Is it possible to connect all the dots with one line without touching the same point, going diagonally, or touching the black line?
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If it is, how can we show this mathematically?
No, it is not possible. Color the dots in a check pattern. You'll see the number of the dots in one color is $18$, and the number of the dots in the other color is $16$. Note that one always has to go from one color to the other color.