Good Pattern For Decorate House
I call the upper figure a window , forming by a square with its sides' midpoints joining to each other. Now, I connect 2016 windows side-by-side to make a chain of windows like this:
How many triangles are there in the chain?
Too easy? Try this .
The answer is 40310.
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1 solution
Each window consists of 1 2 triangles; that is, 8 small triangles and 4 , joining 2 of the previous. So, in the chain of windows there are 2 0 1 6 ⋅ 1 2 = 2 4 1 9 2 triangles, from considering each single window .
When you connect 2 windows , you form 6 new triangles. As in the chain of windows we have 2 0 1 5 unions of 2 side-by-side windows , the number of triangles resulting from this unions is 2 0 1 5 ⋅ 6 = 1 2 0 9 0 .
Finally, when you connect 3 windows , you form 2 new triangles. As in the chain of windows we have 2 0 1 4 unions of 3 side-by-side windows , the number of triangles resulting from this unions is 2 0 1 4 ⋅ 2 = 4 0 2 8 .
Therefore, adding up the results, the total number of triangles in the chain of windows is 4 0 3 1 0 .
I really liked this problem. I hope you create more problems like this which remembers me about fractals. I expect to see soon a harder one, forming weirder and more complicated figures.