Consider an $8 \times 8$ chessboard. We choose a starting edge and call a zigzag a path that reaches the opposite edge by passing through exactly 8 white squares and no black ones. How many zigzags are there?
We will more generally count recursively the number of distinct paths (on white squares!) that reach a given white square. Consider B a white square; since zigzags cannot "go back" (otherwise it would take more than 8 squares to cross the chessboard), there are at most two white squares that can lead to $B$. If there are two, $B_{1}$ and $B_{2}$, the number of paths arriving at $B$ is then the sum of the numbers of the squares $B_{1}$ and $B_{2}$. If there is only one, $B_{1}$, then as many paths arrive at $B_{1}$ as at B.

Thus, the number of zigzags will be the sum of the 4 numbers of paths of the 4 white squares on the arrival edge. We of course write 1 on the 4 squares of the starting edge, and we go up line by line according to the recursive process described above. We find 296 zigzags.