A friend of mine gave me this problem from a european olympiad:
Suppose we have a $8\times8$ chessboard. Each edge has a number; the number of ways of dividing this chessboard into $1\times2$ and $2\times 1$ domino pieces, such that the edge is part of the division. Find out the last digit of the sum of all these numbers.
Only the outer edges count.
(the question is uncleared, so I assume that the outer edges do not count, otherwise it would be much harder)
There are a total of $7\times 8\times 2=112$ edges. Each way of tiling the dominoes, since there are exactly $32$ dominoes used, each failed to involve $1$ edge, there are a total of $32$ edge not involved in each way of tiling. So each way of tiling contribute to the sum an amount of $112-32=80$. Thus the final digit is $0$.