
3. Let us call by an "edge" any segment of length 1 which is common to two adjacent fields of a given chessboard $8 \times 8$. Consider all possible cuttings of the chessboard into 32 pieces $2 \times 1$ and denote by $n(e)$ the total number of such cuttings that involve the given edge $e$. Determine the last digit of the sum of the numbers $n(e)$ over all the edges $e$.

(Michal Rolínek)


Solution. The total number of the vertical edges is $7 \cdot 8=56$ as well as the total number of the horizontal edges. Thus the number of all the edges under consideration is $56 \cdot 2=112$.

The number of edges, which are not involved in a given cutting, is equal to 32, because each of these edges must coincide with the common segment of the two fields forming one of the 32 resulting pieces $2 \times 1$. Thus each cutting gives a contribution $112-32=80$ to the sum $S$ of all the numbers $n(e)$. Consequently, the sum $S$ is a multiple of 80 and thus its last digit is zero.
