What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? 
The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif

Laczkovich gave a solution with many hundreds of triangles, but this was just an demonstration of existence, and not a minimal solution. ( Laczkovich, M. "Tilings of Polygons with Similar Triangles." Combinatorica 10, 281-306, 1990. )
I've offered a prize for this problem: In US dollars, (\$200-number of triangles). 
NEW:  The prize is won, with a 50 triangle solution by Lew Baxter.

I improved on Laczkovich's solution by using a different orientation of the 4 small central triangles, by choosing better parameters (x, y) and using fewer triangles for a total of 64 triangles. The original Laczkovich solution uses about 7 trillion triangles.

Here's one with 50 triangles:

