  Tiling models are classical statistical models in which different geometric
shapes, the tiles, are packed together such that they cover space completely.
In this paper we discuss a class of two-dimensional tiling models in which the
tiles are rectangles and isosceles triangles. Some of these models have been
solved recently by means of Bethe Ansatz. We discuss the question why only
these models in a larger family are solvable, and we search for the Yang-Baxter
structure behind their integrablity. In this quest we find the Bethe Ansatz
solution of the problem of coloring the edges of the square lattice in four
colors, such that edges of the same color never meet in the same vertex.
