If a chessboard is covered with dominoes, the number of horizontal dominoes is even An $8 \times 8$ chessboard is covered with $32$ dominoes. Show
that the number of horizontal dominoes that cover the board is even. 
 A: We will prove the result for any chessboard with an even number of rows and $c$ columns.
Call a horizontal domino a W-domino if has a white square on the left and a B-domino if it has a black square on the left. Number the columns $1$ to $c$ from left to right and say a horizontal domino is in column $k$ if its left square is in column $k$. Obviously no horizontal domino is in column $c$.
We prove by induction that the number of W-dominoes in column $k$ is the same as the number of B-dominoes in the column $k$ for $1 \leq k \leq c-1$.
Every vertical domino covers a white square and a black square and the number of rows is even so every column has the same number of white squares as black squares. Hence in each column the same number of white squares as black squares are covered by the horizontal dominoes.
Every horizontal domino that covers a square in column $1$ is in column $1$, so the number of W-dominoes in column $1$ is the same as the number of B-dominoes in column $1$.
Suppose the number of W-dominoes in column $k$ is the same as the number of B-dominoes in column $k$, for some $k < c-1$. Then the horizontal dominoes in column $k$ cover the same number of white squares as black squares in column $k+1$, so the horizontal dominoes in column $k+1$ must also cover the same number of white squares as black squares in column $k+1$. Therefore the number of W-dominoes in column $k+1$ is the same as the number of B-dominoes in column $k+1$.
A: If you have an even number of rows, then each column must have an even number of horizontal dominoes that project from it to the left and an even number that project from it to the right.  This is easily seen for the leftmost and rightmost columns, from which horizontal dominoes project only to the right and left, respectively:  If there were an odd number of horizontal dominoes, you be left trying to cover an odd number of squares with vertical dominoes.  It follows by induction for the remaining, interior columns -- just sweep, say from left to right, noting that each column's left-projecting dominoes is the previous column's right-projecting dominoes.
The result now follows by totalling the numbers of horizontal dominoes that project to the left:  there are none for the leftmost column and some other even number for all the rest.
