OMA 2008 Problem 6

Consider a board of $a \times b$ , with $a$ and $b$ integers greater than or equal to $2$ . Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of $a$ and $b$ it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white.

Clarification: Initially there are no green squares, but they appear after the first operation.