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Suppose you want to fill the edges of an $n \times n$ square grid using the following shapes:

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Here is an example for $n=3$:

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The problem was to prove that the number of $B$'s is equal to $C$'s.

While trying to prove this, I observed that in every diagonal with slope $-1$, (diagonals that pass through $\{3\}, \{2, 6\}, \{1 ,5 , 9\}, \{4, 8\}, \{7\}$) number of $B$'s and $C$'s are equal.

Even more surprising, their order of appearance (starting from the top) always makes a valid parentheses syntax. for example, in $\{1,5,9\}$, black $B$, green $C$, brown $B$ and blue $C$, form $$\Large ()()$$

I also observed that the same is true for $A$'s and $D$'s in diagonals with slope $1$.

I solved the original problem, however, was unable to prove any of my further observations. I would appreciate any suggestions.

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    Just out of curiosity, how did you hear about this problem? In general, I am interested in tiling problems. Do you happen to know anything about how many ways you can cover a grid with those shapes? – 2017-02-02
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    @RusMay I did prove that the number of $B$'s and $C$'s are equal in the whole grid, using double counting. However this doesn't seem to help proving the stronger claims, and so far I've found nothing useful. The part that I proved was a homework assignment, and the part I'm asking is my own problem, and I realized it trying to prove the original problem, and was unable to prove or disprove it, and so were my classmates and my teacher. – 2017-02-03
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    @RusMay Also, I don't know in how many ways one can cover the grid with these shapes :) – 2017-02-03
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    Many simple sounding tiling problems are fiendishly difficult. I honestly don't know where this one is in the spectrum of difficulty. In any case, I'll keep thinking about your observation about the parentheses here. – 2017-02-03

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