You are to color each of the squares of an *m*×*n*
board black or white so that the four corner squares of any rectangle
formed by the intersection of *a* adjacent rows and *b* adjacent
columns (*a*, *b* > 1) are __not__ all the same color. An
example of such a coloring is shown below for a 3×4 board.

- For a fixed
*m*, what is the largest*n*for which there is such a coloring of an*m*×*n*board? - Suppose that you use
*k*colors (instead of two) and still require the corner squares of any rectangle not to all be the same color. Now what are the largest possible board sizes?