In a grid a corner cell is colored red and all the other cells are colored blue. Consider the following recoloring operation: In any row or column colors of the cells are interchanged – red cells are colored blue and blue cells are colored red. Which of the following can be done through the recoloring operation? I. All the corner cells red. II. All the corner cells blue. III. One of the corner cells blue and the other corner cells red.
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.