Let $m,n$ be even numbers. Let a $m$ by $n$ grid be covered with 2x1 dominoes (horizontally or vertically), colored in the standard chessboard method.

Question

Show that the number of horizontal dominoes with the white square to the left of the black square is equal to the number of dominoes with the white square to the right of the black square.

I tried to use borders (the line that separates two squares) and fault lines (a line that cuts through no dominoes), but it doesn't work. Any hint would be appreciated!

Answer 1

Denote the grid by coordinate system $(x,y)$, where $1 \leq x \leq m$, $1 \leq y \leq n$. Suppose the checkerboard is painted white when $x+y$ is even; and black when it is odd.

Give each grid point a value: $(x,y)$ has value $(-1)^{x+y}x$.

The summation of the value throughout the grid is $0$. A BW horizontal domino has total value $1$; a WB horizontal domino has total value $-1$; and a vertical domino has total value $0$.