{{{1,1},{Sqrt/2,4},{(5 Sqrt)/2,1}},6},{{{1,1},{2 Sqrt,1},{Sqrt,4}},6},{{{3,5},{3 Sqrt,1}},6},{{{Sqrt,4},{4,1},{2 Sqrt,1}},5},{{{Sqrt,1},{Sqrt,4},{4,1}},5},{{{Sqrt,1},{4,1},{Sqrt,4}},5},{{{Sqrt/2,4},{5,1},{6 Sqrt,1}},4},{{{Sqrt/2,4},{1,1},{6 Sqrt,1}},4},{{{(3 Sqrt)/2,4},{3,1},{6 Sqrt,1}},4},{{{Sqrt,4},{3 Sqrt,1},{1,1}},4},{{{Sqrt,4},{2 Sqrt,1},{4,1}},4},{{{Sqrt,4},{5,1},{3 Sqrt,1}},4},{{{Sqrt,4},{4,1},{3 Sqrt,1}},4},{{{Sqrt,4},{Sqrt,1},{5,1}},3},{{{Sqrt,4},{5,1},{Sqrt,1}},3},{{{2 Sqrt,1},{5,1},{Sqrt,4}},3},{{{2 Sqrt,5},{6,1}},3},{{{Sqrt/2,4},{6 Sqrt,1},{5,1}},2},{{{(3 Sqrt)/2,4},{6 Sqrt,1},{3,1}},2},{{{2 Sqrt,1},{Sqrt,4},{5,1}},2},{{{Sqrt,1},{2,1},{1,4}},2},{{{3,1},{2 Sqrt,1},{Sqrt,4}},2},{{{2,1},{1,4},{Sqrt,1}},2},{{{1,1},{Sqrt/2,4},{6 Sqrt,1}},2},{{{3 Sqrt,1},{Sqrt,4},{1,1}},1},{{{2 Sqrt,1},{3,1},{Sqrt,4}},1},{{{5,1},{Sqrt,4},{3 Sqrt,1}},1},{{{5,1},{3 Sqrt,1},{Sqrt,4}},1},{{{4,1},{Sqrt,4},{3 Sqrt,1}},1},{{{4,1},{3 Sqrt,1},{Sqrt,4}},1},{{{1,1},{Sqrt,4},{3 Sqrt,1}},1},{{{1,1},{Sqrt,1},{Sqrt,4}},1},{{{3,1},{Sqrt,5}},1}}}

northwolves 发表于 3 天前

去掉三点共线情景，余417种组合：

{417,{{{{1,5},{Sqrt,1}},30},{{{Sqrt,4},{2 Sqrt,1},{1,1}},24},{{{Sqrt,4},{Sqrt,1},{1,1}},20},{{{Sqrt,4},{4,1},{Sqrt,1}},19},{{{1,1},{Sqrt,4},{Sqrt,1}},18},{{{Sqrt,5},{3,1}},17},{{{Sqrt,4},{5,1},{2 Sqrt,1}},13},{{{Sqrt/2,4},{1,1},{(7 Sqrt)/2,1}},12},{{{Sqrt/2,4},{1,1},{(5 Sqrt)/2,1}},12},{{{Sqrt/2,5},{Sqrt/2,1}},12},{{{Sqrt,5},{Sqrt,1}},12},{{{Sqrt,4},{1,1},{Sqrt,1}},11},{{{Sqrt,1},{Sqrt,4},{1,1}},10},{{{1,1},{Sqrt,4},{2 Sqrt,1}},10},{{{2,5},{2 Sqrt,1}},9},{{{Sqrt,4},{1,1},{2 Sqrt,1}},8},{{{Sqrt,4},{Sqrt,1},{4,1}},7},{{{Sqrt/2,4},{6 Sqrt,1},{19/2,1}},6},{{{Sqrt/2,4},{6 Sqrt,1},{17/2,1}},6},{{{Sqrt/2,4},{(7 Sqrt)/2,1},{19/2,1}},6},{{{Sqrt/2,4},{(7 Sqrt)/2,1},{1,1}},6},{{{Sqrt/2,4},{(5 Sqrt)/2,1},{17/2,1}},6},{{{Sqrt/2,4},{(5 Sqrt)/2,1},{1,1}},6},{{{Sqrt/2,4},{19/2,1},{6 Sqrt,1}},6},{{{Sqrt/2,4},{19/2,1},{(7 Sqrt)/2,1}},6},{{{Sqrt/2,4},{17/2,1},{6 Sqrt,1}},6},{{{Sqrt/2,4},{17/2,1},{(5 Sqrt)/2,1}},6},{{{Sqrt,4},{3 Sqrt,1},{5,1}},6},{{{Sqrt,4},{3 Sqrt,1},{4,1}},6},{{{Sqrt,4},{2 Sqrt,1},{5,1}},6},{{{1,1},{Sqrt/2,4},{(7 Sqrt)/2,1}},6},{{{1,1},{Sqrt/2,4},{(5 Sqrt)/2,1}},6},{{{1,1},{2 Sqrt,1},{Sqrt,4}},6},{{{3,5},{3 Sqrt,1}},6},{{{Sqrt,4},{4,1},{2 Sqrt,1}},5},{{{Sqrt,1},{Sqrt,4},{4,1}},5},{{{Sqrt,1},{4,1},{Sqrt,4}},5},{{{Sqrt/2,4},{5,1},{6 Sqrt,1}},4},{{{Sqrt/2,4},{1,1},{6 Sqrt,1}},4},{{{(3 Sqrt)/2,4},{3,1},{6 Sqrt,1}},4},{{{Sqrt,4},{3 Sqrt,1},{1,1}},4},{{{Sqrt,4},{2 Sqrt,1},{4,1}},4},{{{Sqrt,4},{5,1},{3 Sqrt,1}},4},{{{Sqrt,4},{4,1},{3 Sqrt,1}},4},{{{Sqrt,4},{Sqrt,1},{5,1}},3},{{{Sqrt,4},{5,1},{Sqrt,1}},3},{{{2 Sqrt,1},{5,1},{Sqrt,4}},3},{{{2 Sqrt,5},{6,1}},3},{{{Sqrt/2,4},{6 Sqrt,1},{5,1}},2},{{{(3 Sqrt)/2,4},{6 Sqrt,1},{3,1}},2},{{{2 Sqrt,1},{Sqrt,4},{5,1}},2},{{{1,1},{Sqrt/2,4},{6 Sqrt,1}},2},{{{3 Sqrt,1},{Sqrt,4},{1,1}},1},{{{5,1},{Sqrt,4},{3 Sqrt,1}},1},{{{5,1},{3 Sqrt,1},{Sqrt,4}},1},{{{4,1},{Sqrt,4},{3 Sqrt,1}},1},{{{4,1},{3 Sqrt,1},{Sqrt,4}},1},{{{1,1},{Sqrt,4},{3 Sqrt,1}},1},{{{1,1},{Sqrt,1},{Sqrt,4}},1},{{{3,1},{Sqrt,5}},1}}}

qsq 发表于 3 天前

楼主你好

clion1988 发表于 前天 11:56

本帖最后由 clion1988 于 2024-11-19 14:22 编辑

应该是一共246个吧。
以两点之间距离为L，找到共10类：
【1】最小菱形，边长L，按中垂线对称算，左边右边分别9个；中间蓝色菱形上下对称，上下分别3个。再按整体对称轴3个轴对称计算，共72个。
9+9+3+3=24,24*3=72。
https://picx.zhimg.com/80/v2-c2f78c50b2f72ea4c915f48436bbed2a_720w.webp?source=2c26e567
【2】第二种菱形，边长2L。
左图左边右边都是4个，左图共8个；右图中间共4个；整图3个轴对称。
8+4=12,12*3=36。
https://picx.zhimg.com/80/v2-86c25807b8fca8546bfe8f4e8e6b8b0d_720w.webp?source=2c26e567

【3】第三种边长3L。
左图2个；右图中间共2个；整图3个轴对称。
2+2=4,4*3=12。
https://pica.zhimg.com/80/v2-6fa3d86f079ab06bddffb651bdff2956_720w.webp?source=2c26e567


【4】边长√3L。
左图上下对称，上下分别5个；右图中间共4个；整图3个轴对称。
5+5+4=14,14*3=42。
https://picx.zhimg.com/80/v2-f1e771e4c4d217530fea14e104126b35_720w.webp?source=2c26e567


【5】一种角度的菱形，边长√7L。
左图4个；右图中间共3个；整图3个轴对称。
4+3=7,7*3=21。
https://pic1.zhimg.com/80/v2-2047f5644d4eee35820ea098d5b43a9c_720w.webp?source=2c26e567


【6】边长√13L。
图示2个；整图3个轴对称。
2*3=6。
https://pica.zhimg.com/80/v2-1214d7909bf207f5e6aa363581926d9d_720w.webp?source=2c26e567


【7】第二种角度的菱形，边长√7L。
图中共4个；整图3个轴对称。
4*3=12。
https://picx.zhimg.com/80/v2-d41e2ff682bbf596e33f67b0b7f06aa3_720w.webp?source=2c26e567


【8】第三种角度的菱形，边长√7L。
图中共2个；整图3个轴对称。
2*3=6。
https://picx.zhimg.com/80/v2-f0f7531c1a2d8bfa5168b51eba8890d9_720w.webp?source=2c26e567

【9】第四种角度的菱形，边长√7L。

图中共6个；整图6个轴对称。 6*6=36。
https://picx.zhimg.com/80/v2-ae508bcd687350871d1e33cb896e8ff0_720w.webp?source=2c26e567

【10】边长2√3L。
图中共1个；整图3个轴对称。
1*3=3。
https://pic1.zhimg.com/80/v2-6bd0f9a2b0f2a50f8a7b219cd3b42123_720w.webp?source=2c26e567

查看完整版本: 有多少个菱形