Walks on tilings of polygons (Q1682861)

The proof of the well-known algebraic zig-zag theorem is based on the existence of some special paths into rectangular tilings. More precisely, consider a tiling of some plane rectangle \(R\) by rectangles of unit height coloured into white or grey. Suppose that the top edge of \(R\) is coloured white and the bottom edge of \(R\) is coloured grey. Then there exist a path from the left edge to the right edge of \(R\) using only bordering edges between different colours. In this paper, a new proof of this result is given. Also, the theorem is generalized to tilings of \(R\) by arbitrary polygons.