6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$, where $k$ is an inner edge.
Solution: First, we calculate along how many inner edges a cut is made during a decomposition. In total, there are $7 \cdot 8$ horizontal and vertical inner edges each. During a decomposition, the board is cut along all inner edges except the 32 within a domino tile. Therefore, in each decomposition, cuts are made along $2 \cdot (7 \cdot 8) - 32 = 80$ inner edges. This means that each decomposition is counted 80 times. Therefore, we know that the sum of all $N(k)$ is divisible by 80. It follows that the last digit of the sum is zero.