NO-ABEL 2006 Problem 1

Each square in an $n \times n$ table is painted black or white. The routes where two rows meet two columns, called a quartet if the remaining squares are the same color. (a) What is the largest possible number of black squares in a $4 \times 4$ table without quartets? (b) Is it possible to paint a $5 \times 5$ table so that it has no quartets?