Every entry in Pascal's triangle counts the number of non-backtracking paths through some rectangular lattice. Furthermore, in each row of the triangle all the paths have the same total length. In the bottom row shown here, for example, all the paths are exactly four blocks long. In the case of the 0-by-4 grid and the 4-by-0 grid there is only one such path. The 1-by-3 and 3-by-1 lattices each offer four possible routes. The 2-by-2 lattice, as noted in Figure 3, has six diagonal paths.