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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.

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  • $\begingroup$ Could you please reformulate the problem? It's quite hard to parse in its current form. $\endgroup$
    – dtldarek
    Jun 23, 2014 at 18:37

1 Answer 1

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(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 $0$.

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  • $\begingroup$ The outer edges don't count $\endgroup$
    – Asinomás
    Jun 23, 2014 at 18:53
  • $\begingroup$ Wow, damn !!, that was sweet. $\endgroup$
    – Asinomás
    Jun 23, 2014 at 18:54

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