$\textbf{Problem:}$ The player has cut a square into $57$ triangles and painted a blue dot at all their vertices. It turned out that the blue dots are only inside the square (not on the sides) and in the corners. How many blue dots could there be?
$\textbf{My ideas of solving:}$ It is clear that we need to find the range. That is, find the smallest and the largest number of blue points.
I think the lowest number of points corresponds to the following splitting (see picture). Where $57 = 28 + 29$ and number of blue points is equal to $32 = 29 + 1 + 2$.
To the greatest number of points come up with such a split (see picture).
Where $57 = 4 + 13 + 13 + 13 + 14$ and number of blue points is equal to $58$.
Do these pictures show the boundary cases?