A diagonalised lattice is a two dimensional grid, where we add exactly one arbitrary diagonal in each square, and color each vertex black or white.We show that for every diagonalised lattice there is a walk from the left to the right, using only black vertices, if and only if there is no walk from the top to the bottom, using only white vertices.
Citation
Bertran Steinsky. Diagonalised Lattices and the Steinhaus Chessboard Theorem[J], Ars Combinatoria, Volume 097-A. 59-63. .