Question 1
Determine whether it is possible to distribute $60$ zeros and $61$ ones in the cells of an $11 \times 11$ board, one number in each cell, so that the sum of the numbers in each row is odd, the sum of the numbers in each column is odd, and the sum of the numbers on each of the two diagonals is odd. What if the board is $12 \times 12$ and one wants to distribute $72$ zeros and $72$ ones?