It is 44.
Here is the corrected explanation of all 44 triangles...
If we think of this mathematically...
Look at the largest square.
First lets break it up into halves to get 2 main triangles.
We can rotate these triangles so there are actually 4 possible halfves
depending on which 2 positions we choose.
Even further we can break it up into 4ths
Which will produce 4 triangles, one in each of the possible positions.
So from the largest square we have a total so far of 8 triangles
Now look at one of the smaller squares
The same applies rules apply.
If we break it up into halves, we have 4 possible triangles
If we break it up into fourths we have 4 possible triangles.
So for each smaller square we have 8 triangles.
Since there are 4 smaller squares 4*8 = 32 Triangles.
This puts our total at 40.
Now look at the diamond inside the largest square.
This is just another large square.
The fourths we have already accounted for in the smaller squares.
But the halves are new.
Thus we add the last 4 halves and our grand total is 44.