NL-NWO 2012 Problem 2

We number the columns of an $n\times n$ -board from $1$ to $n$ . In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$ .

(a) Suppose that $n = 5$ . Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that $n = 10$ . Can the numbers be placed in such a way that each row contains the same number of grey cells?