3. Let's have an $8 \times 8$ chessboard and for each "edge" that separates two of its squares, write a natural number that indicates the number of ways the entire chessboard can be cut into $2 \times 1$ rectangles such that the given edge is part of the cut. Determine the last digit of the sum of all such written numbers.

(Michal Rolínek)
Solution. There are $7 \cdot 8=56$ vertical edges and the same number of horizontal edges, making a total of $56 \cdot 2=112$. When the chessboard is cut in any way, 32 rectangles of $2 \times 1$ are formed, so each such cut touches exactly $112-32=80$ edges, contributing the number 80 to the total sum. Therefore, the resulting sum is divisible by 80, meaning its decimal representation ends in zero.