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 have no answer to the question, but here's a picture resulting from some initial attempts to understand the constraints that exist on any solution.
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This image was generated by considering what seemed to be the simplest possible configuration that might produce a tiling of a rectangle. Starting with the two “split pentagons” in the centre, the rest of the configuration is produced by triangulation. In this image, all the additional triangles are “forced”, and the configuration can be extended no further without violating the contraints of triangulation. If I had time, I'd move on to investigating the use of “split hexagons”.
The forcing criterion is that triangulation requires every vertex to be surrounded either (a) by six $60^\circ$ angles, three triangles being oriented one way and three the other, or else (b) by two $45^\circ$ angles, two $60^\circ$ angles and two $75^\circ$ angles, the triangles in each pair being of opposite orientations.
