Tiles and colors

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 integrability. 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.