There is only one point that can be an apex for any triangles i.e. the top point. Then there are four different horizontal base lines you, can use with this apex.
Each of these base lines form 6 different triangles from the apex. 1 using outside lines, 3 separate internal ones and 2 overlapping internal ones.
4 levels X 6 triangles = 24. That is all I found here.
The squares question I guess is still open.......