Consider a white square of size , that consists of unit squares of size . In a turn one can choose any
a) square;
b) rectangle,
that consists of a whole number of unit squares and contains at least one of the vertices of the square , and change a color of every unit square in it to the opposite one (white is changed to black and vice versa). Is it possible to obtain arbitrary coloring of the square using the turns described above?
a) Clearly, choosing the same square and acting in it twice is equivalent to not acting at all. Thus, we can assume that every square can be chosen not more than once. There are squares for which one can make a turn. Thus, there are not more than different colorings, which is less than – the amount of all possible colorings of unit squares in two colors.
b) Choose any unit square (depicted in black) and change color of every highlighted rectangle into the opposite one. There are either two (Fig. 20a), three (Fig. 20b) or four (Fig. 20c) such rectangles. After that change the color of the whole square into the opposite one. This way, only the color of the chosen unit square is changed. Thus, it is possible to obtain any coloring of the original square in two colors.