*Problem #3*

Alternately color the squares of an *a* × *b* rectangle
black and white beginning with black in the upper left-hand corner. If you
are allowed to designate any subrectangle of the board and reverse the
colors inside that rectangle, what is the minimal number of steps it takes
to change all of the squares to black? For example, the figure below gives
a non-minimal procedure which converts a 3 × 3 checkerboard to all
black in four steps [the designated subrectangles are outlined in red].

Source: A generalization of a question from the USA Mathematical Olympiad